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2-dimensional random walk of a silver adatom on an Ag(111) surface [ 1 ] Simulation of the Brownian motion of a large particle, analogous to a dust particle, that collides with a large set of smaller particles, analogous to molecules of a gas, which move with different velocities in different random directions Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas ). [ 2 ] The traditional mathematical formulation of Brownian motion is that of the Wiener process , which is often called Brownian motion, even in mathematical sources. This motion pattern typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium , defined by a given temperature . Within such a fluid, there exists no preferential direction of flow (as in transport phenomena ). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem ). [ 3 ] This motion is named after the Scottish botanist Robert Brown , who first described the phenomenon in 1827, while looking through a microscope at pollen of the plant Clarkia pulchella immersed in water. In 1900, the French mathematician Louis Bachelier modeled the stochastic process now called Brownian motion in his doctoral thesis, The Theory of Speculation (ThĂ©orie de la spĂ©culation), prepared under the supervision of Henri PoincarĂ© . Then, in 1905, theoretical physicist Albert Einstein published a paper in which he modelled the motion of the pollen particles as being moved by individual water molecules , making one of his first major scientific contributions. [ 4 ] The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. This explanation of Brownian motion served as convincing evidence that atoms and molecules exist and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter". [ 5 ] The many-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. Consequently, only probabilistic models applied to molecular populations can be employed to describe it. [ 6 ] Two such models of the statistical mechanics , due to Einstein and Smoluchowski, are presented below. Another, pure probabilistic class of models is the class of the stochastic process models. There exist sequences of both simpler and more complicated stochastic processes which converge (in the limit ) to Brownian motion (see random walk and Donsker's theorem ). [ 7 ] [ 8 ] Reproduced from the Jean Baptiste Perrin book Les Atomes : tracings of the motions of three colloidal particles of radius 0.53 Όm, as seen under the microscope, with each point representing that particle's successive position every 30   seconds; the points are then joined by straight line segments (mesh size = 3.2 Όm) [ 9 ] The Roman philosopher-poet Lucretius ' scientific poem On the Nature of Things ( c.  60 BC ) has a remarkable description of the motion of dust particles in verses 113–140 from Book II. He uses this as a proof of the existence of atoms: Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves [i.e., spontaneously]. Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible. [ 10 ] Although the mingling, tumbling motion of dust particles is caused largely by macroscopic air currents and convection, the glittering, microscopic jiggling motion of small dust particles is caused chiefly by true Brownian dynamics ; Lucretius "perfectly describes and explains the Brownian movement by a wrong example". [ 11 ] The formal scientific discovery of this phenomenon is credited to the botanist Robert Brown in 1827. Brown was studying plant reproduction when he observed pollen grains of the plant Clarkia pulchella in water under a simple microscope. These grains contain minute particles on the order of 1/4,000 th of an inch (6.4   microns) in size. He observed these particles executing a continuous, jittery motion. By repeating the experiment with particles of inorganic matter, such as glass and rock dust, he was able to rule out that the motion was life-related, although its physical origin was yet to be explained. [ 12 ] [ 13 ] The mathematics of much of stochastic analysis, including the mathematics of Brownian motion, was introduced by Louis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented an innovative probabilistic analysis of the stock and option markets. However, this pioneering mathematical work connecting random walks to continuous time was largely unknown until the 1950s. [ 14 ] [ 15 ] : 33  The early 20th century saw the theoretical formalization of Brownian motion bridging the gap between thermodynamics and atomic theory: Albert Einstein (in one of his 1905 papers ) provided an explanation of Brownian motion in terms of atoms and molecules at a time when their physical existence was still fiercely debated by scientists. Einstein proved the mathematical relation between the probability distribution of a Brownian particle and the macroscopic diffusion equation . [ 15 ] : 33  These predictive equations describing Brownian motion were subsequently verified by the meticulous experimental work of Jean Baptiste Perrin in 1908, leading to his Nobel prize and settling the atomic debate. [ 16 ] Norbert Wiener gave the first complete and rigorous mathematical analysis of the phenomenon in 1923, leading to the underlying mathematical concept being permanently called a Wiener process . [ 15 ] The instantaneous velocity of the Brownian motion can be defined as v = Δ x /Δ t , when Δ t << τ , where τ is the momentum relaxation time. Advancements in modern physics have allowed this to be directly measured: In 2010, the instantaneous velocity of a single Brownian particle (a glass microsphere trapped in air with optical tweezers ) was measured successfully for the first time. The velocity data perfectly verified the Maxwell–Boltzmann velocity distribution and confirmed the equipartition theorem for a Brownian particle at microscopic timescales. [ 17 ] Statistical mechanics theories [ edit ] There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian particles, in which the diffusion coefficient is related to the mean squared displacement of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities. [ 18 ] In this way Einstein was able to determine the size of atoms, and how many atoms there are in a mole, or the molecular weight in grams, of a gas. [ 19 ] In accordance to Avogadro's law , this volume is the same for all ideal gases, namely 22.414   liters at standard temperature and pressure. The number of atoms contained in this volume is referred to as the Avogadro constant or as Avogadro's number (approximately 6.02 × 10 23   mol −1 ), and the determination of this number is tantamount to the knowledge of the mass of an atom, since the latter is obtained by dividing the molar mass of the gas by the Avogadro constant . The characteristic bell-shaped curves of the diffusion of Brownian particles. The distribution begins as a Dirac delta function , indicating that all the particles are located at the origin at time t = 0. As t increases, the distribution flattens (though remains bell-shaped), and ultimately becomes uniform in the limit that time goes to infinity. The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval. [ 4 ] Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 10 14 collisions per second. [ 2 ] He regarded the increment of particle positions in time in a one-dimensional ( x ) space (with the coordinates chosen so that the origin lies at the initial position of the particle) as a random variable ( ) with some probability density function (i.e., is the probability density for a jump of magnitude , i.e., the probability density of the particle incrementing its position from to in the time interval ). Further, assuming conservation of particle number, he expanded the number density (number of particles per unit volume around ) at time in a Taylor series , where the second equality is by definition of . The integral in the first term is equal to one by the definition of probability, and the second and other even terms (i.e. first and other odd moments ) vanish because of space symmetry. What is left gives rise to the following relation: Where the coefficient after the Laplacian , the second moment of probability of displacement , is interpreted as mass diffusivity D : Then the density of Brownian particles ρ at point x at time t satisfies the diffusion equation : Assuming that N particles start from the origin at the initial time t = 0, the diffusion equation has the solution: [ 20 ] This expression (which is a normal distribution with the mean and variance usually called Brownian motion ) allowed Einstein to calculate the moments directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right. The second moment is, however, non-vanishing, being given by This equation expresses the mean squared displacement in terms of the time elapsed and the diffusivity. From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root. [ 18 ] His argument is based on a conceptual switch from the "ensemble" of Brownian particles to the "single" Brownian particle: we can speak of the relative number of particles at a single instant just as well as of the time it takes a Brownian particle to reach a given point. [ 21 ] The second part of Einstein's theory relates the diffusion constant to physically measurable quantities, such as the mean squared displacement of a particle in a given time interval. This result enables the experimental determination of the Avogadro number and therefore the size of molecules. Einstein analyzed a dynamic equilibrium being established between opposing forces. The beauty of his argument is that the final result does not depend upon which forces are involved in setting up the dynamic equilibrium. In his original treatment, Einstein considered an osmotic pressure experiment, but the same conclusion can be reached in other ways. Consider, for instance, particles suspended in a viscous fluid in a gravitational field. Gravity tends to make the particles settle, whereas diffusion acts to homogenize them, driving them into regions of smaller concentration. Under the action of gravity, a particle acquires a downward speed of v = ÎŒmg , where m is the mass of the particle, g is the acceleration due to gravity, and ÎŒ is the particle's mobility in the fluid. George Stokes had shown that the mobility for a spherical particle with radius r is , where η is the dynamic viscosity of the fluid. In a state of dynamic equilibrium, and under the hypothesis of isothermal fluid, the particles are distributed according to the barometric distribution where ρ − ρ o is the difference in density of particles separated by a height difference, of , k B is the Boltzmann constant (the ratio of the universal gas constant , R , to the Avogadro constant , N A ), and T is the absolute temperature . Perrin examined the equilibrium ( barometric distribution ) of granules (0.6 microns ) of gamboge , a viscous substance, under the microscope. The granules move against gravity to regions of lower concentration. The relative change in density observed in 10 microns of suspension is equivalent to that occurring in 6 km of air. Dynamic equilibrium is established because the more that particles are pulled down by gravity , the greater the tendency for the particles to migrate to regions of lower concentration. The flux is given by Fick's law , where J = ρv . Introducing the formula for ρ , we find that In a state of dynamical equilibrium, this speed must also be equal to v = ÎŒmg . Both expressions for v are proportional to mg , reflecting that the derivation is independent of the type of forces considered. Similarly, one can derive an equivalent formula for identical charged particles of charge q in a uniform electric field of magnitude E , where mg is replaced with the electrostatic force qE . Equating these two expressions yields the Einstein relation for the diffusivity, independent of mg or qE or other such forces: Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of the Boltzmann constant as k B = R / N A , and the fourth equality follows from Stokes's formula for the mobility. By measuring the mean squared displacement over a time interval along with the universal gas constant R , the temperature T , the viscosity η , and the particle radius r , the Avogadro constant N A can be determined. The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously by J. J. Thomson [ 22 ] in his series of lectures at Yale University in May 1903 that the dynamic equilibrium between the velocity generated by a concentration gradient given by Fick's law and the velocity due to the variation of the partial pressure caused when ions are set in motion "gives us a method of determining Avogadro's constant which is independent of any hypothesis as to the shape or size of molecules, or of the way in which they act upon each other". [ 22 ] An identical expression to Einstein's formula for the diffusion coefficient was also found by Walther Nernst in 1888 [ 23 ] in which he expressed the diffusion coefficient as the ratio of the osmotic pressure to the ratio of the frictional force and the velocity to which it gives rise. The former was equated to the law of van 't Hoff while the latter was given by Stokes's law . He writes for the diffusion coefficient kâ€Č , where is the osmotic pressure and k is the ratio of the frictional force to the molecular viscosity which he assumes is given by Stokes's formula for the viscosity. Introducing the ideal gas law per unit volume for the osmotic pressure, the formula becomes identical to that of Einstein's. [ 24 ] The use of Stokes's law in Nernst's case, as well as in Einstein and Smoluchowski, is not strictly applicable since it does not apply to the case where the radius of the sphere is small in comparison with the mean free path . [ 25 ] Confirming Einstein's formula experimentally proved difficult. Initial attempts by Theodor Svedberg in 1906 and 1907 were critiqued by Einstein and by Perrin as not measuring a quantity directly comparable to the formula. Victor Henri in 1908 took cinematographic shots through a microscope and found quantitative disagreement with the formula but again the analysis was uncertain. [ 26 ] Einstein's predictions were finally confirmed in a series of experiments carried out by Chaudesaigues in 1908 and Perrin in 1909. [ 27 ] [ 12 ] The confirmation of Einstein's theory constituted empirical progress for the kinetic theory of heat . In essence, Einstein showed that the motion can be predicted directly from the kinetic model of thermal equilibrium . The importance of the theory lay in the fact that it confirmed the kinetic theory's account of the second law of thermodynamics as being an essentially statistical law. [ 28 ] Brownian motion model of the trajectory of a particle of dye in water Smoluchowski 's theory of Brownian motion, [ 29 ] later contextualized in comprehensive reviews of stochastic physics, [ 30 ] starts from the same premise as that of Einstein and derives the same probability distribution ρ ( x , t ) for the displacement of a Brownian particle along the x axis in time t . He therefore gets the same expression for the mean squared displacement: . However, when he relates it to a particle of mass m moving at a velocity u which is the result of a frictional force governed by Stokes's law, he finds where ÎŒ is the viscosity coefficient, and a is the radius of the particle. Associating the kinetic energy with the thermal energy , the expression for the mean squared displacement is 64/27 times that found by Einstein. This discrepancy arises from differing theoretical approaches: Einstein assumed Stokes drag applied directly to the macroscopic drift velocity, while Smoluchowski performed a more detailed kinematic collision analysis but introduced a slight calculation variance when averaging over the Maxwellian velocity distribution. The fraction 27/64 was commented on by Arnold Sommerfeld in his necrology on Smoluchowski: "The numerical coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt." [ 31 ] Smoluchowski [ 32 ] attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are exactly equal. To address this paradox, he relies on the inevitability of statistical fluctuations: If the probability of m gains and losses follows a binomial distribution , with equal a priori probabilities of 1/2, the mean total gain is If n is large enough so that Stirling's approximation can be used in the form , then the expected total absolute gain representing the net drift can be approximated. [ 33 ] This expected gain will be: showing that the net displacement increases proportionally to the square root of the total population of collision events. Suppose that a Brownian particle of mass M is surrounded by lighter particles of mass m which are traveling at a speed u . Smoluchowski reasons that the mechanics of these interactions produce a macroscopically observable effect: In any single collision between a surrounding fluid particle and the Brownian particle, the velocity transmitted to the latter will be . This ratio is exceedingly small, on the order of 10 −7 cm/s. However, in a gas there will be more than 10 16 collisions in a second, and even more in a liquid, where there are expected to be roughly 10 20 collisions in one second. Because of the immense scale of collisions, statistical imbalances are inevitable. While some collisions will accelerate the Brownian particle, others will decelerate it. If there is a mean excess of one kind of collision (e.g., more impacts from the left than the right) on the order of 10 8 to 10 10 collisions in a single second, then the instantaneous velocity of the Brownian particle may be anywhere between 10 and 1000 cm/s. Thus, even though there are equal probabilities for forward and backward collisions, the sheer volume of events creates a net tendency to keep the Brownian particle in erratic, continuous motion, much like the fluctuations predicted by the ballot theorem. These orders of magnitude do not take into consideration the velocity of the Brownian particle, U , which actively depends on the collisions that tend to accelerate and decelerate it. The larger U is, the greater will be the resistive drag of collisions that will retard it, so that the velocity of a Brownian particle can never increase without limit. If such an unbounded process could occur, it would be tantamount to a perpetual motion machine of the second kind. Since the equipartition of energy applies to this system in thermal equilibrium, the kinetic energy of the Brownian particle, , will be equal, on the average, to the kinetic energy of the surrounding fluid particle, . In 1906, Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion. [ 34 ] The model assumes collisions with where M is the test particle's mass and m the mass of one of the individual particles composing the fluid. To build this theoretical framework, several simplifying assumptions are made: The particle collisions are confined to one dimension along a single axis. It is equally probable for the test particle to be hit from the left as from the right. Every collision always imparts the exact same discrete magnitude of velocity change, . If is the number of collisions from the right and the number of collisions from the left then after N collisions the particle's velocity will have changed by . The multiplicity is then simply given by: and the total number of possible states is given by . Therefore, the probability of the particle being hit from the right times is: As a result of its simplicity, Smoluchowski's 1D model can only qualitatively describe Brownian motion. For a realistic particle undergoing Brownian motion in a fluid, many of the assumptions do not apply. For example, the assumption that on average there are an equal number of collisions from the right as from the left falls apart once the particle is in motion, as it will sweep into particles in its path and create a velocity-dependent drag force. Furthermore, there would be a continuous statistical distribution of different possible s governed by the Maxwell-Boltzmann distribution of fluid molecule velocities, rather than a single discrete value in a physical liquid or gas. The diffusion equation yields an approximation of the time evolution of the probability density function associated with the position of the particle going under a Brownian movement under the physical definition. The approximation becomes valid on timescales much larger than the timescale of individual atomic collisions, since it does not include a term to describe the acceleration of particles during collision. The time evolution of the position of the Brownian particle over all time scales described using the Langevin equation , an equation that involves a random force field representing the effect of the thermal fluctuations of the solvent on the particle. [ 17 ] At longer times scales, where acceleration is negligible, individual particle dynamics can be approximated using Brownian dynamics in place of Langevin dynamics . Astrophysics: star motion within galaxies [ edit ] In stellar dynamics , a massive body (star, black hole , etc.) can experience Brownian motion as it responds to gravitational forces from surrounding stars. [ 35 ] The rms velocity V of the massive object, of mass M , is related to the rms velocity of the background stars by where is the mass of the background stars. The gravitational force from the massive object causes nearby stars to move faster than they otherwise would, increasing both and V . [ 35 ] The Brownian velocity of Sgr A* , the supermassive black hole at the center of the Milky Way galaxy , is predicted from this formula to be less than 1 km s −1 . [ 36 ] An animated example of a Brownian motion-like random walk on a 2D surface with periodic boundary conditions. In the scaling limit , random walk approaches the Wiener process according to Donsker's theorem . In mathematics , Brownian motion is described by the Wiener process , a continuous-time stochastic process named in honor of Norbert Wiener . It is one of the best known LĂ©vy processes ( cĂ dlĂ g stochastic processes with stationary independent increments ) and occurs frequently in pure and applied mathematics, economics and physics . A single realisation of three-dimensional Brownian motion for times 0 ≀ t ≀ 2 The Wiener process W t is characterized by four facts: [ 37 ] W 0 = 0 W t is almost surely continuous W t has independent increments (for ). denotes the normal distribution with expected value ÎŒ and variance σ 2 . The condition that it has independent increments means that if then and are independent random variables. In addition, for some filtration , is measurable for all . An alternative characterisation of the Wiener process is the so-called LĂ©vy characterisation that says that the Wiener process is an almost surely continuous martingale with W 0 = 0 and quadratic variation . A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent random variables. This representation can be obtained using the Kosambi–Karhunen–LoĂšve theorem . The Wiener process can be constructed as the scaling limit of a random walk , or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem . Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is scale invariant . A d-dimensional Gaussian free field has been described as "a d-dimensional-time analog of Brownian motion." [ 38 ] The Brownian motion can be modeled by a random walk . [ 39 ] In the general case, Brownian motion is a Markov process and described by stochastic integral equations . [ 40 ] LĂ©vy characterisation [ edit ] The French mathematician Paul LĂ©vy proved the following theorem, which gives a necessary and sufficient condition for a continuous R n -valued stochastic process X to actually be n -dimensional Brownian motion. Hence, LĂ©vy's condition can actually be used as an alternative definition of Brownian motion. Let X = ( X 1 , ..., X n ) be a continuous stochastic process on a probability space (Ω, ÎŁ, P ) taking values in R n . Then the following are equivalent: X is a Brownian motion with respect to P , i.e., the law of X with respect to P is the same as the law of an n -dimensional Brownian motion, i.e., the push-forward measure X ∗ ( P ) is classical Wiener measure on C 0 ( [0, ∞) ; R n ) . both X is a martingale with respect to P (and its own natural filtration ); and for all 1 ≀ i , j ≀ n , X i ( t ) X j ( t ) − ÎŽ ij t is a martingale with respect to P (and its own natural filtration ), where ÎŽ ij denotes the Kronecker delta . The spectral content of a stochastic process can be found from the power spectral density , formally defined as where stands for the expected value . The power spectral density of Brownian motion is found to be [ 41 ] where D is the diffusion coefficient of X t . For naturally occurring signals, the spectral content can be found from the power spectral density of a single realization, with finite available time, i.e., which for an individual realization of a Brownian motion trajectory, [ 42 ] it is found to have expected value and variance [ 42 ] For sufficiently long realization times, the expected value of the power spectrum of a single trajectory converges to the formally defined power spectral density , but its coefficient of variation tends to . This implies the distribution of is broad even in the infinite time limit. Riemannian manifolds [ edit ] Brownian motion on a sphere Brownian motion is usually considered to take place in Euclidean space . It is natural to consider how such motion generalizes to more complex shapes, such as surfaces or higher dimensional manifolds . The formalization requires the space to possess some form of a derivative , as well as a metric , so that a Laplacian can be defined. Both of these are available on Riemannian manifolds . Riemannian manifolds have the property that geodesics can be described in polar coordinates ; that is, displacements are always in a radial direction, at some given angle. Uniform random motion is then described by Gaussians along the radial direction, independent of the angle, the same as in Euclidean space. The infinitesimal generator (and hence characteristic operator ) of Brownian motion on Euclidean R n is ⁠ 1 / 2 ⁠ Δ , where Δ denotes the Laplace operator . Brownian motion on an m -dimensional Riemannian manifold ( M , g ) can be defined as diffusion on M with the characteristic operator given by ⁠ 1 / 2 ⁠ Δ LB , half the Laplace–Beltrami operator Δ LB . One of the topics of study is a characterization of the PoincarĂ© recurrence time for such systems. [ 16 ] The narrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology which has the following formulation: a Brownian particle ( ion , molecule , or protein ) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is that of calculating the mean escape time. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem. Brownian bridge  – Stochastic process in physics Brownian covariance  – Statistical measure Brownian dynamics  – Ideal molecular motion where no average acceleration takes place Brownian motor  – Nanoscale machine Brownian noise  – Type of noise produced by Brownian motion Brownian ratchet  – Perpetual motion device Brownian surface Brownian tree  – Concept in probability theory Brownian web Fractional Brownian motion  – Probability theory concept Geometric Brownian motion  – Continuous stochastic process ItĂŽ diffusion  – Solution to a specific type of stochastic differential equation LĂ©vy arcsine law  – Collection of results for one-dimensional random walks and Brownian motion Marangoni effect  – Physical phenomenon between two fluids Nanoparticle tracking analysis  – Method for visualizing and analyzing particles in liquids Reflected Brownian motion  – Wiener process with reflecting spatial boundaries Rotational Brownian motion Schramm–Loewner evolution  – Concept in probability theory Single particle tracking Single particle trajectories Surface diffusion  – Physical Process Tyndall effect  – Scattering of light by tiny particles in a colloidal suspension ^ Meyburg, Jan Philipp; Diesing, Detlef (2017). 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ISSN   0273-0979 . ^ a b Li, Tongcang; Kheifets, Simon; Medellin, David; Raizen, Mark (2010). "Measurement of the instantaneous velocity of a Brownian particle" (PDF) . Science . 328 (5986): 1673– 1675. Bibcode : 2010Sci...328.1673L . CiteSeerX   10.1.1.167.8245 . doi : 10.1126/science.1189403 . PMID   20488989 . S2CID   45828908 . Archived from the original (PDF) on 31 March 2011. ^ a b Einstein, Albert (1956) [1926]. Investigations on the Theory of the Brownian Movement (PDF) . Dover Publications. Archived (PDF) from the original on 9 October 2022 . Retrieved 25 December 2013 . ^ Stachel, J., ed. (1989). "Einstein's Dissertation on the Determination of Molecular Dimensions" (PDF) . The Collected Papers of Albert Einstein, Volume 2 . Princeton University Press. Archived (PDF) from the original on 9 October 2022. ^ Kozdron, Michael J. (2008). "Brownian Motion and the Heat Equation – chpt. 3 Albert Einstein's proof of the existence of Brownian motion" (PDF) . University of Regina . Archived from the original (PDF) on 26 November 2024 . Retrieved 3 November 2025 . ^ Lavenda, Bernard H. (1985). Nonequilibrium Statistical Thermodynamics . John Wiley & Sons. p.  20 . ISBN   978-0-471-90670-4 . ^ a b Thomson, J. J. (1904). Electricity and Matter . Yale University Press. pp.  80 –83. ^ Nernst, Walther (1888). "Zur Kinetik der in Lösung befindlichen Körper". Zeitschrift fĂŒr Physikalische Chemie (in German). 9 : 613– 637. ^ Leveugle, J. (2004). La RelativitĂ©, PoincarĂ© et Einstein, Planck, Hilbert . Harmattan. p. 181. ^ Townsend, J.E.S. (1915). Electricity in Gases . Clarendon Press. p.  254 . ^ Maiocchi, Roberto (September 1990). "The case of Brownian motion" . The British Journal for the History of Science . 23 (3): 257– 283. doi : 10.1017/S0007087400043983 . ISSN   1474-001X . ^ Haw, M D (26 August 2002). "Colloidal suspensions, Brownian motion, molecular reality: a short history" . Journal of Physics: Condensed Matter . 14 (33): 7769– 7779. doi : 10.1088/0953-8984/14/33/315 . ^ See P. Clark 1976, p. 97 ^ Smoluchowski, M. M. (1906). "Sur le chemin moyen parcouru par les molĂ©cules d'un gaz et sur son rapport avec la thĂ©orie de la diffusion" [On the average path taken by gas molecules and its relation with the theory of diffusion]. Bulletin International de l'AcadĂ©mie des Sciences de Cracovie (in French): 202. ^ Chandrasekhar, Subrahmanyan (1943). "Stochastic Problems in Physics and Astronomy". Reviews of Modern Physics . 15 (1): 1– 89. doi : 10.1103/RevModPhys.15.1 . ^ See p. 535 in Sommerfeld, A. (1917). "Zum Andenken an Marian von Smoluchowski" [In Memory of Marian von Smoluchowski]. Physikalische Zeitschrift (in German). 18 (22): 533– 539. ^ Smoluchowski, M. M. (1906). "Essai d'une thĂ©orie cinĂ©tique du mouvement Brownien et des milieux troubles" [Test of a kinetic theory of Brownian motion and turbid media]. Bulletin International de l'AcadĂ©mie des Sciences de Cracovie (in French): 577. ^ Feller, William (1968). An Introduction to Probability Theory and Its Applications . Vol. 1 (3rd ed.). John Wiley & Sons. pp.  73– 75. ISBN   978-0471257080 . ^ von Smoluchowski, M. (1906). "Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen" . Annalen der Physik (in German). 326 (14): 756– 780. Bibcode : 1906AnP...326..756V . doi : 10.1002/andp.19063261405 . ^ a b Merritt, David (2013). Dynamics and Evolution of Galactic Nuclei . Princeton University Press. p. 575. ISBN   978-1-4008-4612-2 . OL   16802359W . ^ Reid, M. J.; Brunthaler, A. (2004). "The Proper Motion of Sagittarius A*. II. The Mass of Sagittarius A*". The Astrophysical Journal . 616 (2): 872– 884. arXiv : astro-ph/0408107 . Bibcode : 2004ApJ...616..872R . doi : 10.1086/424960 . S2CID   16568545 . ^ Bass, Richard F. (2011). Stochastic Processes . Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press. doi : 10.1017/cbo9780511997044 . ISBN   978-1-107-00800-7 . ^ Sheffield, Scott (9 May 2007). "Gaussian free fields for mathematicians" . Probability Theory and Related Fields . 139 ( 3– 4): 521– 541. arXiv : math/0312099 . doi : 10.1007/s00440-006-0050-1 . ISSN   0178-8051 . ^ Weiss, G. H. (1994). Aspects and applications of the random walk . North Holland. ^ Morozov, A. N.; Skripkin, A. V. (2011). "Spherical particle Brownian motion in viscous medium as non-Markovian random process". Physics Letters A . 375 (46): 4113– 4115. Bibcode : 2011PhLA..375.4113M . doi : 10.1016/j.physleta.2011.10.001 . ^ Karczub, D. G.; Norton, M. P. (2003). Fundamentals of Noise and Vibration Analysis for Engineers by M. P. Norton . doi : 10.1017/cbo9781139163927 . ISBN   978-1-139-16392-7 . ^ a b Krapf, Diego; Marinari, Enzo; Metzler, Ralf; Oshanin, Gleb; Xu, Xinran; Squarcini, Alessio (2018). "Power spectral density of a single Brownian trajectory: what one can and cannot learn from it" . New Journal of Physics . 20 (2): 023029. arXiv : 1801.02986 . Bibcode : 2018NJPh...20b3029K . doi : 10.1088/1367-2630/aaa67c . ISSN   1367-2630 . S2CID   485685 . Brown, Robert (1828). "A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies" (PDF) . Philosophical Magazine . 4 (21): 161– 173. doi : 10.1080/14786442808674769 . Archived (PDF) from the original on 9 October 2022. Also includes a subsequent defense by Brown of his original observations, Additional remarks on active molecules . Chaudesaigues, M. (1908). "Le mouvement brownien et la formule d'Einstein" [Brownian motion and Einstein's formula]. Comptes Rendus (in French). 147 : 1044– 6. Clark, P. (1976). "Atomism versus thermodynamics". In Howson, Colin (ed.). Method and appraisal in the physical sciences . Cambridge University Press. ISBN   978-0-521-21110-9 . Cohen, Ruben D. (1986). "Self Similarity in Brownian Motion and Other Ergodic Phenomena" (PDF) . Journal of Chemical Education . 63 (11): 933– 934. Bibcode : 1986JChEd..63..933C . doi : 10.1021/ed063p933 . Archived (PDF) from the original on 9 October 2022. Dubins, Lester E.; Schwarz, Gideon (15 May 1965). "On Continuous Martingales" . Proceedings of the National Academy of Sciences of the United States of America . 53 (3): 913– 916. Bibcode : 1965PNAS...53..913D . doi : 10.1073/pnas.53.5.913 . JSTOR   72837 . PMC   301348 . PMID   16591279 . Einstein, A. (1956). Investigations on the Theory of Brownian Movement . New York: Dover. ISBN   978-0-486-60304-9 . Retrieved 6 January 2014 . Henri, V. (1908). "Études cinĂ©matographique du mouvement brownien" [Cinematographic studies of Brownian motion]. Comptes Rendus (in French) (146): 1024– 6. Lucretius , On The Nature of Things , translated by William Ellery Leonard . ( on-line version , from Project Gutenberg . See the heading 'Atomic Motions'; this translation differs slightly from the one quoted). Nelson, Edward , (1967). Dynamical Theories of Brownian Motion . (PDF version of this out-of-print book, from the author's webpage.) This is primarily a mathematical work, but the first four chapters discuss the history of the topic, in the era from Brown to Einstein. Pearle, P.; Collett, B.; Bart, K.; Bilderback, D.; Newman, D.; Samuels, S. (2010). "What Brown saw and you can too". American Journal of Physics . 78 (12): 1278– 1289. arXiv : 1008.0039 . Bibcode : 2010AmJPh..78.1278P . doi : 10.1119/1.3475685 . S2CID   12342287 . Perrin, J. (1909). "Mouvement brownien et rĂ©alitĂ© molĂ©culaire" [Brownian movement and molecular reality]. Annales de chimie et de physique . 8th series. 18 : 5– 114. See also Perrin's book "Les Atomes" (1914). von Smoluchowski, M. (1906). "Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen" . Annalen der Physik . 21 (14): 756– 780. Bibcode : 1906AnP...326..756V . doi : 10.1002/andp.19063261405 . Svedberg, T. (1907). Studien zur Lehre von den kolloiden Losungen . Theile, T. N . Danish version: "Om Anvendelse af mindste Kvadraters Methode i nogle TilfĂŠlde, hvor en Komplikation af visse Slags uensartede tilfĂŠldige Fejlkilder giver Fejlene en 'systematisk' Karakter". French version: "Sur la compensation de quelques erreurs quasi-systĂ©matiques par la mĂ©thodes de moindre carrĂ©s" published simultaneously in Vidensk. Selsk. Skr. 5. Rk., naturvid. og mat. Afd. , 12:381–408, 1880. English Wikisource has original text related to this article: Einstein on Brownian Motion Discusses history, botany and physics of Brown's original observations, with videos "Einstein's prediction finally witnessed one century later"  : a test to observe the velocity of Brownian motion Large-Scale Brownian Motion Demonstration
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[Jump to content](https://en.wikipedia.org/wiki/Brownian_motion#bodyContent) Main menu Main menu move to sidebar hide Navigation - [Main page](https://en.wikipedia.org/wiki/Main_Page "Visit the main page [z]") - [Contents](https://en.wikipedia.org/wiki/Wikipedia:Contents "Guides to browsing Wikipedia") - [Current events](https://en.wikipedia.org/wiki/Portal:Current_events "Articles related to current events") - [Random article](https://en.wikipedia.org/wiki/Special:Random "Visit a randomly selected article [x]") - [About Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:About "Learn about Wikipedia and how it works") - [Contact us](https://en.wikipedia.org/wiki/Wikipedia:Contact_us "How to contact Wikipedia") Contribute - [Help](https://en.wikipedia.org/wiki/Help:Contents "Guidance on how to use and edit Wikipedia") - [Learn to edit](https://en.wikipedia.org/wiki/Help:Introduction "Learn how to edit Wikipedia") - [Community portal](https://en.wikipedia.org/wiki/Wikipedia:Community_portal "The hub for editors") - [Recent changes](https://en.wikipedia.org/wiki/Special:RecentChanges "A list of recent changes to Wikipedia [r]") - [Upload file](https://en.wikipedia.org/wiki/Wikipedia:File_upload_wizard "Add images or other media for use on Wikipedia") - [Special pages](https://en.wikipedia.org/wiki/Special:SpecialPages "A list of all special pages [q]") [![](https://en.wikipedia.org/static/images/icons/enwiki-25.svg) ![Wikipedia](https://en.wikipedia.org/static/images/mobile/copyright/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https://en.wikipedia.org/static/images/mobile/copyright/wikipedia-tagline-en-25.svg)](https://en.wikipedia.org/wiki/Main_Page) [Search](https://en.wikipedia.org/wiki/Special:Search "Search Wikipedia [f]") Appearance - [Donate](https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en) - [Create account](https://en.wikipedia.org/w/index.php?title=Special:CreateAccount&returnto=Brownian+motion "You are encouraged to create an account and log in; however, it is not mandatory") - [Log in](https://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Brownian+motion "You're encouraged to log in; however, it's not mandatory. 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[o]") ## Contents move to sidebar hide - [(Top)](https://en.wikipedia.org/wiki/Brownian_motion) - [1 History](https://en.wikipedia.org/wiki/Brownian_motion#History) - [2 Statistical mechanics theories](https://en.wikipedia.org/wiki/Brownian_motion#Statistical_mechanics_theories) Toggle Statistical mechanics theories subsection - [2\.1 Einstein's theory](https://en.wikipedia.org/wiki/Brownian_motion#Einstein's_theory) - [2\.2 Smoluchowski model](https://en.wikipedia.org/wiki/Brownian_motion#Smoluchowski_model) - [2\.3 Langevin equation](https://en.wikipedia.org/wiki/Brownian_motion#Langevin_equation) - [2\.4 Astrophysics: star motion within galaxies](https://en.wikipedia.org/wiki/Brownian_motion#Astrophysics:_star_motion_within_galaxies) - [3 Mathematics](https://en.wikipedia.org/wiki/Brownian_motion#Mathematics) Toggle Mathematics subsection - [3\.1 Statistics](https://en.wikipedia.org/wiki/Brownian_motion#Statistics) - [3\.2 LĂ©vy characterisation](https://en.wikipedia.org/wiki/Brownian_motion#L%C3%A9vy_characterisation) - [3\.3 Spectral content](https://en.wikipedia.org/wiki/Brownian_motion#Spectral_content) - [3\.4 Riemannian manifolds](https://en.wikipedia.org/wiki/Brownian_motion#Riemannian_manifolds) - [4 Narrow escape](https://en.wikipedia.org/wiki/Brownian_motion#Narrow_escape) - [5 See also](https://en.wikipedia.org/wiki/Brownian_motion#See_also) - [6 References](https://en.wikipedia.org/wiki/Brownian_motion#References) - [7 Further reading](https://en.wikipedia.org/wiki/Brownian_motion#Further_reading) - [8 External links](https://en.wikipedia.org/wiki/Brownian_motion#External_links) Toggle the table of contents # Brownian motion 71 languages - [Afrikaans](https://af.wikipedia.org/wiki/Brownse_beweging "Brownse beweging – Afrikaans") - [Ű§Ù„ŰčŰ±ŰšÙŠŰ©](https://ar.wikipedia.org/wiki/%D8%AD%D8%B1%D9%83%D8%A9_%D8%A8%D8%B1%D8%A7%D9%88%D9%86%D9%8A%D8%A9 "Ű­Ű±ÙƒŰ© ŰšŰ±Ű§ÙˆÙ†ÙŠŰ© – Arabic") - [Asturianu](https://ast.wikipedia.org/wiki/Movimientu_brownianu "Movimientu brownianu – Asturian") - [Azərbaycanca](https://az.wikipedia.org/wiki/Broun_h%C9%99r%C9%99k%C9%99ti "Broun hərəkəti – Azerbaijani") - [Đ‘Đ”Đ»Đ°Ń€ŃƒŃĐșая](https://be.wikipedia.org/wiki/%D0%91%D1%80%D0%BE%D1%9E%D0%BD%D0%B0%D1%9E%D1%81%D0%BA%D1%96_%D1%80%D1%83%D1%85 "Đ‘Ń€ĐŸŃžĐœĐ°ŃžŃĐșі рух – Belarusian") - [БългарсĐșĐž](https://bg.wikipedia.org/wiki/%D0%91%D1%80%D0%B0%D1%83%D0%BD%D0%BE%D0%B2%D0%BE_%D0%B4%D0%B2%D0%B8%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5 "Đ‘Ń€Đ°ŃƒĐœĐŸĐČĐŸ ĐŽĐČĐžĐ¶Đ”ĐœĐžĐ” – Bulgarian") - [àŠŹàŠŸàŠ‚àŠČàŠŸ](https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%8D%E0%A6%B0%E0%A6%BE%E0%A6%89%E0%A6%A8%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%97%E0%A6%A4%E0%A6%BF "àŠŹà§àŠ°àŠŸàŠ‰àŠšà§€àŠŻàŠŒ àŠ—àŠ€àŠż – Bangla") - [CatalĂ ](https://ca.wikipedia.org/wiki/Moviment_browni%C3%A0 "Moviment browniĂ  – Catalan") - [ČeĆĄtina](https://cs.wikipedia.org/wiki/Brown%C5%AFv_pohyb "BrownĆŻv pohyb – Czech") - [ЧӑĐČашла](https://cv.wikipedia.org/wiki/%D0%91%D1%80%D0%BE%D1%83%D0%BD_%D0%BA%D1%83%C3%A7%C4%83%D0%BC%C4%95 "Đ‘Ń€ĐŸŃƒĐœ ĐșŃƒĂ§ÄƒĐŒÄ• – Chuvash") - [Cymraeg](https://cy.wikipedia.org/wiki/Symudedd_Brown "Symudedd Brown – Welsh") - [Dansk](https://da.wikipedia.org/wiki/Brownske_bev%C3%A6gelser "Brownske bevĂŠgelser – Danish") - [Deutsch](https://de.wikipedia.org/wiki/Brownsche_Bewegung "Brownsche Bewegung – German") - [ΕλληΜÎčÎșÎŹ](https://el.wikipedia.org/wiki/%CE%9A%CE%AF%CE%BD%CE%B7%CF%83%CE%B7_%CE%9C%CF%80%CF%81%CE%AC%CE%BF%CF%85%CE%BD "ÎšÎŻÎœÎ·ÏƒÎ· ÎœÏ€ÏÎŹÎżÏ…Îœ – Greek") - [Esperanto](https://eo.wikipedia.org/wiki/Brown-a_movado "Brown-a movado – Esperanto") - [Español](https://es.wikipedia.org/wiki/Movimiento_browniano "Movimiento browniano – Spanish") - [Eesti](https://et.wikipedia.org/wiki/Browni_liikumine "Browni liikumine – Estonian") - [Euskara](https://eu.wikipedia.org/wiki/Browndar_higidura "Browndar higidura – Basque") - [ÙŰ§Ű±ŰłÛŒ](https://fa.wikipedia.org/wiki/%D8%AD%D8%B1%DA%A9%D8%AA_%D8%A8%D8%B1%D8%A7%D9%88%D9%86%DB%8C "ۭ۱کŰȘ ŰšŰ±Ű§ÙˆÙ†ÛŒ – Persian") - [Suomi](https://fi.wikipedia.org/wiki/Brownin_liike "Brownin liike – Finnish") - [Français](https://fr.wikipedia.org/wiki/Mouvement_brownien "Mouvement brownien – French") - [Gaeilge](https://ga.wikipedia.org/wiki/Gluaisne_de_Br%C3%BAn "Gluaisne de BrĂșn – Irish") - [Galego](https://gl.wikipedia.org/wiki/Movemento_browniano "Movemento browniano – Galician") - [ŚąŚ‘ŚšŚ™ŚȘ](https://he.wikipedia.org/wiki/%D7%AA%D7%A0%D7%95%D7%A2%D7%94_%D7%91%D7%A8%D7%90%D7%95%D7%A0%D7%99%D7%AA "ŚȘŚ Ś•ŚąŚ” Ś‘ŚšŚŚ•Ś Ś™ŚȘ – Hebrew") - [à€čà€żà€šà„à€Šà„€](https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%89%E0%A4%A8%E0%A5%80_%E0%A4%97%E0%A4%A4%E0%A4%BF "à€Źà„à€°à€Ÿà€‰à€šà„€ à€—à€€à€ż – Hindi") - [Hrvatski](https://hr.wikipedia.org/wiki/Brownovo_gibanje "Brownovo gibanje – Croatian") - [KreyĂČl ayisyen](https://ht.wikipedia.org/wiki/Mouvman_bwonyen "Mouvman bwonyen – Haitian Creole") - [Magyar](https://hu.wikipedia.org/wiki/Brown-mozg%C3%A1s "Brown-mozgĂĄs – Hungarian") - [Ő€ŐĄŐ”Ő„Ö€Ő„Ő¶](https://hy.wikipedia.org/wiki/%D4%B2%D6%80%D5%B8%D5%B8%D6%82%D5%B6%D5%B5%D5%A1%D5%B6_%D5%B7%D5%A1%D6%80%D5%AA%D5%B8%D6%82%D5%B4 "ÔČÖ€ŐžŐžÖ‚Ő¶Ő”ŐĄŐ¶ Ő·ŐĄÖ€ŐȘվւծ – Armenian") - [Bahasa Indonesia](https://id.wikipedia.org/wiki/Gerak_Brown "Gerak Brown – Indonesian") - [Ido](https://io.wikipedia.org/wiki/Browniana_movemento "Browniana movemento – Ido") - [Italiano](https://it.wikipedia.org/wiki/Moto_browniano "Moto browniano – Italian") - [æ—„æœŹèȘž](https://ja.wikipedia.org/wiki/%E3%83%96%E3%83%A9%E3%82%A6%E3%83%B3%E9%81%8B%E5%8B%95 "ăƒ–ăƒ©ă‚Šăƒłé‹ć‹• – Japanese") - [ÒšĐ°Đ·Đ°Ò›ŃˆĐ°](https://kk.wikipedia.org/wiki/%D0%91%D1%80%D0%BE%D1%83%D0%BD%D0%B4%D1%8B%D2%9B_%D2%9B%D0%BE%D0%B7%D2%93%D0%B0%D0%BB%D1%8B%D1%81 "Đ‘Ń€ĐŸŃƒĐœĐŽŃ‹Ò› Ò›ĐŸĐ·Ò“Đ°Đ»Ń‹Ń – Kazakh") - [àȕàČšàłàČšàČĄ](https://kn.wikipedia.org/wiki/%E0%B2%AC%E0%B3%8D%E0%B2%B0%E0%B3%8C%E0%B2%A8%E0%B2%BF%E0%B2%AF%E0%B2%A8%E0%B3%8D_%E0%B2%9A%E0%B2%B2%E0%B2%A8%E0%B3%86 "àČŹàłàČ°àłŒàČšàČżàČŻàČšàł àȚàČČàČšàł† – Kannada") - [한ꔭ얎](https://ko.wikipedia.org/wiki/%EB%B8%8C%EB%9D%BC%EC%9A%B4_%EC%9A%B4%EB%8F%99 "람띌욎 욎동 – Korean") - [Кыргызча](https://ky.wikipedia.org/wiki/%D0%91%D1%80%D0%BE%D1%83%D0%BD_%D0%BA%D1%8B%D0%B9%D0%BC%D1%8B%D0%BB%D1%8B "Đ‘Ń€ĐŸŃƒĐœ ĐșыĐčĐŒŃ‹Đ»Ń‹ – Kyrgyz") - [LietuviĆł](https://lt.wikipedia.org/wiki/Brauno_jud%C4%97jimas "Brauno judėjimas – Lithuanian") - [LatvieĆĄu](https://lv.wikipedia.org/wiki/Brauna_kust%C4%ABba "Brauna kustÄ«ba – Latvian") - [МаĐșĐ”ĐŽĐŸĐœŃĐșĐž](https://mk.wikipedia.org/wiki/%D0%91%D1%80%D0%B0%D1%83%D0%BD%D0%BE%D0%B2%D0%BE_%D0%B4%D0%B2%D0%B8%D0%B6%D0%B5%D1%9A%D0%B5 "Đ‘Ń€Đ°ŃƒĐœĐŸĐČĐŸ ĐŽĐČОжДњД – Macedonian") - [ĐœĐŸĐœĐłĐŸĐ»](https://mn.wikipedia.org/wiki/%D0%91%D1%80%D0%B0%D1%83%D0%BD%D1%8B_%D1%85%D3%A9%D0%B4%D3%A9%D0%BB%D0%B3%D3%A9%D3%A9%D0%BD "Đ‘Ń€Đ°ŃƒĐœŃ‹ Ń…Ó©ĐŽÓ©Đ»ĐłÓ©Ó©Đœ – Mongolian") - [Bahasa Melayu](https://ms.wikipedia.org/wiki/Pergerakan_Brown "Pergerakan Brown – Malay") - [Malti](https://mt.wikipedia.org/wiki/Moviment_Brownjan "Moviment Brownjan – Maltese") - [Nederlands](https://nl.wikipedia.org/wiki/Brownse_beweging_\(natuurkunde\) "Brownse beweging (natuurkunde) – Dutch") - [Norsk nynorsk](https://nn.wikipedia.org/wiki/Brownsk_r%C3%B8rsle "Brownsk rĂžrsle – Norwegian Nynorsk") - [Norsk bokmĂ„l](https://no.wikipedia.org/wiki/Brownsk_bevegelse "Brownsk bevegelse – Norwegian BokmĂ„l") - [Polski](https://pl.wikipedia.org/wiki/Ruchy_Browna "Ruchy Browna – Polish") - [ÙŸÙ†ŰŹŰ§ŰšÛŒ](https://pnb.wikipedia.org/wiki/%D8%A8%D8%B1%D8%A7%D8%A4%D9%86%DB%8C_%D9%B9%D9%88%D8%B1 "ŰšŰ±Ű§Ű€Ù†ÛŒ ÙčÙˆŰ± – Western Punjabi") - [ÙŸÚšŰȘو](https://ps.wikipedia.org/wiki/%D8%A8%D8%B1%D8%A7%D9%88%D9%86%D9%8A_%D8%AD%D8%B1%DA%A9%D8%AA "ŰšŰ±Ű§ÙˆÙ†ÙŠ ۭ۱کŰȘ – Pashto") - [PortuguĂȘs](https://pt.wikipedia.org/wiki/Movimento_browniano "Movimento browniano – Portuguese") - [RomĂąnă](https://ro.wikipedia.org/wiki/Mi%C8%99care_brownian%C4%83 "Mișcare browniană – Romanian") - [РуссĐșĐžĐč](https://ru.wikipedia.org/wiki/%D0%91%D1%80%D0%BE%D1%83%D0%BD%D0%BE%D0%B2%D1%81%D0%BA%D0%BE%D0%B5_%D0%B4%D0%B2%D0%B8%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5 "Đ‘Ń€ĐŸŃƒĐœĐŸĐČсĐșĐŸĐ” ĐŽĐČĐžĐ¶Đ”ĐœĐžĐ” – Russian") - [Srpskohrvatski / српсĐșĐŸŃ…Ń€ĐČатсĐșĐž](https://sh.wikipedia.org/wiki/Brownovo_kretanje "Brownovo kretanje – Serbo-Croatian") - [Simple English](https://simple.wikipedia.org/wiki/Brownian_motion "Brownian motion – Simple English") - [Slovenčina](https://sk.wikipedia.org/wiki/Brownov_pohyb "Brownov pohyb – Slovak") - [Slovenơčina](https://sl.wikipedia.org/wiki/Brownovo_gibanje "Brownovo gibanje – Slovenian") - [Shqip](https://sq.wikipedia.org/wiki/Levizja_Brauniane "Levizja Brauniane – Albanian") - [СрпсĐșĐž / srpski](https://sr.wikipedia.org/wiki/%D0%91%D1%80%D0%B0%D1%83%D0%BD%D0%BE%D0%B2%D0%BE_%D0%BA%D1%80%D0%B5%D1%82%D0%B0%D1%9A%D0%B5 "Đ‘Ń€Đ°ŃƒĐœĐŸĐČĐŸ ĐșрДтањД – Serbian") - [Sunda](https://su.wikipedia.org/wiki/Gerak_Brown "Gerak Brown – Sundanese") - [Svenska](https://sv.wikipedia.org/wiki/Brownsk_r%C3%B6relse "Brownsk rörelse – Swedish") - [àź€àźźàźżàźŽàŻ](https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%BF%E0%AE%B0%E0%AF%8C%E0%AE%A9%E0%AE%BF%E0%AE%AF%E0%AE%A9%E0%AF%8D_%E0%AE%87%E0%AE%AF%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%AE%E0%AF%8D "àźȘàźżàź°àŻŒàź©àźżàźŻàź©àŻ àź‡àźŻàź•àŻàź•àźźàŻ – Tamil") - [ఀెà°Čుగు](https://te.wikipedia.org/wiki/%E0%B0%95%E0%B1%8D%E0%B0%B0%E0%B0%AE%E0%B0%B0%E0%B0%B9%E0%B0%BF%E0%B0%A4_%E0%B0%9A%E0%B0%B2%E0%B0%A8%E0%B0%82 "à°•à±à°°à°źà°°à°čà°żà°€ చà°Čà°šà°‚ – Telugu") - [àč„àž—àžą](https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%84%E0%B8%A5%E0%B8%B7%E0%B9%88%E0%B8%AD%E0%B8%99%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%9A%E0%B8%A3%E0%B8%B2%E0%B8%A7%E0%B8%99%E0%B9%8C "àžàžČàžŁàč€àž„àž„àž·àčˆàž­àž™àž—àž”àčˆàčàžšàžšàžšàžŁàžČàž§àž™àčŒ â€“ Thai") - [Tagalog](https://tl.wikipedia.org/wiki/Galaw_Brownian "Galaw Brownian – Tagalog") - [TĂŒrkçe](https://tr.wikipedia.org/wiki/Brown_hareketi "Brown hareketi – Turkish") - [ĐŁĐșŃ€Đ°Ń—ĐœŃŃŒĐșĐ°](https://uk.wikipedia.org/wiki/%D0%91%D1%80%D0%BE%D1%83%D0%BD%D1%96%D0%B2%D1%81%D1%8C%D0%BA%D0%B8%D0%B9_%D1%80%D1%83%D1%85 "Đ‘Ń€ĐŸŃƒĐœŃ–ĐČсьĐșĐžĐč рух – Ukrainian") - [OÊ»zbekcha / ўзбДĐșча](https://uz.wikipedia.org/wiki/Brown_harakati "Brown harakati – Uzbek") - [Tiáșżng Việt](https://vi.wikipedia.org/wiki/Chuy%E1%BB%83n_%C4%91%E1%BB%99ng_Brown "Chuyển động Brown – Vietnamese") - [ćŽèŻ­](https://wuu.wikipedia.org/wiki/%E5%B8%83%E6%9C%97%E8%BF%90%E5%8A%A8 "ćžƒæœ—èżćŠš – Wu") - [çČ”èȘž](https://zh-yue.wikipedia.org/wiki/%E5%B8%83%E6%9C%97%E9%81%8B%E5%8B%95 "ćžƒæœ—é‹ć‹• – Cantonese") - [äž­æ–‡](https://zh.wikipedia.org/wiki/%E5%B8%83%E6%9C%97%E8%BF%90%E5%8A%A8 "ćžƒæœ—èżćŠš – Chinese") [Edit links](https://www.wikidata.org/wiki/Special:EntityPage/Q178036#sitelinks-wikipedia "Edit interlanguage links") - [Article](https://en.wikipedia.org/wiki/Brownian_motion "View the content page [c]") - [Talk](https://en.wikipedia.org/wiki/Talk:Brownian_motion "Discuss improvements to the content page [t]") English - [Read](https://en.wikipedia.org/wiki/Brownian_motion) - [Edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit "Edit this page [e]") - [View history](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=history "Past revisions of this page [h]") Tools Tools move to sidebar hide Actions - [Read](https://en.wikipedia.org/wiki/Brownian_motion) - [Edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit "Edit this page [e]") - [View history](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=history) General - [What links here](https://en.wikipedia.org/wiki/Special:WhatLinksHere/Brownian_motion "List of all English Wikipedia pages containing links to this page [j]") - [Related changes](https://en.wikipedia.org/wiki/Special:RecentChangesLinked/Brownian_motion "Recent changes in pages linked from this page [k]") - 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[Wikimedia Commons](https://commons.wikimedia.org/wiki/Category:Brownian_motion) - [Wikibooks](https://en.wikibooks.org/wiki/Wikijunior:Particles/Brownian_motion) - [Wikidata item](https://www.wikidata.org/wiki/Special:EntityPage/Q178036 "Structured data on this page hosted by Wikidata [g]") Appearance move to sidebar hide From Wikipedia, the free encyclopedia Random motion of particles suspended in a fluid [![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/97/2d_random_walk_ag_adatom_ag111.gif/330px-2d_random_walk_ag_adatom_ag111.gif)](https://en.wikipedia.org/wiki/File:2d_random_walk_ag_adatom_ag111.gif) 2-dimensional random walk of a silver [adatom](https://en.wikipedia.org/wiki/Adatom "Adatom") on an Ag(111) surface[\[1\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-1) [![](https://upload.wikimedia.org/wikipedia/commons/c/c2/Brownian_motion_large.gif)](https://en.wikipedia.org/wiki/File:Brownian_motion_large.gif) [Simulation](https://en.wikipedia.org/wiki/Brownian_dynamics "Brownian dynamics") of the Brownian motion of a large particle, analogous to a dust particle, that collides with a large set of smaller particles, analogous to molecules of a gas, which move with different velocities in different random directions **Brownian motion** is the random motion of [particles](https://en.wikipedia.org/wiki/Particle "Particle") suspended in a medium (a [liquid](https://en.wikipedia.org/wiki/Liquid "Liquid") or a [gas](https://en.wikipedia.org/wiki/Gas "Gas")).[\[2\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Feynman-1964-2) The traditional mathematical formulation of Brownian motion is that of the [Wiener process](https://en.wikipedia.org/wiki/Wiener_process "Wiener process"), which is often called Brownian motion, even in mathematical sources. This motion pattern typically consists of [random](https://en.wikipedia.org/wiki/Randomness "Randomness") fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at [thermal equilibrium](https://en.wikipedia.org/wiki/Thermal_equilibrium "Thermal equilibrium"), defined by a given [temperature](https://en.wikipedia.org/wiki/Temperature "Temperature"). Within such a fluid, there exists no preferential direction of flow (as in [transport phenomena](https://en.wikipedia.org/wiki/Transport_phenomena "Transport phenomena")). More specifically, the fluid's overall [linear](https://en.wikipedia.org/wiki/Linear_momentum "Linear momentum") and [angular](https://en.wikipedia.org/wiki/Angular_momentum "Angular momentum") momenta remain null over time. The [kinetic energies](https://en.wikipedia.org/wiki/Kinetic_energy "Kinetic energy") of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's [internal energy](https://en.wikipedia.org/wiki/Internal_energy "Internal energy") (the [equipartition theorem](https://en.wikipedia.org/wiki/Equipartition_theorem "Equipartition theorem")).[\[3\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-3) This motion is named after the Scottish botanist [Robert Brown](https://en.wikipedia.org/wiki/Robert_Brown_\(botanist,_born_1773\) "Robert Brown (botanist, born 1773)"), who first described the phenomenon in 1827, while looking through a microscope at [pollen](https://en.wikipedia.org/wiki/Pollen "Pollen") of the plant *[Clarkia pulchella](https://en.wikipedia.org/wiki/Clarkia_pulchella "Clarkia pulchella")* immersed in water. In 1900, the French mathematician [Louis Bachelier](https://en.wikipedia.org/wiki/Louis_Bachelier "Louis Bachelier") modeled the stochastic process now called Brownian motion in his doctoral thesis, The Theory of Speculation (ThĂ©orie de la spĂ©culation), prepared under the supervision of [Henri PoincarĂ©](https://en.wikipedia.org/wiki/Henri_Poincar%C3%A9 "Henri PoincarĂ©"). Then, in 1905, theoretical physicist [Albert Einstein](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein") published [a paper](https://en.wikipedia.org/wiki/%C3%9Cber_die_von_der_molekularkinetischen_Theorie_der_W%C3%A4rme_geforderte_Bewegung_von_in_ruhenden_Fl%C3%BCssigkeiten_suspendierten_Teilchen "Über die von der molekularkinetischen Theorie der WĂ€rme geforderte Bewegung von in ruhenden FlĂŒssigkeiten suspendierten Teilchen") in which he modelled the motion of the pollen particles as being moved by individual water [molecules](https://en.wikipedia.org/wiki/Molecule "Molecule"), making one of his first major scientific contributions.[\[4\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Einstein-1905-4) The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. This explanation of Brownian motion served as convincing evidence that [atoms](https://en.wikipedia.org/wiki/Atom "Atom") and molecules exist and was further verified experimentally by [Jean Perrin](https://en.wikipedia.org/wiki/Jean_Baptiste_Perrin "Jean Baptiste Perrin") in 1908. Perrin was awarded the [Nobel Prize in Physics](https://en.wikipedia.org/wiki/Nobel_Prize_in_Physics "Nobel Prize in Physics") in 1926 "for his work on the discontinuous structure of matter".[\[5\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-5) The [many-body interactions](https://en.wikipedia.org/wiki/Many-body_problem "Many-body problem") that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. Consequently, only probabilistic models applied to [molecular populations](https://en.wikipedia.org/wiki/Statistical_ensemble "Statistical ensemble") can be employed to describe it.[\[6\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-6) Two such models of the [statistical mechanics](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics"), due to Einstein and Smoluchowski, are presented below. Another, pure probabilistic class of models is the class of the [stochastic process](https://en.wikipedia.org/wiki/Stochastic_process "Stochastic process") models. There exist sequences of both simpler and more complicated stochastic processes which converge (in the [limit](https://en.wikipedia.org/wiki/Limit_of_a_function "Limit of a function")) to Brownian motion (see [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk") and [Donsker's theorem](https://en.wikipedia.org/wiki/Donsker%27s_theorem "Donsker's theorem")).[\[7\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-7)[\[8\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-8) ## History \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=1 "Edit section: History")\] [![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/44/PerrinPlot2.svg/250px-PerrinPlot2.svg.png)](https://en.wikipedia.org/wiki/File:PerrinPlot2.svg) Reproduced from the [Jean Baptiste Perrin](https://en.wikipedia.org/wiki/Jean_Baptiste_Perrin "Jean Baptiste Perrin") book *Les Atomes*: tracings of the motions of three colloidal particles of radius 0.53 ÎŒm, as seen under the microscope, with each point representing that particle's successive position every 30 seconds; the points are then joined by straight line segments (mesh size = 3.2 ÎŒm)[\[9\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-9) The Roman philosopher-poet [Lucretius](https://en.wikipedia.org/wiki/Lucretius "Lucretius")' scientific poem *[On the Nature of Things](https://en.wikipedia.org/wiki/On_the_Nature_of_Things "On the Nature of Things")* (c. 60 BC) has a remarkable description of the motion of [dust](https://en.wikipedia.org/wiki/Dust "Dust") particles in verses 113–140 from Book II. He uses this as a proof of the existence of atoms: > Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves \[i.e., spontaneously\]. Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible.[\[10\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-10) Although the mingling, tumbling motion of dust particles is caused largely by macroscopic air currents and convection, the glittering, microscopic jiggling motion of small dust particles is caused chiefly by true [Brownian dynamics](https://en.wikipedia.org/wiki/Brownian_dynamics "Brownian dynamics"); Lucretius "perfectly describes and explains the Brownian movement by a wrong example".[\[11\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-11) The formal scientific discovery of this phenomenon is credited to the botanist [Robert Brown](https://en.wikipedia.org/wiki/Robert_Brown_\(botanist,_born_1773\) "Robert Brown (botanist, born 1773)") in 1827. Brown was studying plant reproduction when he observed [pollen](https://en.wikipedia.org/wiki/Pollen "Pollen") grains of the plant *[Clarkia pulchella](https://en.wikipedia.org/wiki/Clarkia_pulchella "Clarkia pulchella")* in water under a simple microscope. These grains contain minute particles on the order of 1/4,000th of an inch (6.4 microns) in size. He observed these particles executing a continuous, jittery motion. By repeating the experiment with particles of inorganic matter, such as glass and rock dust, he was able to rule out that the motion was life-related, although its physical origin was yet to be explained.[\[12\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Brush-1968-12)[\[13\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-13) The mathematics of much of stochastic analysis, including the mathematics of Brownian motion, was introduced by [Louis Bachelier](https://en.wikipedia.org/wiki/Louis_Bachelier "Louis Bachelier") in 1900 in his PhD thesis "The theory of speculation", in which he presented an innovative probabilistic analysis of the stock and option markets. However, this pioneering mathematical work connecting random walks to continuous time was largely unknown until the 1950s.[\[14\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-14)[\[15\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Morters-2001-15): 33 The early 20th century saw the theoretical formalization of Brownian motion bridging the gap between thermodynamics and atomic theory: - [Albert Einstein](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein") (in one of his [1905 papers](https://en.wikipedia.org/wiki/%C3%9Cber_die_von_der_molekularkinetischen_Theorie_der_W%C3%A4rme_geforderte_Bewegung_von_in_ruhenden_Fl%C3%BCssigkeiten_suspendierten_Teilchen "Über die von der molekularkinetischen Theorie der WĂ€rme geforderte Bewegung von in ruhenden FlĂŒssigkeiten suspendierten Teilchen")) provided an explanation of Brownian motion in terms of atoms and molecules at a time when their physical existence was still fiercely debated by scientists. Einstein proved the mathematical relation between the probability distribution of a Brownian particle and the macroscopic [diffusion equation](https://en.wikipedia.org/wiki/Diffusion_equation "Diffusion equation").[\[15\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Morters-2001-15): 33 - These predictive equations describing Brownian motion were subsequently verified by the meticulous experimental work of [Jean Baptiste Perrin](https://en.wikipedia.org/wiki/Jean_Baptiste_Perrin "Jean Baptiste Perrin") in 1908, leading to his Nobel prize and settling the atomic debate.[\[16\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Grigoryan-1999-16) - [Norbert Wiener](https://en.wikipedia.org/wiki/Norbert_Wiener "Norbert Wiener") gave the first complete and rigorous mathematical analysis of the phenomenon in 1923, leading to the underlying mathematical concept being permanently called a [Wiener process](https://en.wikipedia.org/wiki/Wiener_process "Wiener process").[\[15\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Morters-2001-15) The instantaneous velocity of the Brownian motion can be defined as *v* = Δ*x*/Δ*t*, when Δ*t* \<\< *τ*, where τ is the momentum relaxation time. Advancements in modern physics have allowed this to be directly measured: - In 2010, the instantaneous velocity of a single Brownian particle (a glass microsphere trapped in air with [optical tweezers](https://en.wikipedia.org/wiki/Optical_tweezers "Optical tweezers")) was measured successfully for the first time. - The velocity data perfectly verified the [Maxwell–Boltzmann velocity distribution](https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution "Maxwell–Boltzmann distribution") and confirmed the equipartition theorem for a Brownian particle at microscopic timescales.[\[17\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Velocity-2010-17) ## Statistical mechanics theories \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=2 "Edit section: Statistical mechanics theories")\] ### Einstein's theory \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=3 "Edit section: Einstein's theory")\] There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian particles, in which the diffusion coefficient is related to the [mean squared displacement](https://en.wikipedia.org/wiki/Mean_squared_displacement "Mean squared displacement") of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities.[\[18\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Einstein-1956-18) In this way Einstein was able to determine the size of atoms, and how many atoms there are in a mole, or the [molecular weight](https://en.wikipedia.org/wiki/Molecular_weight "Molecular weight") in grams, of a gas.[\[19\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-19) In accordance to [Avogadro's law](https://en.wikipedia.org/wiki/Avogadro%27s_law "Avogadro's law"), this volume is the same for all ideal gases, namely 22.414 liters at standard temperature and pressure. The number of atoms contained in this volume is referred to as the [Avogadro constant](https://en.wikipedia.org/wiki/Avogadro_constant "Avogadro constant") or as [Avogadro's number](https://en.wikipedia.org/wiki/Avogadro%27s_number "Avogadro's number") (approximately 6\.02×1023 mol−1), and the determination of this number is tantamount to the knowledge of the mass of an atom, since the latter is obtained by dividing the [molar mass](https://en.wikipedia.org/wiki/Molar_mass "Molar mass") of the gas by the [Avogadro constant](https://en.wikipedia.org/wiki/Avogadro_constant "Avogadro constant"). [![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Diffusion_of_Brownian_particles.svg/330px-Diffusion_of_Brownian_particles.svg.png)](https://en.wikipedia.org/wiki/File:Diffusion_of_Brownian_particles.svg) The characteristic bell-shaped curves of the diffusion of Brownian particles. The distribution begins as a [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function "Dirac delta function"), indicating that all the particles are located at the origin at time *t* = 0. As *t* increases, the distribution flattens (though remains bell-shaped), and ultimately becomes uniform in the limit that time goes to infinity. The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval.[\[4\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Einstein-1905-4) Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 1014 collisions per second.[\[2\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Feynman-1964-2) He regarded the increment of particle positions in time τ {\\displaystyle \\tau } ![{\\displaystyle \\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c) in a one-dimensional (*x*) space (with the coordinates chosen so that the origin lies at the initial position of the particle) as a [random variable](https://en.wikipedia.org/wiki/Random_variable "Random variable") (q {\\displaystyle q} ![{\\displaystyle q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d)) with some [probability density function](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") φ ( q ) {\\displaystyle \\varphi (q)} ![{\\displaystyle \\varphi (q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c84e62844cff8763f4f35a5ac00be1d4ccb691f3) (i.e., φ ( q ) {\\displaystyle \\varphi (q)} ![{\\displaystyle \\varphi (q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c84e62844cff8763f4f35a5ac00be1d4ccb691f3) is the probability density for a jump of magnitude q {\\displaystyle q} ![{\\displaystyle q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d), i.e., the probability density of the particle incrementing its position from x {\\displaystyle x} ![{\\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4) to x \+ q {\\displaystyle x+q} ![{\\displaystyle x+q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb88adc883f9d0bd380d68976d6db693aa2a938e) in the time interval τ {\\displaystyle \\tau } ![{\\displaystyle \\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c)). Further, assuming conservation of particle number, he expanded the [number density](https://en.wikipedia.org/wiki/Number_density "Number density") ρ ( x , t \+ τ ) {\\displaystyle \\rho (x,t+\\tau )} ![{\\displaystyle \\rho (x,t+\\tau )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/095bf2247c50785be4cf836d85367a73e163ce55) (number of particles per unit volume around x {\\displaystyle x} ![{\\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)) at time t \+ τ {\\displaystyle t+\\tau } ![{\\displaystyle t+\\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2517ff18a2091c903aa8d144e7e60535e4f673b) in a [Taylor series](https://en.wikipedia.org/wiki/Taylor_series "Taylor series"), ρ ( x , t \+ τ ) \= ρ ( x , t ) \+ τ ∂ ρ ( x , t ) ∂ t \+ ⋯ \= ∫ − ∞ ∞ ρ ( x − q , t ) φ ( q ) d q \= E q \[ ρ ( x − q , t ) \] \= ρ ( x , t ) ∫ − ∞ ∞ φ ( q ) d q − ∂ ρ ∂ x ∫ − ∞ ∞ q φ ( q ) d q \+ ∂ 2 ρ ∂ x 2 ∫ − ∞ ∞ q 2 2 φ ( q ) d q \+ ⋯ \= ρ ( x , t ) ⋅ 1 − 0 \+ ∂ 2 ρ ∂ x 2 ∫ − ∞ ∞ q 2 2 φ ( q ) d q \+ ⋯ {\\displaystyle {\\begin{aligned}\\rho (x,t+\\tau )={}&\\rho (x,t)+\\tau {\\frac {\\partial \\rho (x,t)}{\\partial t}}+\\cdots \\\\\[2ex\]={}&\\int \_{-\\infty }^{\\infty }\\rho (x-q,t)\\,\\varphi (q)\\,dq=\\mathbb {E} \_{q}{\\left\[\\rho (x-q,t)\\right\]}\\\\\[1ex\]={}&\\rho (x,t)\\,\\int \_{-\\infty }^{\\infty }\\varphi (q)\\,dq-{\\frac {\\partial \\rho }{\\partial x}}\\,\\int \_{-\\infty }^{\\infty }q\\,\\varphi (q)\\,dq+{\\frac {\\partial ^{2}\\rho }{\\partial x^{2}}}\\,\\int \_{-\\infty }^{\\infty }{\\frac {q^{2}}{2}}\\varphi (q)\\,dq+\\cdots \\\\\[1ex\]={}&\\rho (x,t)\\cdot 1-0+{\\cfrac {\\partial ^{2}\\rho }{\\partial x^{2}}}\\,\\int \_{-\\infty }^{\\infty }{\\frac {q^{2}}{2}}\\varphi (q)\\,dq+\\cdots \\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\rho (x,t+\\tau )={}&\\rho (x,t)+\\tau {\\frac {\\partial \\rho (x,t)}{\\partial t}}+\\cdots \\\\\[2ex\]={}&\\int \_{-\\infty }^{\\infty }\\rho (x-q,t)\\,\\varphi (q)\\,dq=\\mathbb {E} \_{q}{\\left\[\\rho (x-q,t)\\right\]}\\\\\[1ex\]={}&\\rho (x,t)\\,\\int \_{-\\infty }^{\\infty }\\varphi (q)\\,dq-{\\frac {\\partial \\rho }{\\partial x}}\\,\\int \_{-\\infty }^{\\infty }q\\,\\varphi (q)\\,dq+{\\frac {\\partial ^{2}\\rho }{\\partial x^{2}}}\\,\\int \_{-\\infty }^{\\infty }{\\frac {q^{2}}{2}}\\varphi (q)\\,dq+\\cdots \\\\\[1ex\]={}&\\rho (x,t)\\cdot 1-0+{\\cfrac {\\partial ^{2}\\rho }{\\partial x^{2}}}\\,\\int \_{-\\infty }^{\\infty }{\\frac {q^{2}}{2}}\\varphi (q)\\,dq+\\cdots \\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/155f9b33c2bde783f0cf06e55dc69d7a705118e9) where the second equality is by definition of φ {\\displaystyle \\varphi } ![{\\displaystyle \\varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e). The [integral](https://en.wikipedia.org/wiki/Integral "Integral") in the first term is equal to one by the definition of probability, and the second and other even terms (i.e. first and other odd [moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)")) vanish because of space symmetry. What is left gives rise to the following relation: ∂ ρ ∂ t \= ∂ 2 ρ ∂ x 2 ⋅ ∫ − ∞ ∞ q 2 2 τ φ ( q ) d q \+ higher-order even moments. {\\displaystyle {\\frac {\\partial \\rho }{\\partial t}}={\\frac {\\partial ^{2}\\rho }{\\partial x^{2}}}\\cdot \\int \_{-\\infty }^{\\infty }{\\frac {q^{2}}{2\\tau }}\\varphi (q)\\,dq+{\\text{higher-order even moments.}}} ![{\\displaystyle {\\frac {\\partial \\rho }{\\partial t}}={\\frac {\\partial ^{2}\\rho }{\\partial x^{2}}}\\cdot \\int \_{-\\infty }^{\\infty }{\\frac {q^{2}}{2\\tau }}\\varphi (q)\\,dq+{\\text{higher-order even moments.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/045362993238e9286f197c2d08c0d7da7cdf3070) Where the coefficient after the [Laplacian](https://en.wikipedia.org/wiki/Laplacian "Laplacian"), the second moment of probability of displacement q {\\displaystyle q} ![{\\displaystyle q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d), is interpreted as [mass diffusivity](https://en.wikipedia.org/wiki/Mass_diffusivity "Mass diffusivity") *D*: D \= ∫ − ∞ ∞ q 2 2 τ φ ( q ) d q . {\\displaystyle D=\\int \_{-\\infty }^{\\infty }{\\frac {q^{2}}{2\\tau }}\\varphi (q)\\,dq.} ![{\\displaystyle D=\\int \_{-\\infty }^{\\infty }{\\frac {q^{2}}{2\\tau }}\\varphi (q)\\,dq.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb16997b332ff1ac71b0a99fa6e4aea7e466124) Then the density of Brownian particles ρ at point x at time t satisfies the [diffusion equation](https://en.wikipedia.org/wiki/Diffusion_equation "Diffusion equation"): ∂ ρ ∂ t \= D ⋅ ∂ 2 ρ ∂ x 2 , {\\displaystyle {\\frac {\\partial \\rho }{\\partial t}}=D\\cdot {\\frac {\\partial ^{2}\\rho }{\\partial x^{2}}},} ![{\\displaystyle {\\frac {\\partial \\rho }{\\partial t}}=D\\cdot {\\frac {\\partial ^{2}\\rho }{\\partial x^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/317b275652b2f923d52340b08570553d3c843879) Assuming that *N* particles start from the origin at the initial time *t* = 0, the diffusion equation has the solution:[\[20\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-20) ρ ( x , t ) \= N 4 π D t exp ⁥ ( − x 2 4 D t ) . {\\displaystyle \\rho (x,t)={\\frac {N}{\\sqrt {4\\pi Dt}}}\\exp {\\left(-{\\frac {x^{2}}{4Dt}}\\right)}.} ![{\\displaystyle \\rho (x,t)={\\frac {N}{\\sqrt {4\\pi Dt}}}\\exp {\\left(-{\\frac {x^{2}}{4Dt}}\\right)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/baef5b3d13bb5b6626a641937ac0d18318a1a548) This expression (which is a [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution") with the mean ÎŒ \= 0 {\\displaystyle \\mu =0} ![{\\displaystyle \\mu =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3753282c0ad2ea1e7d63f39425efd13c37da3169) and variance σ 2 \= 2 D t {\\displaystyle \\sigma ^{2}=2Dt} ![{\\displaystyle \\sigma ^{2}=2Dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d88a84c4b4144f34e1a879fc7ec7d25b236c2809) usually called Brownian motion B t {\\displaystyle B\_{t}} ![{\\displaystyle B\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f92553a3c8585519e540724dffe9306c83ca2f8)) allowed Einstein to calculate the [moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right. The second moment is, however, non-vanishing, being given by E \[ x 2 \] \= 2 D t . {\\displaystyle \\mathbb {E} {\\left\[x^{2}\\right\]}=2Dt.} ![{\\displaystyle \\mathbb {E} {\\left\[x^{2}\\right\]}=2Dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f808eb010dadd7cb5da992f62b95334223e9f4) This equation expresses the mean squared displacement in terms of the time elapsed and the diffusivity. From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root.[\[18\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Einstein-1956-18) His argument is based on a conceptual switch from the "ensemble" of Brownian particles to the "single" Brownian particle: we can speak of the relative number of particles at a single instant just as well as of the time it takes a Brownian particle to reach a given point.[\[21\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-21) The second part of Einstein's theory relates the diffusion constant to physically measurable quantities, such as the mean squared displacement of a particle in a given time interval. This result enables the experimental determination of the Avogadro number and therefore the size of molecules. Einstein analyzed a dynamic equilibrium being established between opposing forces. The beauty of his argument is that the final result does not depend upon which forces are involved in setting up the dynamic equilibrium. In his original treatment, Einstein considered an [osmotic pressure](https://en.wikipedia.org/wiki/Osmotic_pressure "Osmotic pressure") experiment, but the same conclusion can be reached in other ways. Consider, for instance, particles suspended in a viscous fluid in a gravitational field. Gravity tends to make the particles settle, whereas diffusion acts to homogenize them, driving them into regions of smaller concentration. Under the action of gravity, a particle acquires a downward speed of *v* = *ÎŒmg*, where m is the mass of the particle, g is the acceleration due to gravity, and ÎŒ is the particle's [mobility](https://en.wikipedia.org/wiki/Einstein_relation_\(kinetic_theory\) "Einstein relation (kinetic theory)") in the fluid. [George Stokes](https://en.wikipedia.org/wiki/Sir_George_Stokes,_1st_Baronet "Sir George Stokes, 1st Baronet") had shown that the mobility for a spherical particle with radius r is ÎŒ \= 1 6 π η r {\\displaystyle \\mu ={\\tfrac {1}{6\\pi \\eta r}}} ![{\\displaystyle \\mu ={\\tfrac {1}{6\\pi \\eta r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be24ee78d91ecf2f45fba9b087986821463a7858), where η is the [dynamic viscosity](https://en.wikipedia.org/wiki/Dynamic_viscosity "Dynamic viscosity") of the fluid. In a state of dynamic equilibrium, and under the hypothesis of isothermal fluid, the particles are distributed according to the [barometric distribution](https://en.wikipedia.org/wiki/Barometric_formula "Barometric formula") ρ \= ρ o exp ⁥ ( − m g h k B T ) , {\\displaystyle \\rho =\\rho \_{o}\\,\\exp \\left({-{\\frac {mgh}{k\_{\\text{B}}T}}}\\right),} ![{\\displaystyle \\rho =\\rho \_{o}\\,\\exp \\left({-{\\frac {mgh}{k\_{\\text{B}}T}}}\\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de96f9973a262aa9c33869b220ae9f1b76307452) where *ρ* − *ρ*o is the difference in density of particles separated by a height difference, of h \= z − z o {\\displaystyle h=z-z\_{o}} ![{\\displaystyle h=z-z\_{o}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f968d881d543c1ece3483bb6788b61da38908e4), *k*B is the [Boltzmann constant](https://en.wikipedia.org/wiki/Boltzmann_constant "Boltzmann constant") (the ratio of the [universal gas constant](https://en.wikipedia.org/wiki/Universal_gas_constant "Universal gas constant"), *R*, to the [Avogadro constant](https://en.wikipedia.org/wiki/Avogadro_constant "Avogadro constant"), *N*A), and *T* is the [absolute temperature](https://en.wikipedia.org/wiki/Thermodynamic_temperature "Thermodynamic temperature"). [![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Brownian_motion_gamboge.jpg/250px-Brownian_motion_gamboge.jpg)](https://en.wikipedia.org/wiki/File:Brownian_motion_gamboge.jpg) [Perrin](https://en.wikipedia.org/wiki/Jean_Baptiste_Perrin "Jean Baptiste Perrin") examined the equilibrium ([barometric distribution](https://en.wikipedia.org/wiki/Barometric_formula "Barometric formula")) of granules (0.6 [microns](https://en.wikipedia.org/wiki/Micron "Micron")) of [gamboge](https://en.wikipedia.org/wiki/Gamboge "Gamboge"), a viscous substance, under the microscope. The granules move against gravity to regions of lower concentration. The relative change in density observed in 10 microns of suspension is equivalent to that occurring in 6 km of air. [Dynamic equilibrium](https://en.wikipedia.org/wiki/Dynamic_equilibrium "Dynamic equilibrium") is established because the more that particles are pulled down by [gravity](https://en.wikipedia.org/wiki/Gravity "Gravity"), the greater the tendency for the particles to migrate to regions of lower concentration. The flux is given by [Fick's law](https://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion "Fick's laws of diffusion"), J \= − D d ρ d h , {\\displaystyle J=-D{\\frac {d\\rho }{dh}},} ![{\\displaystyle J=-D{\\frac {d\\rho }{dh}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2729e7641f9c691261fb9ddc635e161f61dd63) where *J* = *ρv*. Introducing the formula for ρ, we find that v \= D m g k B T . {\\displaystyle v={\\frac {Dmg}{k\_{\\text{B}}T}}.} ![{\\displaystyle v={\\frac {Dmg}{k\_{\\text{B}}T}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d70f9b4b3f064d2b3edf6ab1e39ed33e02c6095) In a state of dynamical equilibrium, this speed must also be equal to *v* = *ÎŒmg*. Both expressions for v are proportional to *mg*, reflecting that the derivation is independent of the type of forces considered. Similarly, one can derive an equivalent formula for identical [charged particles](https://en.wikipedia.org/wiki/Charged_particle "Charged particle") of charge q in a uniform [electric field](https://en.wikipedia.org/wiki/Electric_field "Electric field") of magnitude E, where *mg* is replaced with the [electrostatic force](https://en.wikipedia.org/wiki/Electrostatic_force "Electrostatic force") *qE*. Equating these two expressions yields the [Einstein relation](https://en.wikipedia.org/wiki/Einstein_relation_\(kinetic_theory\) "Einstein relation (kinetic theory)") for the diffusivity, independent of *mg* or *qE* or other such forces: E \[ x 2 \] 2 t \= D \= ÎŒ k B T \= ÎŒ R T N A \= R T 6 π η r N A . {\\displaystyle {\\frac {\\mathbb {E} {\\left\[x^{2}\\right\]}}{2t}}=D=\\mu k\_{\\text{B}}T={\\frac {\\mu RT}{N\_{\\text{A}}}}={\\frac {RT}{6\\pi \\eta rN\_{\\text{A}}}}.} ![{\\displaystyle {\\frac {\\mathbb {E} {\\left\[x^{2}\\right\]}}{2t}}=D=\\mu k\_{\\text{B}}T={\\frac {\\mu RT}{N\_{\\text{A}}}}={\\frac {RT}{6\\pi \\eta rN\_{\\text{A}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d19b78ac840afe66ba1ad694390512cf72bd90d2) Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of the [Boltzmann constant](https://en.wikipedia.org/wiki/Boltzmann_constant "Boltzmann constant") as *k*B = *R* / *N*A, and the fourth equality follows from Stokes's formula for the mobility. By measuring the mean squared displacement over a time interval along with the universal gas constant *R*, the temperature T, the viscosity η, and the particle radius r, the Avogadro constant *N*A can be determined. The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously by [J. J. Thomson](https://en.wikipedia.org/wiki/J._J._Thomson "J. J. Thomson")[\[22\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Thomson-1904-22) in his series of lectures at Yale University in May 1903 that the dynamic equilibrium between the velocity generated by a [concentration gradient](https://en.wikipedia.org/wiki/Concentration_gradient "Concentration gradient") given by Fick's law and the velocity due to the variation of the partial pressure caused when ions are set in motion "gives us a method of determining Avogadro's constant which is independent of any hypothesis as to the shape or size of molecules, or of the way in which they act upon each other".[\[22\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Thomson-1904-22) An identical expression to Einstein's formula for the diffusion coefficient was also found by [Walther Nernst](https://en.wikipedia.org/wiki/Walther_Nernst "Walther Nernst") in 1888[\[23\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-23) in which he expressed the diffusion coefficient as the ratio of the [osmotic pressure](https://en.wikipedia.org/wiki/Osmotic_pressure "Osmotic pressure") to the ratio of the [frictional](https://en.wikipedia.org/wiki/Friction "Friction") force and the velocity to which it gives rise. The former was equated to the law of van 't Hoff while the latter was given by [Stokes's law](https://en.wikipedia.org/wiki/Stokes%27s_law "Stokes's law"). He writes k â€Č \= p o / k {\\displaystyle k'=p\_{o}/k} ![{\\displaystyle k'=p\_{o}/k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d61c5206a0db1e99ba184d2a3649cf4ad38db948) for the diffusion coefficient kâ€Č, where p o {\\displaystyle p\_{o}} ![{\\displaystyle p\_{o}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c058013bfdb87708b775d59bda857fd5e6d53b3) is the osmotic pressure and k is the ratio of the frictional force to the molecular viscosity which he assumes is given by Stokes's formula for the viscosity. Introducing the [ideal gas law](https://en.wikipedia.org/wiki/Ideal_gas_law "Ideal gas law") per unit volume for the osmotic pressure, the formula becomes identical to that of Einstein's.[\[24\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-24) The use of Stokes's law in Nernst's case, as well as in Einstein and Smoluchowski, is not strictly applicable since it does not apply to the case where the radius of the sphere is small in comparison with the [mean free path](https://en.wikipedia.org/wiki/Mean_free_path "Mean free path").[\[25\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-25) Confirming Einstein's formula experimentally proved difficult. Initial attempts by [Theodor Svedberg](https://en.wikipedia.org/wiki/Theodor_Svedberg "Theodor Svedberg") in 1906 and 1907 were critiqued by Einstein and by Perrin as not measuring a quantity directly comparable to the formula. [Victor Henri](https://en.wikipedia.org/wiki/Victor_Henri "Victor Henri") in 1908 took cinematographic shots through a microscope and found quantitative disagreement with the formula but again the analysis was uncertain.[\[26\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-26) Einstein's predictions were finally confirmed in a series of experiments carried out by Chaudesaigues in 1908 and Perrin in 1909.[\[27\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-27)[\[12\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Brush-1968-12) The confirmation of Einstein's theory constituted empirical progress for the [kinetic theory of heat](https://en.wikipedia.org/wiki/Kinetic_theory_of_gases "Kinetic theory of gases"). In essence, Einstein showed that the motion can be predicted directly from the kinetic model of [thermal equilibrium](https://en.wikipedia.org/wiki/Thermal_equilibrium "Thermal equilibrium"). The importance of the theory lay in the fact that it confirmed the kinetic theory's account of the [second law of thermodynamics](https://en.wikipedia.org/wiki/Second_law_of_thermodynamics "Second law of thermodynamics") as being an essentially statistical law.[\[28\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-28) Brownian motion model of the trajectory of a particle of dye in water ### Smoluchowski model \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=4 "Edit section: Smoluchowski model")\] [Smoluchowski](https://en.wikipedia.org/wiki/Marian_Smoluchowski "Marian Smoluchowski")'s theory of Brownian motion,[\[29\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-29) later contextualized in comprehensive reviews of stochastic physics,[\[30\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-30) starts from the same premise as that of Einstein and derives the same probability distribution *ρ*(*x*, *t*) for the displacement of a Brownian particle along the x axis in time t. He therefore gets the same expression for the mean squared displacement: E \[ ( Δ x ) 2 \] {\\textstyle \\mathbb {E} {\\left\[(\\Delta x)^{2}\\right\]}} ![{\\textstyle \\mathbb {E} {\\left\[(\\Delta x)^{2}\\right\]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb176c581545de90d31f1ffea58ef73ed686093d). However, when he relates it to a particle of mass m moving at a velocity u which is the result of a frictional force governed by Stokes's law, he finds E \[ ( Δ x ) 2 \] \= 2 D t \= t 32 81 m u 2 π ÎŒ a \= t 64 27 1 2 m u 2 3 π ÎŒ a {\\displaystyle \\mathbb {E} {\\left\[(\\Delta x)^{2}\\right\]}=2Dt=t{\\frac {32}{81}}{\\frac {mu^{2}}{\\pi \\mu a}}=t{\\frac {64}{27}}{\\frac {{\\frac {1}{2}}mu^{2}}{3\\pi \\mu a}}} ![{\\displaystyle \\mathbb {E} {\\left\[(\\Delta x)^{2}\\right\]}=2Dt=t{\\frac {32}{81}}{\\frac {mu^{2}}{\\pi \\mu a}}=t{\\frac {64}{27}}{\\frac {{\\frac {1}{2}}mu^{2}}{3\\pi \\mu a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86135c60133697844b01ee4c06a9d5fa84470b14) where ÎŒ is the viscosity coefficient, and a is the radius of the particle. Associating the kinetic energy m u 2 / 2 {\\textstyle mu^{2}/2} ![{\\textstyle mu^{2}/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27984b6f95bafd1da2add2193f974aec53bc671c) with the thermal energy R T / N {\\textstyle RT/N} ![{\\textstyle RT/N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24cac37f97fde56f0713eab5107b57a509c68b50), the expression for the mean squared displacement is 64/27 times that found by Einstein. This discrepancy arises from differing theoretical approaches: Einstein assumed Stokes drag applied directly to the macroscopic drift velocity, while Smoluchowski performed a more detailed kinematic collision analysis but introduced a slight calculation variance when averaging over the Maxwellian velocity distribution. The fraction 27/64 was commented on by [Arnold Sommerfeld](https://en.wikipedia.org/wiki/Arnold_Sommerfeld "Arnold Sommerfeld") in his necrology on Smoluchowski: "The numerical coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt."[\[31\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-31) Smoluchowski[\[32\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-32) attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are exactly equal. To address this paradox, he relies on the inevitability of statistical fluctuations: - If the probability of m gains and n − m {\\textstyle n-m} ![{\\textstyle n-m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9408cf61fe6c72a387c31a444cfc7ac3137ddd5d) losses follows a [binomial distribution](https://en.wikipedia.org/wiki/Binomial_distribution "Binomial distribution"), P m , n \= ( n m ) 2 − n {\\displaystyle P\_{m,n}={\\binom {n}{m}}2^{-n}} ![{\\displaystyle P\_{m,n}={\\binom {n}{m}}2^{-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98367f0a10e66d5b02b9998d80de5cbe77416e9e) with equal *a priori* probabilities of 1/2, the mean total gain is E \[ 2 m − n \] \= ∑ m \= n 2 n ( 2 m − n ) P m , n \= n n \! 2 n \+ 1 \[ ( n 2 ) \! \] 2 {\\displaystyle \\mathbb {E} {\\left\[2m-n\\right\]}=\\sum \_{m={\\frac {n}{2}}}^{n}(2m-n)P\_{m,n}={\\frac {nn!}{2^{n+1}\\left\[\\left({\\frac {n}{2}}\\right)!\\right\]^{2}}}} ![{\\displaystyle \\mathbb {E} {\\left\[2m-n\\right\]}=\\sum \_{m={\\frac {n}{2}}}^{n}(2m-n)P\_{m,n}={\\frac {nn!}{2^{n+1}\\left\[\\left({\\frac {n}{2}}\\right)!\\right\]^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72f3259e6d8b5f8b588ba14dbae36071836cc493) - If n is large enough so that Stirling's approximation can be used in the form n \! ≈ ( n e ) n 2 π n {\\textstyle n!\\approx \\left({\\frac {n}{e}}\\right)^{n}{\\sqrt {2\\pi n}}} ![{\\textstyle n!\\approx \\left({\\frac {n}{e}}\\right)^{n}{\\sqrt {2\\pi n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/487096849d06aea7ffa11b9be260dedd53c3391e) , then the expected total absolute gain representing the net drift can be approximated.[\[33\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-33) This expected gain will be: E \[ 2 m − n \] ≈ n 2 π {\\displaystyle \\mathbb {E} {\\left\[2m-n\\right\]}\\approx {\\sqrt {\\frac {n}{2\\pi }}}} ![{\\displaystyle \\mathbb {E} {\\left\[2m-n\\right\]}\\approx {\\sqrt {\\frac {n}{2\\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e27b8f4165bc2e0c804df161fd4c33d6fd50d3e) showing that the net displacement increases proportionally to the square root of the total population of collision events. Suppose that a Brownian particle of mass M is surrounded by lighter particles of mass m which are traveling at a speed u. Smoluchowski reasons that the mechanics of these interactions produce a macroscopically observable effect: - In any single collision between a surrounding fluid particle and the Brownian particle, the velocity transmitted to the latter will be m u / M {\\textstyle mu/M} ![{\\textstyle mu/M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57ce044c79db00202c9dbf9ea6aaaf88336a1d8b) . This ratio is exceedingly small, on the order of 10−7 cm/s. - However, in a gas there will be more than 1016 collisions in a second, and even more in a liquid, where there are expected to be roughly 1020 collisions in one second. - Because of the immense scale of collisions, statistical imbalances are inevitable. While some collisions will accelerate the Brownian particle, others will decelerate it. - If there is a mean excess of one kind of collision (e.g., more impacts from the left than the right) on the order of 108 to 1010 collisions in a single second, then the instantaneous velocity of the Brownian particle may be anywhere between 10 and 1000 cm/s. - Thus, even though there are equal probabilities for forward and backward collisions, the sheer volume of events creates a net tendency to keep the Brownian particle in erratic, continuous motion, much like the fluctuations predicted by the ballot theorem. These orders of magnitude do not take into consideration the velocity of the Brownian particle, U, which actively depends on the collisions that tend to accelerate and decelerate it. The larger U is, the greater will be the resistive drag of collisions that will retard it, so that the velocity of a Brownian particle can never increase without limit. If such an unbounded process could occur, it would be tantamount to a perpetual motion machine of the second kind. Since the equipartition of energy applies to this system in thermal equilibrium, the kinetic energy of the Brownian particle, M U 2 / 2 {\\textstyle MU^{2}/2} ![{\\textstyle MU^{2}/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e04abbc311be0a74c39f0d76dd428d46979282c6), will be equal, on the average, to the kinetic energy of the surrounding fluid particle, m u 2 / 2 {\\textstyle mu^{2}/2} ![{\\textstyle mu^{2}/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27984b6f95bafd1da2add2193f974aec53bc671c). In 1906, Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion.[\[34\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-34) The model assumes collisions with M ≫ m {\\textstyle M\\gg m} ![{\\textstyle M\\gg m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de9fc63624060a1a77eb908a9a3909dbf706c0ba) where M is the test particle's mass and m the mass of one of the individual particles composing the fluid. To build this theoretical framework, several simplifying assumptions are made: - The particle collisions are confined to one dimension along a single axis. - It is equally probable for the test particle to be hit from the left as from the right. - Every collision always imparts the exact same discrete magnitude of velocity change, Δ V {\\textstyle \\Delta V} ![{\\textstyle \\Delta V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6098f6ae65a71ec75c77595a1ce6c6b844a81972) . If N R {\\textstyle N\_{\\text{R}}} ![{\\textstyle N\_{\\text{R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d52c69d029e1517130dc6283d10a4fb12b257f7) is the number of collisions from the right and N L {\\textstyle N\_{\\text{L}}} ![{\\textstyle N\_{\\text{L}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f92bed4f749b91147fadfa9164cba3b03141ff94) the number of collisions from the left then after N collisions the particle's velocity will have changed by Δ V ( 2 N R − N ) {\\textstyle \\Delta V(2N\_{\\text{R}}-N)} ![{\\textstyle \\Delta V(2N\_{\\text{R}}-N)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4944e34eda114f9c1b868debd9b33950c658822). The [multiplicity](https://en.wikipedia.org/wiki/Multiplicity_\(mathematics\) "Multiplicity (mathematics)") is then simply given by: ( N N R ) \= N \! N R \! ( N − N R ) \! {\\displaystyle {\\binom {N}{N\_{\\text{R}}}}={\\frac {N!}{N\_{\\text{R}}!(N-N\_{\\text{R}})!}}} ![{\\displaystyle {\\binom {N}{N\_{\\text{R}}}}={\\frac {N!}{N\_{\\text{R}}!(N-N\_{\\text{R}})!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb98a3f680bae27365c32c7a87b1cf1ef660969f) and the total number of possible states is given by 2 N {\\textstyle 2^{N}} ![{\\textstyle 2^{N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef00a629421e96c78395ff90b9cf4d57b0176782). Therefore, the probability of the particle being hit from the right N R {\\textstyle N\_{\\text{R}}} ![{\\textstyle N\_{\\text{R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d52c69d029e1517130dc6283d10a4fb12b257f7) times is: P N ( N R ) \= N \! 2 N N R \! ( N − N R ) \! {\\displaystyle P\_{N}(N\_{\\text{R}})={\\frac {N!}{2^{N}N\_{\\text{R}}!(N-N\_{\\text{R}})!}}} ![{\\displaystyle P\_{N}(N\_{\\text{R}})={\\frac {N!}{2^{N}N\_{\\text{R}}!(N-N\_{\\text{R}})!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/711bd4ce858b9e6ba3a2f6c6bac3ca4de26d5de2) As a result of its simplicity, Smoluchowski's 1D model can only qualitatively describe Brownian motion. For a realistic particle undergoing Brownian motion in a fluid, many of the assumptions do not apply. For example, the assumption that on average there are an equal number of collisions from the right as from the left falls apart once the particle is in motion, as it will sweep into particles in its path and create a velocity-dependent drag force. Furthermore, there would be a continuous statistical distribution of different possible Δ V {\\textstyle \\Delta V} ![{\\textstyle \\Delta V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6098f6ae65a71ec75c77595a1ce6c6b844a81972)s governed by the Maxwell-Boltzmann distribution of fluid molecule velocities, rather than a single discrete value in a physical liquid or gas. ### Langevin equation \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=5 "Edit section: Langevin equation")\] Main article: [Langevin equation](https://en.wikipedia.org/wiki/Langevin_equation "Langevin equation") The [diffusion equation](https://en.wikipedia.org/wiki/Diffusion_equation "Diffusion equation") yields an approximation of the time evolution of the [probability density function](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") associated with the position of the particle going under a Brownian movement under the physical definition. The approximation becomes valid on timescales much larger than the timescale of individual atomic collisions, since it does not include a term to describe the acceleration of particles during collision. The time evolution of the position of the Brownian particle over all time scales described using the [Langevin equation](https://en.wikipedia.org/wiki/Langevin_equation "Langevin equation"), an equation that involves a random force field representing the effect of the [thermal fluctuations](https://en.wikipedia.org/wiki/Thermal_fluctuations "Thermal fluctuations") of the solvent on the particle.[\[17\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Velocity-2010-17) At longer times scales, where acceleration is negligible, individual particle dynamics can be approximated using [Brownian dynamics](https://en.wikipedia.org/wiki/Brownian_dynamics "Brownian dynamics") in place of [Langevin dynamics](https://en.wikipedia.org/wiki/Langevin_dynamics "Langevin dynamics"). ### Astrophysics: star motion within galaxies \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=6 "Edit section: Astrophysics: star motion within galaxies")\] In [stellar dynamics](https://en.wikipedia.org/wiki/Stellar_dynamics "Stellar dynamics"), a massive body (star, [black hole](https://en.wikipedia.org/wiki/Black_hole "Black hole"), etc.) can experience Brownian motion as it responds to [gravitational](https://en.wikipedia.org/wiki/Gravitational "Gravitational") forces from surrounding stars.[\[35\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Merritt-2013-35) The rms velocity V of the massive object, of mass M, is related to the rms velocity v ⋆ {\\displaystyle v\_{\\star }} ![{\\displaystyle v\_{\\star }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/913c9f2b0add99440a59ec96a769759b57ab05ed) of the background stars by M V 2 ≈ m v ⋆ 2 {\\displaystyle MV^{2}\\approx mv\_{\\star }^{2}} ![{\\displaystyle MV^{2}\\approx mv\_{\\star }^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc109205eeeaa8d536a127bf22ff3e716d6fe29e) where m â‰Ș M {\\displaystyle m\\ll M} ![{\\displaystyle m\\ll M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9913ddecf5981f7796c73b2b166d3f4e6fb9e39e) is the mass of the background stars. The gravitational force from the massive object causes nearby stars to move faster than they otherwise would, increasing both v ⋆ {\\displaystyle v\_{\\star }} ![{\\displaystyle v\_{\\star }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/913c9f2b0add99440a59ec96a769759b57ab05ed) and V.[\[35\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Merritt-2013-35) The Brownian velocity of [Sgr A\*](https://en.wikipedia.org/wiki/Sagittarius_A* "Sagittarius A*"), the [supermassive black hole](https://en.wikipedia.org/wiki/Supermassive_black_hole "Supermassive black hole") at the center of the [Milky Way galaxy](https://en.wikipedia.org/wiki/Milky_Way_galaxy "Milky Way galaxy"), is predicted from this formula to be less than 1 km s−1.[\[36\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Reid-2004-36) ## Mathematics \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=7 "Edit section: Mathematics")\] Main article: [Wiener process](https://en.wikipedia.org/wiki/Wiener_process "Wiener process") An animated example of a Brownian motion-like [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk") on a 2D surface with periodic boundary conditions. In the [scaling limit](https://en.wikipedia.org/wiki/Scaling_limit "Scaling limit"), random walk approaches the Wiener process according to [Donsker's theorem](https://en.wikipedia.org/wiki/Donsker%27s_theorem "Donsker's theorem"). In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), Brownian motion is described by the **Wiener process**, a continuous-time [stochastic process](https://en.wikipedia.org/wiki/Stochastic_process "Stochastic process") named in honor of [Norbert Wiener](https://en.wikipedia.org/wiki/Norbert_Wiener "Norbert Wiener"). It is one of the best known [LĂ©vy processes](https://en.wikipedia.org/wiki/L%C3%A9vy_process "LĂ©vy process") ([cĂ dlĂ g](https://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g "CĂ dlĂ g") stochastic processes with [stationary](https://en.wikipedia.org/wiki/Stationary_increments "Stationary increments") [independent increments](https://en.wikipedia.org/wiki/Independent_increments "Independent increments")) and occurs frequently in pure and applied mathematics, [economics](https://en.wikipedia.org/wiki/Economy "Economy") and [physics](https://en.wikipedia.org/wiki/Physics "Physics"). [![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wiener_process_3d.png/250px-Wiener_process_3d.png)](https://en.wikipedia.org/wiki/File:Wiener_process_3d.png) A single realisation of three-dimensional Brownian motion for times 0 ≀ *t* ≀ 2 The Wiener process *Wt* is characterized by four facts:[\[37\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-37) 1. *W*0 = 0 2. *Wt* is [almost surely](https://en.wikipedia.org/wiki/Almost_surely "Almost surely") continuous 3. *Wt* has independent increments 4. W t − W s ∌ N ( 0 , t − s ) {\\displaystyle W\_{t}-W\_{s}\\sim {\\mathcal {N}}(0,t-s)} ![{\\displaystyle W\_{t}-W\_{s}\\sim {\\mathcal {N}}(0,t-s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c502ae53a6726372cce17efdd787e8a791abf88) (for 0 ≀ s ≀ t {\\displaystyle 0\\leq s\\leq t} ![{\\displaystyle 0\\leq s\\leq t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efd3155df471ed05348da5307e4104e085f77d45) ). N ( ÎŒ , σ 2 ) {\\displaystyle {\\mathcal {N}}(\\mu ,\\sigma ^{2})} ![{\\displaystyle {\\mathcal {N}}(\\mu ,\\sigma ^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/863304aaa42a945f2f07d79facc3d2eebc845ce7) denotes the [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution") with [expected value](https://en.wikipedia.org/wiki/Expected_value "Expected value") ÎŒ and [variance](https://en.wikipedia.org/wiki/Variance "Variance") *σ*2. The condition that it has independent increments means that if 0 ≀ s 1 \< t 1 ≀ s 2 \< t 2 {\\displaystyle 0\\leq s\_{1}\<t\_{1}\\leq s\_{2}\<t\_{2}} ![{\\displaystyle 0\\leq s\_{1}\<t\_{1}\\leq s\_{2}\<t\_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/753cdf256564c50198f038805e98885e058e2119) then W t 1 − W s 1 {\\displaystyle W\_{t\_{1}}-W\_{s\_{1}}} ![{\\displaystyle W\_{t\_{1}}-W\_{s\_{1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23dcd43cde7eb8ce6ce229da07b72ec2f98d47cb) and W t 2 − W s 2 {\\displaystyle W\_{t\_{2}}-W\_{s\_{2}}} ![{\\displaystyle W\_{t\_{2}}-W\_{s\_{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5c045f44a9b78e02815793e35af984c270f729) are independent random variables. In addition, for some [filtration](https://en.wikipedia.org/wiki/Filtration_\(probability_theory\) "Filtration (probability theory)") F t {\\displaystyle {\\mathcal {F}}\_{t}} ![{\\displaystyle {\\mathcal {F}}\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5209aeab53016ae720999b2a50e7f0bc27cb33) , W t {\\displaystyle W\_{t}} ![{\\displaystyle W\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50680c5535c83badfa630dba63b583d2eeaa2977) is F t {\\displaystyle {\\mathcal {F}}\_{t}} ![{\\displaystyle {\\mathcal {F}}\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5209aeab53016ae720999b2a50e7f0bc27cb33) [measurable](https://en.wikipedia.org/wiki/Measurable "Measurable") for all t ≄ 0 {\\displaystyle t\\geq 0} ![{\\displaystyle t\\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/248525429e9cd266f53ab8c52d17bc206c546060) . An alternative characterisation of the Wiener process is the so-called *LĂ©vy characterisation* that says that the Wiener process is an almost surely continuous [martingale](https://en.wikipedia.org/wiki/Martingale_\(probability_theory\) "Martingale (probability theory)") with *W*0 = 0 and [quadratic variation](https://en.wikipedia.org/wiki/Quadratic_variation "Quadratic variation") \[ W t , W t \] \= t {\\displaystyle \[W\_{t},W\_{t}\]=t} ![{\\displaystyle \[W\_{t},W\_{t}\]=t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b53fb1bd6ab3cb0b4d2732924e5b654454b11171) . A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent N ( 0 , 1 ) {\\displaystyle {\\mathcal {N}}(0,1)} ![{\\displaystyle {\\mathcal {N}}(0,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3eeb356405c0b33b680b5caa425ada4e9f53e8b) random variables. This representation can be obtained using the [Kosambi–Karhunen–LoĂšve theorem](https://en.wikipedia.org/wiki/Kosambi%E2%80%93Karhunen%E2%80%93Lo%C3%A8ve_theorem "Kosambi–Karhunen–LoĂšve theorem"). The Wiener process can be constructed as the [scaling limit](https://en.wikipedia.org/wiki/Scaling_limit "Scaling limit") of a [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk"), or other discrete-time stochastic processes with stationary independent increments. This is known as [Donsker's theorem](https://en.wikipedia.org/wiki/Donsker%27s_theorem "Donsker's theorem"). Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed [neighborhood](https://en.wikipedia.org/wiki/Neighborhood_\(mathematics\) "Neighborhood (mathematics)") of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is [scale invariant](https://en.wikipedia.org/wiki/Scale_invariance "Scale invariance"). A d-dimensional [Gaussian free field](https://en.wikipedia.org/wiki/Gaussian_free_field "Gaussian free field") has been described as "a d-dimensional-time analog of Brownian motion."[\[38\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-38) ### Statistics \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=8 "Edit section: Statistics")\] The Brownian motion can be modeled by a [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk").[\[39\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-39) In the general case, Brownian motion is a [Markov process](https://en.wikipedia.org/wiki/Markov_process "Markov process") and described by [stochastic integral equations](https://en.wikipedia.org/wiki/Stochastic_calculus "Stochastic calculus").[\[40\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Morozov-2011-40) ### LĂ©vy characterisation \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=9 "Edit section: LĂ©vy characterisation")\] The French mathematician [Paul LĂ©vy](https://en.wikipedia.org/wiki/Paul_L%C3%A9vy_\(mathematician\) "Paul LĂ©vy (mathematician)") proved the following theorem, which gives a necessary and sufficient condition for a continuous **R***n*\-valued stochastic process *X* to actually be n\-dimensional Brownian motion. Hence, LĂ©vy's condition can actually be used as an alternative definition of Brownian motion. Let *X* = (*X*1, ..., *X**n*) be a continuous stochastic process on a [probability space](https://en.wikipedia.org/wiki/Probability_space "Probability space") (Ω, ÎŁ, **P**) taking values in **R***n*. Then the following are equivalent: 1. *X* is a Brownian motion with respect to **P**, i.e., the law of *X* with respect to **P** is the same as the law of an n\-dimensional Brownian motion, i.e., the [push-forward measure](https://en.wikipedia.org/wiki/Push-forward_measure "Push-forward measure") *X*∗(**P**) is [classical Wiener measure](https://en.wikipedia.org/wiki/Classical_Wiener_measure "Classical Wiener measure") on *C*0(\[0, ∞); **R***n*). 2. both 1. *X* is a [martingale](https://en.wikipedia.org/wiki/Martingale_\(probability_theory\) "Martingale (probability theory)") with respect to **P** (and its own [natural filtration](https://en.wikipedia.org/wiki/Natural_filtration "Natural filtration")); and 2. for all 1 ≀ *i*, *j* ≀ *n*, *X**i*(*t*) *X**j*(*t*) − *ÎŽ**ij* *t* is a martingale with respect to **P** (and its own [natural filtration](https://en.wikipedia.org/wiki/Natural_filtration "Natural filtration")), where *ÎŽ**ij* denotes the [Kronecker delta](https://en.wikipedia.org/wiki/Kronecker_delta "Kronecker delta"). ### Spectral content \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=10 "Edit section: Spectral content")\] The spectral content of a stochastic process X t {\\displaystyle X\_{t}} ![{\\displaystyle X\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82120d04dfb3cbadc4912951dd12b5568c9cd8f3) can be found from the [power spectral density](https://en.wikipedia.org/wiki/Power_spectral_density "Power spectral density"), formally defined as S ( ω ) \= lim T → ∞ 1 T E { \| ∫ 0 T e i ω t X t d t \| 2 } , {\\displaystyle S(\\omega )=\\lim \_{T\\to \\infty }{\\frac {1}{T}}\\mathbb {E} \\left\\{\\left\|\\int \_{0}^{T}e^{i\\omega t}X\_{t}dt\\right\|^{2}\\right\\},} ![{\\displaystyle S(\\omega )=\\lim \_{T\\to \\infty }{\\frac {1}{T}}\\mathbb {E} \\left\\{\\left\|\\int \_{0}^{T}e^{i\\omega t}X\_{t}dt\\right\|^{2}\\right\\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a23858585053a9d0db9a71bcafe5970d4e72cd0) where E {\\displaystyle \\mathbb {E} } ![{\\displaystyle \\mathbb {E} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad9faf1fd4a61d36d7f8a2f3204f3805a43c0d4a) stands for the [expected value](https://en.wikipedia.org/wiki/Expected_value "Expected value"). The power spectral density of Brownian motion is found to be[\[41\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-41) S B M ( ω ) \= 4 D ω 2 . {\\displaystyle S\_{BM}(\\omega )={\\frac {4D}{\\omega ^{2}}}.} ![{\\displaystyle S\_{BM}(\\omega )={\\frac {4D}{\\omega ^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6397d3243cdb5b94d63dd1e37b481f61ac8e172) where D is the [diffusion coefficient](https://en.wikipedia.org/wiki/Diffusion_coefficient "Diffusion coefficient") of *Xt*. For naturally occurring signals, the spectral content can be found from the power spectral density of a single realization, with finite available time, i.e., S ( 1 ) ( ω , T ) \= 1 T \| ∫ 0 T e i ω t X t d t \| 2 , {\\displaystyle S^{(1)}(\\omega ,T)={\\frac {1}{T}}\\left\|\\int \_{0}^{T}e^{i\\omega t}X\_{t}dt\\right\|^{2},} ![{\\displaystyle S^{(1)}(\\omega ,T)={\\frac {1}{T}}\\left\|\\int \_{0}^{T}e^{i\\omega t}X\_{t}dt\\right\|^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5353bd441f2a3f690560a9287276afea554172) which for an individual realization of a Brownian motion trajectory,[\[42\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Krapf-2018-42) it is found to have expected value ÎŒ B M ( ω , T ) {\\displaystyle \\mu \_{BM}(\\omega ,T)} ![{\\displaystyle \\mu \_{BM}(\\omega ,T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a192047176a2599314e03efc49fc4d99a91db73) ÎŒ BM ( ω , T ) \= 4 D ω 2 \[ 1 − sin ⁥ ( ω T ) ω T \] {\\displaystyle \\mu \_{\\text{BM}}(\\omega ,T)={\\frac {4D}{\\omega ^{2}}}\\left\[1-{\\frac {\\sin \\left(\\omega T\\right)}{\\omega T}}\\right\]} ![{\\displaystyle \\mu \_{\\text{BM}}(\\omega ,T)={\\frac {4D}{\\omega ^{2}}}\\left\[1-{\\frac {\\sin \\left(\\omega T\\right)}{\\omega T}}\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1620301f27770202e280584b0bf65d3cad88dc59) and [variance](https://en.wikipedia.org/wiki/Variance "Variance") σ BM 2 ( ω , T ) {\\displaystyle \\sigma \_{\\text{BM}}^{2}(\\omega ,T)} ![{\\displaystyle \\sigma \_{\\text{BM}}^{2}(\\omega ,T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f093f46c9333d664e26790ad65ce12242fec46d4)[\[42\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Krapf-2018-42) σ S 2 ( f , T ) \= E { ( S T ( j ) ( f ) ) 2 } − ÎŒ S 2 ( f , T ) \= 20 D 2 f 4 \[ 1 − ( 6 − cos ⁥ ( f T ) ) 2 sin ⁥ ( f T ) 5 f T \+ ( 17 − cos ⁥ ( 2 f T ) − 16 cos ⁥ ( f T ) ) 10 f 2 T 2 \] . {\\displaystyle \\sigma \_{S}^{2}(f,T)=\\mathbb {E} \\left\\{\\left(S\_{T}^{(j)}(f)\\right)^{2}\\right\\}-\\mu \_{S}^{2}(f,T)={\\frac {20D^{2}}{f^{4}}}\\left\[1-{\\Big (}6-\\cos \\left(fT\\right){\\Big )}{\\frac {2\\sin \\left(fT\\right)}{5fT}}+{\\frac {{\\Big (}17-\\cos \\left(2fT\\right)-16\\cos \\left(fT\\right){\\Big )}}{10f^{2}T^{2}}}\\right\].} ![{\\displaystyle \\sigma \_{S}^{2}(f,T)=\\mathbb {E} \\left\\{\\left(S\_{T}^{(j)}(f)\\right)^{2}\\right\\}-\\mu \_{S}^{2}(f,T)={\\frac {20D^{2}}{f^{4}}}\\left\[1-{\\Big (}6-\\cos \\left(fT\\right){\\Big )}{\\frac {2\\sin \\left(fT\\right)}{5fT}}+{\\frac {{\\Big (}17-\\cos \\left(2fT\\right)-16\\cos \\left(fT\\right){\\Big )}}{10f^{2}T^{2}}}\\right\].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8390eff4a72b6d313d0800fee1e33fcb978b784) For sufficiently long realization times, the expected value of the power spectrum of a single trajectory converges to the formally defined power spectral density S ( ω ) {\\displaystyle S(\\omega )} ![{\\displaystyle S(\\omega )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cd569c75ee7dad692b0f20a695130098e18c279) , but its coefficient of variation Îł \= σ / ÎŒ {\\displaystyle \\gamma =\\sigma /\\mu } ![{\\displaystyle \\gamma =\\sigma /\\mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/076b60902718422c76d5985bbffa9f9a4ef10b68) tends to 5 / 2 {\\displaystyle {\\sqrt {5}}/2} ![{\\displaystyle {\\sqrt {5}}/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97f3998e64a3d3ca91193be8e617198eb1a83346) . This implies the distribution of S ( 1 ) ( ω , T ) {\\displaystyle S^{(1)}(\\omega ,T)} ![{\\displaystyle S^{(1)}(\\omega ,T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe2f9e122d38c84f693175abcf1a850a4f2d593) is broad even in the infinite time limit. ### Riemannian manifolds \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=11 "Edit section: Riemannian manifolds")\] [![](https://upload.wikimedia.org/wikipedia/commons/4/4e/BMonSphere.jpg)](https://en.wikipedia.org/wiki/File:BMonSphere.jpg) Brownian motion on a sphere Brownian motion is usually considered to take place in [Euclidean space](https://en.wikipedia.org/wiki/Euclidean_space "Euclidean space"). It is natural to consider how such motion generalizes to more complex shapes, such as [surfaces](https://en.wikipedia.org/wiki/Surface "Surface") or higher dimensional [manifolds](https://en.wikipedia.org/wiki/Manifold "Manifold"). The formalization requires the space to possess some form of a [derivative](https://en.wikipedia.org/wiki/Derivative "Derivative"), as well as a [metric](https://en.wikipedia.org/wiki/Metric_space "Metric space"), so that a [Laplacian](https://en.wikipedia.org/wiki/Laplacian "Laplacian") can be defined. Both of these are available on [Riemannian manifolds](https://en.wikipedia.org/wiki/Riemannian_manifold "Riemannian manifold"). Riemannian manifolds have the property that [geodesics](https://en.wikipedia.org/wiki/Geodesic "Geodesic") can be described in [polar coordinates](https://en.wikipedia.org/wiki/Polar_coordinates "Polar coordinates"); that is, displacements are always in a radial direction, at some given angle. Uniform random motion is then described by Gaussians along the radial direction, independent of the angle, the same as in Euclidean space. The [infinitesimal generator](https://en.wikipedia.org/wiki/Infinitesimal_generator_\(stochastic_processes\) "Infinitesimal generator (stochastic processes)") (and hence [characteristic operator](https://en.wikipedia.org/wiki/Characteristic_operator "Characteristic operator")) of Brownian motion on Euclidean **R***n* is ⁠1/2⁠Δ, where Δ denotes the [Laplace operator](https://en.wikipedia.org/wiki/Laplace_operator "Laplace operator"). Brownian motion on an m\-dimensional [Riemannian manifold](https://en.wikipedia.org/wiki/Riemannian_manifold "Riemannian manifold") (*M*, *g*) can be defined as diffusion on M with the characteristic operator given by ⁠1/2⁠ΔLB, half the [Laplace–Beltrami operator](https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator "Laplace–Beltrami operator") ΔLB. One of the topics of study is a characterization of the [PoincarĂ© recurrence time](https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem "PoincarĂ© recurrence theorem") for such systems.[\[16\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Grigoryan-1999-16) ## Narrow escape \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=12 "Edit section: Narrow escape")\] The [narrow escape problem](https://en.wikipedia.org/wiki/Narrow_escape_problem "Narrow escape problem") is a ubiquitous problem in biology, biophysics and cellular biology which has the following formulation: a Brownian particle ([ion](https://en.wikipedia.org/wiki/Ion "Ion"), [molecule](https://en.wikipedia.org/wiki/Molecule "Molecule"), or [protein](https://en.wikipedia.org/wiki/Protein "Protein")) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is that of calculating the mean escape time. This time diverges as the window shrinks, thus rendering the calculation a [singular perturbation](https://en.wikipedia.org/wiki/Singular_perturbation "Singular perturbation") problem. ## See also \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=13 "Edit section: See also")\] - [Brownian bridge](https://en.wikipedia.org/wiki/Brownian_bridge "Brownian bridge") – Stochastic process in physics - [Brownian covariance](https://en.wikipedia.org/wiki/Brownian_covariance "Brownian covariance") – Statistical measurePages displaying short descriptions of redirect targets - [Brownian dynamics](https://en.wikipedia.org/wiki/Brownian_dynamics "Brownian dynamics") – Ideal molecular motion where no average acceleration takes place - [Brownian motor](https://en.wikipedia.org/wiki/Brownian_motor "Brownian motor") – Nanoscale machine - [Brownian noise](https://en.wikipedia.org/wiki/Brownian_noise "Brownian noise") – Type of noise produced by Brownian motion - [Brownian ratchet](https://en.wikipedia.org/wiki/Brownian_ratchet "Brownian ratchet") – Perpetual motion device - [Brownian surface](https://en.wikipedia.org/wiki/Brownian_surface "Brownian surface") - [Brownian tree](https://en.wikipedia.org/wiki/Brownian_tree "Brownian tree") – Concept in probability theory - [Brownian web](https://en.wikipedia.org/wiki/Brownian_web "Brownian web") - [Fractional Brownian motion](https://en.wikipedia.org/wiki/Fractional_Brownian_motion "Fractional Brownian motion") – Probability theory concept - [Geometric Brownian motion](https://en.wikipedia.org/wiki/Geometric_Brownian_motion "Geometric Brownian motion") – Continuous stochastic process - [ItĂŽ diffusion](https://en.wikipedia.org/wiki/It%C3%B4_diffusion "ItĂŽ diffusion") – Solution to a specific type of stochastic differential equation - [LĂ©vy arcsine law](https://en.wikipedia.org/wiki/L%C3%A9vy_arcsine_law "LĂ©vy arcsine law") – Collection of results for one-dimensional random walks and Brownian motionPages displaying short descriptions of redirect targets - [Marangoni effect](https://en.wikipedia.org/wiki/Marangoni_effect "Marangoni effect") – Physical phenomenon between two fluids - [Nanoparticle tracking analysis](https://en.wikipedia.org/wiki/Nanoparticle_tracking_analysis "Nanoparticle tracking analysis") – Method for visualizing and analyzing particles in liquids - [Reflected Brownian motion](https://en.wikipedia.org/wiki/Reflected_Brownian_motion "Reflected Brownian motion") – Wiener process with reflecting spatial boundaries - [Rotational Brownian motion](https://en.wikipedia.org/wiki/Rotational_Brownian_motion "Rotational Brownian motion") - [Schramm–Loewner evolution](https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution "Schramm–Loewner evolution") – Concept in probability theory - [Single particle tracking](https://en.wikipedia.org/wiki/Single_particle_tracking "Single particle tracking") - [Single particle trajectories](https://en.wikipedia.org/wiki/Single_particle_trajectories "Single particle trajectories") - [Surface diffusion](https://en.wikipedia.org/wiki/Surface_diffusion "Surface diffusion") – Physical Process - [Tyndall effect](https://en.wikipedia.org/wiki/Tyndall_effect "Tyndall effect") – Scattering of light by tiny particles in a colloidal suspension ## References \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=14 "Edit section: References")\] 1. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-1)** Meyburg, Jan Philipp; Diesing, Detlef (2017). "Teaching the Growth, Ripening, and Agglomeration of Nanostructures in Computer Experiments". *Journal of Chemical Education*. **94** (9): 1225–1231\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2017JChEd..94.1225M](https://ui.adsabs.harvard.edu/abs/2017JChEd..94.1225M). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1021/acs.jchemed.6b01008](https://doi.org/10.1021%2Facs.jchemed.6b01008). 2. ^ [***a***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Feynman-1964_2-0) [***b***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Feynman-1964_2-1) Feynman, Richard (1964). ["The Brownian Movement"](https://feynmanlectures.caltech.edu/I_41.html). *The Feynman Lectures of Physics, Volume I*. p. 41. 3. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-3)** Pathria, RK (1972). Statistical Mechanics. Pergamon Press. pp. 43–48, 73–74. 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Hardwicke. pp. 463–486\. 14. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-14)** Davis, Mark H. A.; Bachelier, Louis; Etheridge, Alison (2011). *Louis Bachelier's Theory of Speculation: The Origins of Modern Finance*. Princeton: Princeton University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4008-2930-9](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4008-2930-9 "Special:BookSources/978-1-4008-2930-9") . 15. ^ [***a***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Morters-2001_15-0) [***b***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Morters-2001_15-1) [***c***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Morters-2001_15-2) Mörters, Peter; Peres, Yuval (1 January 2001). [*Brownian Motion*](https://www.cambridge.org/core/product/identifier/9780511750489/type/book) (1 ed.). Cambridge University Press. 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[ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0273-0979](https://search.worldcat.org/issn/0273-0979). 17. ^ [***a***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Velocity-2010_17-0) [***b***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Velocity-2010_17-1) Li, Tongcang; Kheifets, Simon; Medellin, David; Raizen, Mark (2010). ["Measurement of the instantaneous velocity of a Brownian particle"](https://wayback.archive-it.org/all/20110331172407/http://chaos.utexas.edu/wp-uploads/2010/06/science.1189403v1.pdf) (PDF). *[Science](https://en.wikipedia.org/wiki/Science_\(journal\) "Science (journal)")*. **328** (5986): 1673–1675\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2010Sci...328.1673L](https://ui.adsabs.harvard.edu/abs/2010Sci...328.1673L). [CiteSeerX](https://en.wikipedia.org/wiki/CiteSeerX_\(identifier\) "CiteSeerX (identifier)") [10\.1.1.167.8245](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.167.8245). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1126/science.1189403](https://doi.org/10.1126%2Fscience.1189403). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [20488989](https://pubmed.ncbi.nlm.nih.gov/20488989). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [45828908](https://api.semanticscholar.org/CorpusID:45828908). Archived from [the original](http://chaos.utexas.edu/wp-uploads/2010/06/science.1189403v1.pdf) (PDF) on 31 March 2011. 18. ^ [***a***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Einstein-1956_18-0) [***b***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Einstein-1956_18-1) Einstein, Albert (1956) \[1926\]. [*Investigations on the Theory of the Brownian Movement*](http://users.physik.fu-berlin.de/~kleinert/files/eins_brownian.pdf) (PDF). Dover Publications. [Archived](https://ghostarchive.org/archive/20221009/http://users.physik.fu-berlin.de/~kleinert/files/eins_brownian.pdf) (PDF) from the original on 9 October 2022. Retrieved 25 December 2013. 19. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-19)** Stachel, J., ed. (1989). ["Einstein's Dissertation on the Determination of Molecular Dimensions"](http://www.csun.edu/~dchoudhary/Physics-Year_files/ed_diss.pdf) (PDF). *The Collected Papers of Albert Einstein, Volume 2*. Princeton University Press. [Archived](https://ghostarchive.org/archive/20221009/http://www.csun.edu/~dchoudhary/Physics-Year_files/ed_diss.pdf) (PDF) from the original on 9 October 2022. 20. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-20)** Kozdron, Michael J. (2008). ["Brownian Motion and the Heat Equation – chpt. 3 Albert Einstein's proof of the existence of Brownian motion"](https://web.archive.org/web/20241126184903/https://uregina.ca/~kozdron/Research/UgradTalks/BM_and_Heat/heat_and_BM.pdf) (PDF). *University of Regina*. Archived from [the original](https://uregina.ca/~kozdron/Research/UgradTalks/BM_and_Heat/heat_and_BM.pdf) (PDF) on 26 November 2024. Retrieved 3 November 2025. 21. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-21)** Lavenda, Bernard H. (1985). [*Nonequilibrium Statistical Thermodynamics*](https://archive.org/details/nonequilibriumst00lave). John Wiley & Sons. p. [20](https://archive.org/details/nonequilibriumst00lave/page/n29). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-471-90670-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-90670-4 "Special:BookSources/978-0-471-90670-4") . 22. ^ [***a***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Thomson-1904_22-0) [***b***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Thomson-1904_22-1) Thomson, J. J. (1904). [*Electricity and Matter*](https://archive.org/details/electricitymatte00thomuoft). Yale University Press. pp. [80](https://archive.org/details/electricitymatte00thomuoft/page/80)–83. 23. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-23)** Nernst, Walther (1888). "Zur Kinetik der in Lösung befindlichen Körper". *[Zeitschrift fĂŒr Physikalische Chemie](https://en.wikipedia.org/wiki/Zeitschrift_f%C3%BCr_Physikalische_Chemie "Zeitschrift fĂŒr Physikalische Chemie")* (in German). **9**: 613–637\. 24. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-24)** Leveugle, J. (2004). *La RelativitĂ©, PoincarĂ© et Einstein, Planck, Hilbert*. Harmattan. p. 181. 25. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-25)** Townsend, J.E.S. (1915). [*Electricity in Gases*](https://archive.org/details/electricityinga00towngoog). Clarendon Press. p. [254](https://archive.org/details/electricityinga00towngoog/page/n282). 26. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-26)** Maiocchi, Roberto (September 1990). ["The case of Brownian motion"](https://www.cambridge.org/core/journals/british-journal-for-the-history-of-science/article/abs/case-of-brownian-motion/7E6FCB8188956D072CC83581B5645099). *The British Journal for the History of Science*. **23** (3): 257–283\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1017/S0007087400043983](https://doi.org/10.1017%2FS0007087400043983). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1474-001X](https://search.worldcat.org/issn/1474-001X). 27. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-27)** Haw, M D (26 August 2002). ["Colloidal suspensions, Brownian motion, molecular reality: a short history"](https://iopscience.iop.org/article/10.1088/0953-8984/14/33/315). *Journal of Physics: Condensed Matter*. **14** (33): 7769–7779\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1088/0953-8984/14/33/315](https://doi.org/10.1088%2F0953-8984%2F14%2F33%2F315). 28. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-28)** See P. Clark 1976, p. 97 29. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-29)** Smoluchowski, M. M. (1906). ["Sur le chemin moyen parcouru par les molĂ©cules d'un gaz et sur son rapport avec la thĂ©orie de la diffusion"](https://archive.org/stream/bulletininternat1906pols#page/202/mode/2up) \[On the average path taken by gas molecules and its relation with the theory of diffusion\]. *[Bulletin International de l'AcadĂ©mie des Sciences de Cracovie](https://en.wikipedia.org/w/index.php?title=Bulletin_International_de_l%27Acad%C3%A9mie_des_Sciences_de_Cracovie&action=edit&redlink=1 "Bulletin International de l'AcadĂ©mie des Sciences de Cracovie (page does not exist)")* (in French): 202. 30. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-30)** Chandrasekhar, Subrahmanyan (1943). "Stochastic Problems in Physics and Astronomy". *Reviews of Modern Physics*. **15** (1): 1–89\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/RevModPhys.15.1](https://doi.org/10.1103%2FRevModPhys.15.1). 31. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-31)** See p. 535 in Sommerfeld, A. (1917). "Zum Andenken an Marian von Smoluchowski" \[In Memory of Marian von Smoluchowski\]. *[Physikalische Zeitschrift](https://en.wikipedia.org/wiki/Physikalische_Zeitschrift "Physikalische Zeitschrift")* (in German). **18** (22): 533–539\. 32. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-32)** Smoluchowski, M. M. (1906). ["Essai d'une thĂ©orie cinĂ©tique du mouvement Brownien et des milieux troubles"](https://archive.org/stream/bulletininternat1906pols#page/577/mode/2up) \[Test of a kinetic theory of Brownian motion and turbid media\]. *[Bulletin International de l'AcadĂ©mie des Sciences de Cracovie](https://en.wikipedia.org/w/index.php?title=Bulletin_International_de_l%27Acad%C3%A9mie_des_Sciences_de_Cracovie&action=edit&redlink=1 "Bulletin International de l'AcadĂ©mie des Sciences de Cracovie (page does not exist)")* (in French): 577. 33. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-33)** Feller, William (1968). *An Introduction to Probability Theory and Its Applications*. Vol. 1 (3rd ed.). John Wiley & Sons. pp. 73–75\. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0471257080](https://en.wikipedia.org/wiki/Special:BookSources/978-0471257080 "Special:BookSources/978-0471257080") . 34. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-34)** von Smoluchowski, M. (1906). ["Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen"](https://zenodo.org/record/1424073). *[Annalen der Physik](https://en.wikipedia.org/wiki/Annalen_der_Physik "Annalen der Physik")* (in German). **326** (14): 756–780\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1906AnP...326..756V](https://ui.adsabs.harvard.edu/abs/1906AnP...326..756V). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/andp.19063261405](https://doi.org/10.1002%2Fandp.19063261405). 35. ^ [***a***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Merritt-2013_35-0) [***b***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Merritt-2013_35-1) Merritt, David (2013). *Dynamics and Evolution of Galactic Nuclei*. Princeton University Press. p. 575. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4008-4612-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4008-4612-2 "Special:BookSources/978-1-4008-4612-2") . [OL](https://en.wikipedia.org/wiki/OL_\(identifier\) "OL (identifier)") [16802359W](https://openlibrary.org/works/OL16802359W). 36. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Reid-2004_36-0)** Reid, M. J.; Brunthaler, A. (2004). "The Proper Motion of Sagittarius A\*. II. The Mass of Sagittarius A\*". *[The Astrophysical Journal](https://en.wikipedia.org/wiki/The_Astrophysical_Journal "The Astrophysical Journal")*. **616** (2): 872–884\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[astro-ph/0408107](https://arxiv.org/abs/astro-ph/0408107). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2004ApJ...616..872R](https://ui.adsabs.harvard.edu/abs/2004ApJ...616..872R). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1086/424960](https://doi.org/10.1086%2F424960). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [16568545](https://api.semanticscholar.org/CorpusID:16568545). 37. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-37)** Bass, Richard F. (2011). [*Stochastic Processes*](https://www.cambridge.org/core/books/stochastic-processes/055A84B1EB586FE3032C0CA7D49598AC). Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1017/cbo9780511997044](https://doi.org/10.1017%2Fcbo9780511997044). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-107-00800-7](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-00800-7 "Special:BookSources/978-1-107-00800-7") . 38. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-38)** Sheffield, Scott (9 May 2007). ["Gaussian free fields for mathematicians"](https://link.springer.com/10.1007/s00440-006-0050-1). *Probability Theory and Related Fields*. **139** (3–4\): 521–541\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[math/0312099](https://arxiv.org/abs/math/0312099). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1007/s00440-006-0050-1](https://doi.org/10.1007%2Fs00440-006-0050-1). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0178-8051](https://search.worldcat.org/issn/0178-8051). 39. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-39)** Weiss, G. H. (1994). *Aspects and applications of the random walk*. North Holland. 40. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Morozov-2011_40-0)** Morozov, A. N.; Skripkin, A. V. (2011). "Spherical particle Brownian motion in viscous medium as non-Markovian random process". *[Physics Letters A](https://en.wikipedia.org/wiki/Physics_Letters_A "Physics Letters A")*. **375** (46): 4113–4115\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2011PhLA..375.4113M](https://ui.adsabs.harvard.edu/abs/2011PhLA..375.4113M). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.physleta.2011.10.001](https://doi.org/10.1016%2Fj.physleta.2011.10.001). 41. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-41)** Karczub, D. G.; Norton, M. P. (2003). *Fundamentals of Noise and Vibration Analysis for Engineers by M. P. Norton*. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1017/cbo9781139163927](https://doi.org/10.1017%2Fcbo9781139163927). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-139-16392-7](https://en.wikipedia.org/wiki/Special:BookSources/978-1-139-16392-7 "Special:BookSources/978-1-139-16392-7") . 42. ^ [***a***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Krapf-2018_42-0) [***b***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Krapf-2018_42-1) Krapf, Diego; Marinari, Enzo; Metzler, Ralf; Oshanin, Gleb; Xu, Xinran; Squarcini, Alessio (2018). ["Power spectral density of a single Brownian trajectory: what one can and cannot learn from it"](https://iopscience.iop.org/article/10.1088/1367-2630/aaa67c). *New Journal of Physics*. **20** (2): 023029. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1801\.02986](https://arxiv.org/abs/1801.02986). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2018NJPh...20b3029K](https://ui.adsabs.harvard.edu/abs/2018NJPh...20b3029K). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1088/1367-2630/aaa67c](https://doi.org/10.1088%2F1367-2630%2Faaa67c). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1367-2630](https://search.worldcat.org/issn/1367-2630). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [485685](https://api.semanticscholar.org/CorpusID:485685). ## Further reading \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=15 "Edit section: Further reading")\] - Brown, Robert (1828). ["A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies"](http://sciweb.nybg.org/science2/pdfs/dws/Brownian.pdf) (PDF). *[Philosophical Magazine](https://en.wikipedia.org/wiki/Philosophical_Magazine "Philosophical Magazine")*. **4** (21): 161–173\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1080/14786442808674769](https://doi.org/10.1080%2F14786442808674769). [Archived](https://ghostarchive.org/archive/20221009/http://sciweb.nybg.org/science2/pdfs/dws/Brownian.pdf) (PDF) from the original on 9 October 2022. Also includes a subsequent defense by Brown of his original observations, *Additional remarks on active molecules*. - Chaudesaigues, M. (1908). "Le mouvement brownien et la formule d'Einstein" \[Brownian motion and Einstein's formula\]. *[Comptes Rendus](https://en.wikipedia.org/wiki/Comptes_Rendus "Comptes Rendus")* (in French). **147**: 1044–6\. - Clark, P. (1976). "Atomism versus thermodynamics". In Howson, Colin (ed.). [*Method and appraisal in the physical sciences*](https://archive.org/details/methodappraisali0000unse). Cambridge University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-521-21110-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-21110-9 "Special:BookSources/978-0-521-21110-9") . - Cohen, Ruben D. (1986). ["Self Similarity in Brownian Motion and Other Ergodic Phenomena"](http://rdcohen.50megs.com/BrownianMotion.pdf) (PDF). *[Journal of Chemical Education](https://en.wikipedia.org/wiki/Journal_of_Chemical_Education "Journal of Chemical Education")*. **63** (11): 933–934\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1986JChEd..63..933C](https://ui.adsabs.harvard.edu/abs/1986JChEd..63..933C). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1021/ed063p933](https://doi.org/10.1021%2Fed063p933). [Archived](https://ghostarchive.org/archive/20221009/http://rdcohen.50megs.com/BrownianMotion.pdf) (PDF) from the original on 9 October 2022. - Dubins, Lester E.; Schwarz, Gideon (15 May 1965). ["On Continuous Martingales"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC301348). *[Proceedings of the National Academy of Sciences of the United States of America](https://en.wikipedia.org/wiki/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America "Proceedings of the National Academy of Sciences of the United States of America")*. **53** (3): 913–916\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1965PNAS...53..913D](https://ui.adsabs.harvard.edu/abs/1965PNAS...53..913D). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1073/pnas.53.5.913](https://doi.org/10.1073%2Fpnas.53.5.913). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [72837](https://www.jstor.org/stable/72837). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [301348](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC301348). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [16591279](https://pubmed.ncbi.nlm.nih.gov/16591279). - Einstein, A. (1956). [*Investigations on the Theory of Brownian Movement*](https://archive.org/details/investigationson00eins). New York: Dover. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-486-60304-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-60304-9 "Special:BookSources/978-0-486-60304-9") . Retrieved 6 January 2014. `{{cite book}}`: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date "Help:CS1 errors")) - Henri, V. (1908). "Études cinĂ©matographique du mouvement brownien" \[Cinematographic studies of Brownian motion\]. *[Comptes Rendus](https://en.wikipedia.org/wiki/Comptes_Rendus "Comptes Rendus")* (in French) (146): 1024–6\. - [Lucretius](https://en.wikipedia.org/wiki/Lucretius "Lucretius"), *On The Nature of Things*, translated by [William Ellery Leonard](https://en.wikipedia.org/wiki/William_Ellery_Leonard "William Ellery Leonard"). (*[on-line version](http://onlinebooks.library.upenn.edu/webbin/gutbook/lookup?num=785)*, from [Project Gutenberg](https://en.wikipedia.org/wiki/Project_Gutenberg "Project Gutenberg"). See the heading 'Atomic Motions'; this translation differs slightly from the one quoted). - [Nelson, Edward](https://en.wikipedia.org/wiki/Edward_Nelson "Edward Nelson"), (1967). *Dynamical Theories of Brownian Motion*. [(PDF version of this out-of-print book, from the author's webpage.)](https://web.math.princeton.edu/~nelson/books/bmotion.pdf) This is primarily a mathematical work, but the first four chapters discuss the history of the topic, in the era from Brown to Einstein. - Pearle, P.; Collett, B.; Bart, K.; Bilderback, D.; Newman, D.; Samuels, S. (2010). "What Brown saw and you can too". *[American Journal of Physics](https://en.wikipedia.org/wiki/American_Journal_of_Physics "American Journal of Physics")*. **78** (12): 1278–1289\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1008\.0039](https://arxiv.org/abs/1008.0039). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2010AmJPh..78.1278P](https://ui.adsabs.harvard.edu/abs/2010AmJPh..78.1278P). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1119/1.3475685](https://doi.org/10.1119%2F1.3475685). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [12342287](https://api.semanticscholar.org/CorpusID:12342287). - Perrin, J. (1909). "Mouvement brownien et rĂ©alitĂ© molĂ©culaire" \[Brownian movement and molecular reality\]. *[Annales de chimie et de physique](https://en.wikipedia.org/wiki/Annales_de_chimie_et_de_physique "Annales de chimie et de physique")*. 8th series. **18**: 5–114\. - See also Perrin's book "Les Atomes" (1914). - von Smoluchowski, M. (1906). ["Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen"](http://gallica.bnf.fr/ark:/12148/bpt6k15328k/f770.chemindefer). *[Annalen der Physik](https://en.wikipedia.org/wiki/Annalen_der_Physik "Annalen der Physik")*. **21** (14): 756–780\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1906AnP...326..756V](https://ui.adsabs.harvard.edu/abs/1906AnP...326..756V). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/andp.19063261405](https://doi.org/10.1002%2Fandp.19063261405). - Svedberg, T. (1907). *Studien zur Lehre von den kolloiden Losungen*. - [Theile, T. N](https://en.wikipedia.org/wiki/Thorvald_N._Thiele "Thorvald N. Thiele"). - Danish version: "Om Anvendelse af mindste Kvadraters Methode i nogle TilfĂŠlde, hvor en Komplikation af visse Slags uensartede tilfĂŠldige Fejlkilder giver Fejlene en 'systematisk' Karakter". - French version: "Sur la compensation de quelques erreurs quasi-systĂ©matiques par la mĂ©thodes de moindre carrĂ©s" published simultaneously in *Vidensk. Selsk. Skr. 5. Rk., naturvid. og mat. Afd.*, 12:381–408, 1880. ## External links \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=16 "Edit section: External links")\] [![Wikimedia Commons logo](https://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png)](https://en.wikipedia.org/wiki/File:Commons-logo.svg) Wikimedia Commons has media related to [Brownian motion](https://commons.wikimedia.org/wiki/Category:Brownian_motion "commons:Category:Brownian motion"). [![Wikisource logo](https://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/40px-Wikisource-logo.svg.png)](https://en.wikipedia.org/wiki/File:Wikisource-logo.svg) English [Wikisource](https://en.wikipedia.org/wiki/Wikisource "Wikisource") has original text related to this article: **[A brief account of microscopical observations made on the particles contained in the pollen of plants](https://en.wikisource.org/wiki/en:A_brief_account_of_microscopical_observations_made_on_the_particles_contained_in_the_pollen_of_plants "s:en:A brief account of microscopical observations made on the particles contained in the pollen of plants")** - [Einstein on Brownian Motion](https://web.archive.org/web/20010222031055/http://www.bun.kyoto-u.ac.jp/~suchii/einsteinBM.html) - [Discusses history, botany and physics of Brown's original observations, with videos](http://physerver.hamilton.edu/Research/Brownian/index.html) - ["Einstein's prediction finally witnessed one century later"](http://www.gizmag.com/einsteins-prediction-finally-witnessed/16212/) : a test to observe the velocity of Brownian motion - [Large-Scale Brownian Motion Demonstration](https://web.archive.org/web/20220331054344/https://demos.smu.ca/demos/thermo/90-brownian-motion) | [v](https://en.wikipedia.org/wiki/Template:Albert_Einstein "Template:Albert Einstein") [t](https://en.wikipedia.org/wiki/Template_talk:Albert_Einstein "Template talk:Albert Einstein") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Albert_Einstein "Special:EditPage/Template:Albert Einstein")[Albert Einstein](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein") | | |---|---| | Physics | [Theory of relativity](https://en.wikipedia.org/wiki/Theory_of_relativity "Theory of relativity") [Special relativity](https://en.wikipedia.org/wiki/Special_relativity "Special relativity") [General relativity](https://en.wikipedia.org/wiki/General_relativity "General relativity") [Mass–energy equivalence (E=mc2)](https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence "Mass–energy equivalence") [Brownian motion]() [Photoelectric effect](https://en.wikipedia.org/wiki/Photoelectric_effect "Photoelectric effect") [Einstein coefficients](https://en.wikipedia.org/wiki/Einstein_coefficients "Einstein coefficients") [Einstein solid](https://en.wikipedia.org/wiki/Einstein_solid "Einstein solid") [Equivalence principle](https://en.wikipedia.org/wiki/Equivalence_principle "Equivalence principle") [Einstein field equations](https://en.wikipedia.org/wiki/Einstein_field_equations "Einstein field equations") [Einstein radius](https://en.wikipedia.org/wiki/Einstein_radius "Einstein radius") [Einstein relation (kinetic theory)](https://en.wikipedia.org/wiki/Einstein_relation_\(kinetic_theory\) "Einstein relation (kinetic theory)") [Einstein ring](https://en.wikipedia.org/wiki/Einstein_ring "Einstein ring") [Cosmological constant](https://en.wikipedia.org/wiki/Cosmological_constant "Cosmological constant") [Bose–Einstein condensate](https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate "Bose–Einstein condensate") [Bose–Einstein statistics](https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics "Bose–Einstein statistics") [Bose–Einstein correlations](https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_correlations "Bose–Einstein correlations") [Einstein–Cartan theory](https://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan_theory "Einstein–Cartan theory") [Einstein–Infeld–Hoffmann equations](https://en.wikipedia.org/wiki/Einstein%E2%80%93Infeld%E2%80%93Hoffmann_equations "Einstein–Infeld–Hoffmann equations") [Einstein–de Haas effect](https://en.wikipedia.org/wiki/Einstein%E2%80%93de_Haas_effect "Einstein–de Haas effect") [EPR paradox](https://en.wikipedia.org/wiki/EPR_paradox "EPR paradox") [Bohr–Einstein debates](https://en.wikipedia.org/wiki/Bohr%E2%80%93Einstein_debates "Bohr–Einstein debates") [Teleparallelism](https://en.wikipedia.org/wiki/Teleparallelism "Teleparallelism") [Thought experiments](https://en.wikipedia.org/wiki/Einstein%27s_thought_experiments "Einstein's thought experiments") [Unsuccessful investigations](https://en.wikipedia.org/wiki/Einstein%27s_unsuccessful_investigations "Einstein's unsuccessful investigations") [Wave–particle duality](https://en.wikipedia.org/wiki/Wave%E2%80%93particle_duality "Wave–particle duality") [Gravitational wave](https://en.wikipedia.org/wiki/Gravitational_wave "Gravitational wave") [Tea leaf paradox](https://en.wikipedia.org/wiki/Tea_leaf_paradox "Tea leaf paradox") | | [Works](https://en.wikipedia.org/wiki/List_of_scientific_publications_by_Albert_Einstein "List of scientific publications by Albert Einstein") | [*Annus mirabilis* papers](https://en.wikipedia.org/wiki/Annus_mirabilis_papers "Annus mirabilis papers") (1905) "[Investigations on the Theory of Brownian Movement](https://en.wikipedia.org/wiki/%C3%9Cber_die_von_der_molekularkinetischen_Theorie_der_W%C3%A4rme_geforderte_Bewegung_von_in_ruhenden_Fl%C3%BCssigkeiten_suspendierten_Teilchen "Über die von der molekularkinetischen Theorie der WĂ€rme geforderte Bewegung von in ruhenden FlĂŒssigkeiten suspendierten Teilchen")" (1905) *[Relativity: The Special and the General Theory](https://en.wikipedia.org/wiki/Relativity:_The_Special_and_the_General_Theory "Relativity: The Special and the General Theory")* (1916) *[The Meaning of Relativity](https://en.wikipedia.org/wiki/The_Meaning_of_Relativity "The Meaning of Relativity")* (1922) *[The World as I See It](https://en.wikipedia.org/wiki/The_World_as_I_See_It_\(book\) "The World as I See It (book)")* (1934) *[The Evolution of Physics](https://en.wikipedia.org/wiki/The_Evolution_of_Physics "The Evolution of Physics")* (1938) "[Why Socialism?](https://en.wikipedia.org/wiki/Why_Socialism%3F "Why Socialism?")" (1949) [Russell–Einstein Manifesto](https://en.wikipedia.org/wiki/Russell%E2%80%93Einstein_Manifesto "Russell–Einstein Manifesto") (1955) | | [In popular culture](https://en.wikipedia.org/wiki/Albert_Einstein_in_popular_culture "Albert Einstein in popular culture") | *[Die Grundlagen der Einsteinschen RelativitĂ€ts-Theorie](https://en.wikipedia.org/wiki/Die_Grundlagen_der_Einsteinschen_Relativit%C3%A4ts-Theorie "Die Grundlagen der Einsteinschen RelativitĂ€ts-Theorie")* (1922 documentary) *[The Einstein Theory of Relativity](https://en.wikipedia.org/wiki/The_Einstein_Theory_of_Relativity "The Einstein Theory of Relativity")* (1923 documentary) *[Relics: Einstein's Brain](https://en.wikipedia.org/wiki/Relics:_Einstein%27s_Brain "Relics: Einstein's Brain")* (1994 documentary) *[Insignificance](https://en.wikipedia.org/wiki/Insignificance_\(film\) "Insignificance (film)")* (1985 film) *[Young Einstein](https://en.wikipedia.org/wiki/Young_Einstein "Young Einstein")* (1988 film) *[Picasso at the Lapin Agile](https://en.wikipedia.org/wiki/Picasso_at_the_Lapin_Agile "Picasso at the Lapin Agile")* (1993 play) *[I.Q.](https://en.wikipedia.org/wiki/I.Q._\(film\) "I.Q. (film)")* (1994 film) *[Einstein's Gift](https://en.wikipedia.org/wiki/Einstein%27s_Gift "Einstein's Gift")* (2003 play) *[Einstein and Eddington](https://en.wikipedia.org/wiki/Einstein_and_Eddington "Einstein and Eddington")* (2008 TV film) *[Genius](https://en.wikipedia.org/wiki/Genius_\(American_TV_series\) "Genius (American TV series)")* (2017 series) *[Oppenheimer](https://en.wikipedia.org/wiki/Oppenheimer_\(film\) "Oppenheimer (film)")* (2023 film) | | Prizes | [Albert Einstein Award](https://en.wikipedia.org/wiki/Albert_Einstein_Award "Albert Einstein Award") [Albert Einstein Medal](https://en.wikipedia.org/wiki/Albert_Einstein_Medal "Albert Einstein Medal") [Kalinga Prize](https://en.wikipedia.org/wiki/Kalinga_Prize "Kalinga Prize") [Albert Einstein Peace Prize](https://en.wikipedia.org/wiki/Albert_Einstein_Peace_Prize "Albert Einstein Peace Prize") [Albert Einstein World Award of Science](https://en.wikipedia.org/wiki/Albert_Einstein_World_Award_of_Science "Albert Einstein World Award of Science") [Einstein Prize for Laser Science](https://en.wikipedia.org/wiki/Einstein_Prize_for_Laser_Science "Einstein Prize for Laser Science") [Einstein Prize (APS)](https://en.wikipedia.org/wiki/Einstein_Prize_\(APS\) "Einstein Prize (APS)") | | Books about Einstein | *[Albert Einstein: Creator and Rebel](https://en.wikipedia.org/wiki/Albert_Einstein:_Creator_and_Rebel "Albert Einstein: Creator and Rebel")* *[Einstein and Religion](https://en.wikipedia.org/wiki/Einstein_and_Religion "Einstein and Religion")* *[Einstein for Beginners](https://en.wikipedia.org/wiki/Einstein_for_Beginners "Einstein for Beginners")* *[Einstein: His Life and Universe](https://en.wikipedia.org/wiki/Einstein:_His_Life_and_Universe "Einstein: His Life and Universe")* *[Einstein in Oxford](https://en.wikipedia.org/wiki/Einstein_in_Oxford "Einstein in Oxford")* *[Einstein on the Run](https://en.wikipedia.org/wiki/Einstein_on_the_Run "Einstein on the Run")* *[Einstein's Cosmos](https://en.wikipedia.org/wiki/Einstein%27s_Cosmos "Einstein's Cosmos")* *[I Am Albert Einstein](https://en.wikipedia.org/wiki/I_Am_Albert_Einstein "I Am Albert Einstein")* *[Introducing Relativity](https://en.wikipedia.org/wiki/Introducing_Relativity "Introducing Relativity")* *[Subtle is the Lord](https://en.wikipedia.org/wiki/Subtle_is_the_Lord "Subtle is the Lord")* | | [Family](https://en.wikipedia.org/wiki/Einstein_family "Einstein family") | [Mileva Marić](https://en.wikipedia.org/wiki/Mileva_Mari%C4%87 "Mileva Marić") (first wife) [Elsa Einstein](https://en.wikipedia.org/wiki/Elsa_Einstein "Elsa Einstein") (second wife; cousin) [Lieserl Einstein](https://en.wikipedia.org/wiki/Lieserl_Einstein "Lieserl Einstein") (daughter) [Hans Albert Einstein](https://en.wikipedia.org/wiki/Hans_Albert_Einstein "Hans Albert Einstein") (son) [Pauline Koch](https://en.wikipedia.org/wiki/Pauline_Koch "Pauline Koch") (mother) [Hermann Einstein](https://en.wikipedia.org/wiki/Hermann_Einstein "Hermann Einstein") (father) [Maja Einstein](https://en.wikipedia.org/wiki/Maja_Einstein "Maja Einstein") (sister) [Eduard Einstein](https://en.wikipedia.org/wiki/Einstein_family#Eduard_"Tete"_Einstein_\(Albert's_second_son\) "Einstein family") (son) [Robert Einstein](https://en.wikipedia.org/wiki/Murder_of_the_family_of_Robert_Einstein "Murder of the family of Robert Einstein") (cousin) [Bernhard Caesar Einstein](https://en.wikipedia.org/wiki/Bernhard_Caesar_Einstein "Bernhard Caesar Einstein") (grandson) [Evelyn Einstein](https://en.wikipedia.org/wiki/Evelyn_Einstein "Evelyn Einstein") (granddaughter) [Thomas Martin Einstein](https://en.wikipedia.org/wiki/Thomas_Martin_Einstein "Thomas Martin Einstein") (great-grandson) [Siegbert Einstein](https://en.wikipedia.org/wiki/Siegbert_Einstein "Siegbert Einstein") (distant cousin) | | Related | [Awards and honors](https://en.wikipedia.org/wiki/List_of_awards_and_honors_received_by_Albert_Einstein "List of awards and honors received by Albert Einstein") [Brain](https://en.wikipedia.org/wiki/Brain_of_Albert_Einstein "Brain of Albert Einstein") [House](https://en.wikipedia.org/wiki/Albert_Einstein_House "Albert Einstein House") [Memorial](https://en.wikipedia.org/wiki/Albert_Einstein_Memorial "Albert Einstein Memorial") [Political views](https://en.wikipedia.org/wiki/Political_views_of_Albert_Einstein "Political views of Albert Einstein") [Religious views](https://en.wikipedia.org/wiki/Religious_and_philosophical_views_of_Albert_Einstein "Religious and philosophical views of Albert Einstein") [Things named after](https://en.wikipedia.org/wiki/List_of_things_named_after_Albert_Einstein "List of things named after Albert Einstein") [Einstein–Oppenheimer relationship](https://en.wikipedia.org/wiki/Einstein%E2%80%93Oppenheimer_relationship "Einstein–Oppenheimer relationship") [Albert Einstein Archives](https://en.wikipedia.org/wiki/Albert_Einstein_Archives "Albert Einstein Archives") [Einstein's Blackboard](https://en.wikipedia.org/wiki/Einstein%27s_Blackboard "Einstein's Blackboard") [Einstein Papers Project](https://en.wikipedia.org/wiki/Einstein_Papers_Project "Einstein Papers Project") [Einstein refrigerator](https://en.wikipedia.org/wiki/Einstein_refrigerator "Einstein refrigerator") [Einsteinhaus](https://en.wikipedia.org/wiki/Einsteinhaus "Einsteinhaus") [Einsteinium](https://en.wikipedia.org/wiki/Einsteinium "Einsteinium") [Max Talmey](https://en.wikipedia.org/wiki/Max_Talmey "Max Talmey") [Emergency Committee of Atomic Scientists](https://en.wikipedia.org/wiki/Emergency_Committee_of_Atomic_Scientists "Emergency Committee of Atomic Scientists") | | ![](https://upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/20px-Symbol_category_class.svg.png) **[Category](https://en.wikipedia.org/wiki/Category:Albert_Einstein "Category:Albert Einstein")** | | | [v](https://en.wikipedia.org/wiki/Template:Fractals "Template:Fractals") [t](https://en.wikipedia.org/wiki/Template_talk:Fractals "Template talk:Fractals") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Fractals "Special:EditPage/Template:Fractals")[Fractals](https://en.wikipedia.org/wiki/Fractal "Fractal") | | |---|---| | Characteristics | [Fractal dimensions](https://en.wikipedia.org/wiki/Fractal_dimension "Fractal dimension") [Assouad](https://en.wikipedia.org/wiki/Assouad_dimension "Assouad dimension") [Box-counting](https://en.wikipedia.org/wiki/Minkowski%E2%80%93Bouligand_dimension "Minkowski–Bouligand dimension") [Higuchi](https://en.wikipedia.org/wiki/Higuchi_dimension "Higuchi dimension") [Correlation](https://en.wikipedia.org/wiki/Correlation_dimension "Correlation dimension") [Hausdorff](https://en.wikipedia.org/wiki/Hausdorff_dimension "Hausdorff dimension") [Packing](https://en.wikipedia.org/wiki/Packing_dimension "Packing dimension") [Topological](https://en.wikipedia.org/wiki/Lebesgue_covering_dimension "Lebesgue covering dimension") [Recursion](https://en.wikipedia.org/wiki/Recursion "Recursion") [Self-similarity](https://en.wikipedia.org/wiki/Self-similarity "Self-similarity") | | [Iterated function system](https://en.wikipedia.org/wiki/Iterated_function_system "Iterated function system") | [Barnsley fern](https://en.wikipedia.org/wiki/Barnsley_fern "Barnsley fern") [Cantor set](https://en.wikipedia.org/wiki/Cantor_set "Cantor set") [Koch snowflake](https://en.wikipedia.org/wiki/Koch_snowflake "Koch snowflake") [Menger sponge](https://en.wikipedia.org/wiki/Menger_sponge "Menger sponge") [SierpiƄski carpet](https://en.wikipedia.org/wiki/Sierpi%C5%84ski_carpet "SierpiƄski carpet") [SierpiƄski triangle](https://en.wikipedia.org/wiki/Sierpi%C5%84ski_triangle "SierpiƄski triangle") [Apollonian gasket](https://en.wikipedia.org/wiki/Apollonian_gasket "Apollonian gasket") [Fibonacci word](https://en.wikipedia.org/wiki/Fibonacci_word_fractal "Fibonacci word fractal") [Space-filling curve](https://en.wikipedia.org/wiki/Space-filling_curve "Space-filling curve") [Blancmange curve](https://en.wikipedia.org/wiki/Blancmange_curve "Blancmange curve") [De Rham curve](https://en.wikipedia.org/wiki/De_Rham_curve "De Rham curve") [Minkowski](https://en.wikipedia.org/wiki/Minkowski_sausage "Minkowski sausage") [Dragon curve](https://en.wikipedia.org/wiki/Dragon_curve "Dragon curve") [Hilbert curve](https://en.wikipedia.org/wiki/Hilbert_curve "Hilbert curve") [Koch curve](https://en.wikipedia.org/wiki/Koch_snowflake "Koch snowflake") [LĂ©vy C curve](https://en.wikipedia.org/wiki/L%C3%A9vy_C_curve "LĂ©vy C curve") [Moore curve](https://en.wikipedia.org/wiki/Moore_curve "Moore curve") [Peano curve](https://en.wikipedia.org/wiki/Peano_curve "Peano curve") [SierpiƄski curve](https://en.wikipedia.org/wiki/Sierpi%C5%84ski_curve "SierpiƄski curve") [Z-order curve](https://en.wikipedia.org/wiki/Z-order_curve "Z-order curve") [String](https://en.wikipedia.org/wiki/Fractal_string "Fractal string") [T-square](https://en.wikipedia.org/wiki/T-square_\(fractal\) "T-square (fractal)") [n-flake](https://en.wikipedia.org/wiki/N-flake "N-flake") [Vicsek fractal](https://en.wikipedia.org/wiki/Vicsek_fractal "Vicsek fractal") [Gosper curve](https://en.wikipedia.org/wiki/Gosper_curve "Gosper curve") [Pythagoras tree](https://en.wikipedia.org/wiki/Pythagoras_tree_\(fractal\) "Pythagoras tree (fractal)") [Weierstrass function](https://en.wikipedia.org/wiki/Weierstrass_function "Weierstrass function") | | [Strange attractor](https://en.wikipedia.org/wiki/Attractor#Strange_attractor "Attractor") | [Multifractal system](https://en.wikipedia.org/wiki/Multifractal_system "Multifractal system") | | [L-system](https://en.wikipedia.org/wiki/L-system "L-system") | [Fractal canopy](https://en.wikipedia.org/wiki/Fractal_canopy "Fractal canopy") [Space-filling curve](https://en.wikipedia.org/wiki/Space-filling_curve "Space-filling curve") [H tree](https://en.wikipedia.org/wiki/H_tree "H tree") | | [Escape-time fractals](https://en.wikipedia.org/wiki/Fractal#Common_techniques_for_generating_fractals "Fractal") | [Burning Ship fractal](https://en.wikipedia.org/wiki/Burning_Ship_fractal "Burning Ship fractal") [Julia set](https://en.wikipedia.org/wiki/Julia_set "Julia set") [Filled](https://en.wikipedia.org/wiki/Filled_Julia_set "Filled Julia set") [Newton fractal](https://en.wikipedia.org/wiki/Newton_fractal "Newton fractal") [Douady rabbit](https://en.wikipedia.org/wiki/Douady_rabbit "Douady rabbit") [Lyapunov fractal](https://en.wikipedia.org/wiki/Lyapunov_fractal "Lyapunov fractal") [Mandelbrot set](https://en.wikipedia.org/wiki/Mandelbrot_set "Mandelbrot set") [Misiurewicz point](https://en.wikipedia.org/wiki/Misiurewicz_point "Misiurewicz point") [Multibrot set](https://en.wikipedia.org/wiki/Multibrot_set "Multibrot set") [Newton fractal](https://en.wikipedia.org/wiki/Newton_fractal "Newton fractal") [Tricorn](https://en.wikipedia.org/wiki/Tricorn_\(mathematics\) "Tricorn (mathematics)") [Mandelbox](https://en.wikipedia.org/wiki/Mandelbox "Mandelbox") [Mandelbulb](https://en.wikipedia.org/wiki/Mandelbulb "Mandelbulb") | | [Rendering](https://en.wikipedia.org/wiki/Rendering_\(computer_graphics\) "Rendering (computer graphics)") techniques | [Buddhabrot](https://en.wikipedia.org/wiki/Buddhabrot "Buddhabrot") [Orbit trap](https://en.wikipedia.org/wiki/Orbit_trap "Orbit trap") [Pickover stalk](https://en.wikipedia.org/wiki/Pickover_stalk "Pickover stalk") | | [Random](https://en.wikipedia.org/wiki/Chaos_game "Chaos game") fractals | [Brownian motion]() [Brownian tree](https://en.wikipedia.org/wiki/Diffusion-limited_aggregation "Diffusion-limited aggregation") [Brownian motor](https://en.wikipedia.org/wiki/Brownian_motor "Brownian motor") [Fractal landscape](https://en.wikipedia.org/wiki/Fractal_landscape "Fractal landscape") [LĂ©vy flight](https://en.wikipedia.org/wiki/L%C3%A9vy_flight "LĂ©vy flight") [Percolation theory](https://en.wikipedia.org/wiki/Percolation_theory "Percolation theory") [Self-avoiding walk](https://en.wikipedia.org/wiki/Self-avoiding_walk "Self-avoiding walk") | | People | [Michael Barnsley](https://en.wikipedia.org/wiki/Michael_Barnsley "Michael Barnsley") [Georg Cantor](https://en.wikipedia.org/wiki/Georg_Cantor "Georg Cantor") [Bill Gosper](https://en.wikipedia.org/wiki/Bill_Gosper "Bill Gosper") [Felix Hausdorff](https://en.wikipedia.org/wiki/Felix_Hausdorff "Felix Hausdorff") [Desmond Paul Henry](https://en.wikipedia.org/wiki/Desmond_Paul_Henry "Desmond Paul Henry") [Gaston Julia](https://en.wikipedia.org/wiki/Gaston_Julia "Gaston Julia") [Niels Fabian Helge von Koch](https://en.wikipedia.org/wiki/Niels_Fabian_Helge_von_Koch "Niels Fabian Helge von Koch") [Paul LĂ©vy](https://en.wikipedia.org/wiki/Paul_L%C3%A9vy_\(mathematician\) "Paul LĂ©vy (mathematician)") [Aleksandr Lyapunov](https://en.wikipedia.org/wiki/Aleksandr_Lyapunov "Aleksandr Lyapunov") [Benoit Mandelbrot](https://en.wikipedia.org/wiki/Benoit_Mandelbrot "Benoit Mandelbrot") [Hamid Naderi Yeganeh](https://en.wikipedia.org/wiki/Hamid_Naderi_Yeganeh "Hamid Naderi Yeganeh") [Lewis Fry Richardson](https://en.wikipedia.org/wiki/Lewis_Fry_Richardson "Lewis Fry Richardson") [WacƂaw SierpiƄski](https://en.wikipedia.org/wiki/Wac%C5%82aw_Sierpi%C5%84ski "WacƂaw SierpiƄski") | | Other | [Coastline paradox](https://en.wikipedia.org/wiki/Coastline_paradox "Coastline paradox") [Fractal art](https://en.wikipedia.org/wiki/Fractal_art "Fractal art") [List of fractals by Hausdorff dimension](https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension "List of fractals by Hausdorff dimension") *[The Fractal Geometry of Nature](https://en.wikipedia.org/wiki/The_Fractal_Geometry_of_Nature "The Fractal Geometry of Nature")* (1982 book) *[The Beauty of Fractals](https://en.wikipedia.org/wiki/The_Beauty_of_Fractals "The Beauty of Fractals")* (1986 book) *[Chaos: Making a New Science](https://en.wikipedia.org/wiki/Chaos:_Making_a_New_Science "Chaos: Making a New Science")* (1987 book) [Kaleidoscope](https://en.wikipedia.org/wiki/Kaleidoscope "Kaleidoscope") [Chaos theory](https://en.wikipedia.org/wiki/Chaos_theory "Chaos theory") | | [Authority control databases](https://en.wikipedia.org/wiki/Help:Authority_control "Help:Authority control") [![Edit this at Wikidata](https://upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png)](https://www.wikidata.org/wiki/Q178036#identifiers "Edit this at Wikidata") | | |---|---| | International | [GND](https://d-nb.info/gnd/4128328-4) | | National | [United States](https://id.loc.gov/authorities/sh85017266) [France](https://catalogue.bnf.fr/ark:/12148/cb11979550d) [BnF data](https://data.bnf.fr/ark:/12148/cb11979550d) [Japan](https://id.ndl.go.jp/auth/ndlna/00560924) [Czech Republic](https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph195850&CON_LNG=ENG) [Israel](https://www.nli.org.il/en/authorities/987007292433205171) | | Other | [Yale LUX](https://lux.collections.yale.edu/view/concept/c48a3542-8f4b-499d-a667-1e1290cb7d0e) | ![](https://en.wikipedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1) Retrieved from "<https://en.wikipedia.org/w/index.php?title=Brownian_motion&oldid=1342051899>" [Categories](https://en.wikipedia.org/wiki/Help:Category "Help:Category"): - [Statistical mechanics](https://en.wikipedia.org/wiki/Category:Statistical_mechanics "Category:Statistical mechanics") - [Wiener process](https://en.wikipedia.org/wiki/Category:Wiener_process "Category:Wiener process") - [Fractals](https://en.wikipedia.org/wiki/Category:Fractals "Category:Fractals") - [Colloidal chemistry](https://en.wikipedia.org/wiki/Category:Colloidal_chemistry "Category:Colloidal chemistry") - [Robert Brown (botanist, born 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[![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/97/2d_random_walk_ag_adatom_ag111.gif/330px-2d_random_walk_ag_adatom_ag111.gif)](https://en.wikipedia.org/wiki/File:2d_random_walk_ag_adatom_ag111.gif) 2-dimensional random walk of a silver [adatom](https://en.wikipedia.org/wiki/Adatom "Adatom") on an Ag(111) surface[\[1\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-1) [![](https://upload.wikimedia.org/wikipedia/commons/c/c2/Brownian_motion_large.gif)](https://en.wikipedia.org/wiki/File:Brownian_motion_large.gif) [Simulation](https://en.wikipedia.org/wiki/Brownian_dynamics "Brownian dynamics") of the Brownian motion of a large particle, analogous to a dust particle, that collides with a large set of smaller particles, analogous to molecules of a gas, which move with different velocities in different random directions **Brownian motion** is the random motion of [particles](https://en.wikipedia.org/wiki/Particle "Particle") suspended in a medium (a [liquid](https://en.wikipedia.org/wiki/Liquid "Liquid") or a [gas](https://en.wikipedia.org/wiki/Gas "Gas")).[\[2\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Feynman-1964-2) The traditional mathematical formulation of Brownian motion is that of the [Wiener process](https://en.wikipedia.org/wiki/Wiener_process "Wiener process"), which is often called Brownian motion, even in mathematical sources. This motion pattern typically consists of [random](https://en.wikipedia.org/wiki/Randomness "Randomness") fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at [thermal equilibrium](https://en.wikipedia.org/wiki/Thermal_equilibrium "Thermal equilibrium"), defined by a given [temperature](https://en.wikipedia.org/wiki/Temperature "Temperature"). Within such a fluid, there exists no preferential direction of flow (as in [transport phenomena](https://en.wikipedia.org/wiki/Transport_phenomena "Transport phenomena")). More specifically, the fluid's overall [linear](https://en.wikipedia.org/wiki/Linear_momentum "Linear momentum") and [angular](https://en.wikipedia.org/wiki/Angular_momentum "Angular momentum") momenta remain null over time. The [kinetic energies](https://en.wikipedia.org/wiki/Kinetic_energy "Kinetic energy") of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's [internal energy](https://en.wikipedia.org/wiki/Internal_energy "Internal energy") (the [equipartition theorem](https://en.wikipedia.org/wiki/Equipartition_theorem "Equipartition theorem")).[\[3\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-3) This motion is named after the Scottish botanist [Robert Brown](https://en.wikipedia.org/wiki/Robert_Brown_\(botanist,_born_1773\) "Robert Brown (botanist, born 1773)"), who first described the phenomenon in 1827, while looking through a microscope at [pollen](https://en.wikipedia.org/wiki/Pollen "Pollen") of the plant *[Clarkia pulchella](https://en.wikipedia.org/wiki/Clarkia_pulchella "Clarkia pulchella")* immersed in water. In 1900, the French mathematician [Louis Bachelier](https://en.wikipedia.org/wiki/Louis_Bachelier "Louis Bachelier") modeled the stochastic process now called Brownian motion in his doctoral thesis, The Theory of Speculation (ThĂ©orie de la spĂ©culation), prepared under the supervision of [Henri PoincarĂ©](https://en.wikipedia.org/wiki/Henri_Poincar%C3%A9 "Henri PoincarĂ©"). Then, in 1905, theoretical physicist [Albert Einstein](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein") published [a paper](https://en.wikipedia.org/wiki/%C3%9Cber_die_von_der_molekularkinetischen_Theorie_der_W%C3%A4rme_geforderte_Bewegung_von_in_ruhenden_Fl%C3%BCssigkeiten_suspendierten_Teilchen "Über die von der molekularkinetischen Theorie der WĂ€rme geforderte Bewegung von in ruhenden FlĂŒssigkeiten suspendierten Teilchen") in which he modelled the motion of the pollen particles as being moved by individual water [molecules](https://en.wikipedia.org/wiki/Molecule "Molecule"), making one of his first major scientific contributions.[\[4\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Einstein-1905-4) The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. This explanation of Brownian motion served as convincing evidence that [atoms](https://en.wikipedia.org/wiki/Atom "Atom") and molecules exist and was further verified experimentally by [Jean Perrin](https://en.wikipedia.org/wiki/Jean_Baptiste_Perrin "Jean Baptiste Perrin") in 1908. Perrin was awarded the [Nobel Prize in Physics](https://en.wikipedia.org/wiki/Nobel_Prize_in_Physics "Nobel Prize in Physics") in 1926 "for his work on the discontinuous structure of matter".[\[5\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-5) The [many-body interactions](https://en.wikipedia.org/wiki/Many-body_problem "Many-body problem") that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. Consequently, only probabilistic models applied to [molecular populations](https://en.wikipedia.org/wiki/Statistical_ensemble "Statistical ensemble") can be employed to describe it.[\[6\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-6) Two such models of the [statistical mechanics](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics"), due to Einstein and Smoluchowski, are presented below. Another, pure probabilistic class of models is the class of the [stochastic process](https://en.wikipedia.org/wiki/Stochastic_process "Stochastic process") models. There exist sequences of both simpler and more complicated stochastic processes which converge (in the [limit](https://en.wikipedia.org/wiki/Limit_of_a_function "Limit of a function")) to Brownian motion (see [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk") and [Donsker's theorem](https://en.wikipedia.org/wiki/Donsker%27s_theorem "Donsker's theorem")).[\[7\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-7)[\[8\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-8) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/4/44/PerrinPlot2.svg/250px-PerrinPlot2.svg.png)](https://en.wikipedia.org/wiki/File:PerrinPlot2.svg) Reproduced from the [Jean Baptiste Perrin](https://en.wikipedia.org/wiki/Jean_Baptiste_Perrin "Jean Baptiste Perrin") book *Les Atomes*: tracings of the motions of three colloidal particles of radius 0.53 ÎŒm, as seen under the microscope, with each point representing that particle's successive position every 30 seconds; the points are then joined by straight line segments (mesh size = 3.2 ÎŒm)[\[9\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-9) The Roman philosopher-poet [Lucretius](https://en.wikipedia.org/wiki/Lucretius "Lucretius")' scientific poem *[On the Nature of Things](https://en.wikipedia.org/wiki/On_the_Nature_of_Things "On the Nature of Things")* (c. 60 BC) has a remarkable description of the motion of [dust](https://en.wikipedia.org/wiki/Dust "Dust") particles in verses 113–140 from Book II. He uses this as a proof of the existence of atoms: > Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves \[i.e., spontaneously\]. Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible.[\[10\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-10) Although the mingling, tumbling motion of dust particles is caused largely by macroscopic air currents and convection, the glittering, microscopic jiggling motion of small dust particles is caused chiefly by true [Brownian dynamics](https://en.wikipedia.org/wiki/Brownian_dynamics "Brownian dynamics"); Lucretius "perfectly describes and explains the Brownian movement by a wrong example".[\[11\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-11) The formal scientific discovery of this phenomenon is credited to the botanist [Robert Brown](https://en.wikipedia.org/wiki/Robert_Brown_\(botanist,_born_1773\) "Robert Brown (botanist, born 1773)") in 1827. Brown was studying plant reproduction when he observed [pollen](https://en.wikipedia.org/wiki/Pollen "Pollen") grains of the plant *[Clarkia pulchella](https://en.wikipedia.org/wiki/Clarkia_pulchella "Clarkia pulchella")* in water under a simple microscope. These grains contain minute particles on the order of 1/4,000th of an inch (6.4 microns) in size. He observed these particles executing a continuous, jittery motion. By repeating the experiment with particles of inorganic matter, such as glass and rock dust, he was able to rule out that the motion was life-related, although its physical origin was yet to be explained.[\[12\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Brush-1968-12)[\[13\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-13) The mathematics of much of stochastic analysis, including the mathematics of Brownian motion, was introduced by [Louis Bachelier](https://en.wikipedia.org/wiki/Louis_Bachelier "Louis Bachelier") in 1900 in his PhD thesis "The theory of speculation", in which he presented an innovative probabilistic analysis of the stock and option markets. However, this pioneering mathematical work connecting random walks to continuous time was largely unknown until the 1950s.[\[14\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-14)[\[15\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Morters-2001-15): 33 The early 20th century saw the theoretical formalization of Brownian motion bridging the gap between thermodynamics and atomic theory: - [Albert Einstein](https://en.wikipedia.org/wiki/Albert_Einstein "Albert Einstein") (in one of his [1905 papers](https://en.wikipedia.org/wiki/%C3%9Cber_die_von_der_molekularkinetischen_Theorie_der_W%C3%A4rme_geforderte_Bewegung_von_in_ruhenden_Fl%C3%BCssigkeiten_suspendierten_Teilchen "Über die von der molekularkinetischen Theorie der WĂ€rme geforderte Bewegung von in ruhenden FlĂŒssigkeiten suspendierten Teilchen")) provided an explanation of Brownian motion in terms of atoms and molecules at a time when their physical existence was still fiercely debated by scientists. Einstein proved the mathematical relation between the probability distribution of a Brownian particle and the macroscopic [diffusion equation](https://en.wikipedia.org/wiki/Diffusion_equation "Diffusion equation").[\[15\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Morters-2001-15): 33 - These predictive equations describing Brownian motion were subsequently verified by the meticulous experimental work of [Jean Baptiste Perrin](https://en.wikipedia.org/wiki/Jean_Baptiste_Perrin "Jean Baptiste Perrin") in 1908, leading to his Nobel prize and settling the atomic debate.[\[16\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Grigoryan-1999-16) - [Norbert Wiener](https://en.wikipedia.org/wiki/Norbert_Wiener "Norbert Wiener") gave the first complete and rigorous mathematical analysis of the phenomenon in 1923, leading to the underlying mathematical concept being permanently called a [Wiener process](https://en.wikipedia.org/wiki/Wiener_process "Wiener process").[\[15\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Morters-2001-15) The instantaneous velocity of the Brownian motion can be defined as *v* = Δ*x*/Δ*t*, when Δ*t* \<\< *τ*, where τ is the momentum relaxation time. Advancements in modern physics have allowed this to be directly measured: - In 2010, the instantaneous velocity of a single Brownian particle (a glass microsphere trapped in air with [optical tweezers](https://en.wikipedia.org/wiki/Optical_tweezers "Optical tweezers")) was measured successfully for the first time. - The velocity data perfectly verified the [Maxwell–Boltzmann velocity distribution](https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution "Maxwell–Boltzmann distribution") and confirmed the equipartition theorem for a Brownian particle at microscopic timescales.[\[17\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Velocity-2010-17) ## Statistical mechanics theories \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=2 "Edit section: Statistical mechanics theories")\] There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian particles, in which the diffusion coefficient is related to the [mean squared displacement](https://en.wikipedia.org/wiki/Mean_squared_displacement "Mean squared displacement") of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities.[\[18\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Einstein-1956-18) In this way Einstein was able to determine the size of atoms, and how many atoms there are in a mole, or the [molecular weight](https://en.wikipedia.org/wiki/Molecular_weight "Molecular weight") in grams, of a gas.[\[19\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-19) In accordance to [Avogadro's law](https://en.wikipedia.org/wiki/Avogadro%27s_law "Avogadro's law"), this volume is the same for all ideal gases, namely 22.414 liters at standard temperature and pressure. The number of atoms contained in this volume is referred to as the [Avogadro constant](https://en.wikipedia.org/wiki/Avogadro_constant "Avogadro constant") or as [Avogadro's number](https://en.wikipedia.org/wiki/Avogadro%27s_number "Avogadro's number") (approximately 6\.02×1023 mol−1), and the determination of this number is tantamount to the knowledge of the mass of an atom, since the latter is obtained by dividing the [molar mass](https://en.wikipedia.org/wiki/Molar_mass "Molar mass") of the gas by the [Avogadro constant](https://en.wikipedia.org/wiki/Avogadro_constant "Avogadro constant"). [![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Diffusion_of_Brownian_particles.svg/330px-Diffusion_of_Brownian_particles.svg.png)](https://en.wikipedia.org/wiki/File:Diffusion_of_Brownian_particles.svg) The characteristic bell-shaped curves of the diffusion of Brownian particles. The distribution begins as a [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function "Dirac delta function"), indicating that all the particles are located at the origin at time *t* = 0. As *t* increases, the distribution flattens (though remains bell-shaped), and ultimately becomes uniform in the limit that time goes to infinity. The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval.[\[4\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Einstein-1905-4) Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 1014 collisions per second.[\[2\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Feynman-1964-2) He regarded the increment of particle positions in time ![{\\displaystyle \\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c) in a one-dimensional (*x*) space (with the coordinates chosen so that the origin lies at the initial position of the particle) as a [random variable](https://en.wikipedia.org/wiki/Random_variable "Random variable") (![{\\displaystyle q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d)) with some [probability density function](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") ![{\\displaystyle \\varphi (q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c84e62844cff8763f4f35a5ac00be1d4ccb691f3) (i.e., ![{\\displaystyle \\varphi (q)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c84e62844cff8763f4f35a5ac00be1d4ccb691f3) is the probability density for a jump of magnitude ![{\\displaystyle q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d), i.e., the probability density of the particle incrementing its position from ![{\\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4) to ![{\\displaystyle x+q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb88adc883f9d0bd380d68976d6db693aa2a938e) in the time interval ![{\\displaystyle \\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c)). Further, assuming conservation of particle number, he expanded the [number density](https://en.wikipedia.org/wiki/Number_density "Number density") ![{\\displaystyle \\rho (x,t+\\tau )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/095bf2247c50785be4cf836d85367a73e163ce55) (number of particles per unit volume around ![{\\displaystyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)) at time ![{\\displaystyle t+\\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2517ff18a2091c903aa8d144e7e60535e4f673b) in a [Taylor series](https://en.wikipedia.org/wiki/Taylor_series "Taylor series"), ![{\\displaystyle {\\begin{aligned}\\rho (x,t+\\tau )={}&\\rho (x,t)+\\tau {\\frac {\\partial \\rho (x,t)}{\\partial t}}+\\cdots \\\\\[2ex\]={}&\\int \_{-\\infty }^{\\infty }\\rho (x-q,t)\\,\\varphi (q)\\,dq=\\mathbb {E} \_{q}{\\left\[\\rho (x-q,t)\\right\]}\\\\\[1ex\]={}&\\rho (x,t)\\,\\int \_{-\\infty }^{\\infty }\\varphi (q)\\,dq-{\\frac {\\partial \\rho }{\\partial x}}\\,\\int \_{-\\infty }^{\\infty }q\\,\\varphi (q)\\,dq+{\\frac {\\partial ^{2}\\rho }{\\partial x^{2}}}\\,\\int \_{-\\infty }^{\\infty }{\\frac {q^{2}}{2}}\\varphi (q)\\,dq+\\cdots \\\\\[1ex\]={}&\\rho (x,t)\\cdot 1-0+{\\cfrac {\\partial ^{2}\\rho }{\\partial x^{2}}}\\,\\int \_{-\\infty }^{\\infty }{\\frac {q^{2}}{2}}\\varphi (q)\\,dq+\\cdots \\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/155f9b33c2bde783f0cf06e55dc69d7a705118e9) where the second equality is by definition of ![{\\displaystyle \\varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e). The [integral](https://en.wikipedia.org/wiki/Integral "Integral") in the first term is equal to one by the definition of probability, and the second and other even terms (i.e. first and other odd [moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)")) vanish because of space symmetry. What is left gives rise to the following relation: ![{\\displaystyle {\\frac {\\partial \\rho }{\\partial t}}={\\frac {\\partial ^{2}\\rho }{\\partial x^{2}}}\\cdot \\int \_{-\\infty }^{\\infty }{\\frac {q^{2}}{2\\tau }}\\varphi (q)\\,dq+{\\text{higher-order even moments.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/045362993238e9286f197c2d08c0d7da7cdf3070) Where the coefficient after the [Laplacian](https://en.wikipedia.org/wiki/Laplacian "Laplacian"), the second moment of probability of displacement ![{\\displaystyle q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d), is interpreted as [mass diffusivity](https://en.wikipedia.org/wiki/Mass_diffusivity "Mass diffusivity") *D*: ![{\\displaystyle D=\\int \_{-\\infty }^{\\infty }{\\frac {q^{2}}{2\\tau }}\\varphi (q)\\,dq.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb16997b332ff1ac71b0a99fa6e4aea7e466124) Then the density of Brownian particles ρ at point x at time t satisfies the [diffusion equation](https://en.wikipedia.org/wiki/Diffusion_equation "Diffusion equation"): ![{\\displaystyle {\\frac {\\partial \\rho }{\\partial t}}=D\\cdot {\\frac {\\partial ^{2}\\rho }{\\partial x^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/317b275652b2f923d52340b08570553d3c843879) Assuming that *N* particles start from the origin at the initial time *t* = 0, the diffusion equation has the solution:[\[20\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-20) ![{\\displaystyle \\rho (x,t)={\\frac {N}{\\sqrt {4\\pi Dt}}}\\exp {\\left(-{\\frac {x^{2}}{4Dt}}\\right)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/baef5b3d13bb5b6626a641937ac0d18318a1a548) This expression (which is a [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution") with the mean ![{\\displaystyle \\mu =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3753282c0ad2ea1e7d63f39425efd13c37da3169) and variance ![{\\displaystyle \\sigma ^{2}=2Dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d88a84c4b4144f34e1a879fc7ec7d25b236c2809) usually called Brownian motion ![{\\displaystyle B\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f92553a3c8585519e540724dffe9306c83ca2f8)) allowed Einstein to calculate the [moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right. The second moment is, however, non-vanishing, being given by ![{\\displaystyle \\mathbb {E} {\\left\[x^{2}\\right\]}=2Dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f808eb010dadd7cb5da992f62b95334223e9f4) This equation expresses the mean squared displacement in terms of the time elapsed and the diffusivity. From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root.[\[18\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Einstein-1956-18) His argument is based on a conceptual switch from the "ensemble" of Brownian particles to the "single" Brownian particle: we can speak of the relative number of particles at a single instant just as well as of the time it takes a Brownian particle to reach a given point.[\[21\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-21) The second part of Einstein's theory relates the diffusion constant to physically measurable quantities, such as the mean squared displacement of a particle in a given time interval. This result enables the experimental determination of the Avogadro number and therefore the size of molecules. Einstein analyzed a dynamic equilibrium being established between opposing forces. The beauty of his argument is that the final result does not depend upon which forces are involved in setting up the dynamic equilibrium. In his original treatment, Einstein considered an [osmotic pressure](https://en.wikipedia.org/wiki/Osmotic_pressure "Osmotic pressure") experiment, but the same conclusion can be reached in other ways. Consider, for instance, particles suspended in a viscous fluid in a gravitational field. Gravity tends to make the particles settle, whereas diffusion acts to homogenize them, driving them into regions of smaller concentration. Under the action of gravity, a particle acquires a downward speed of *v* = *ÎŒmg*, where m is the mass of the particle, g is the acceleration due to gravity, and ÎŒ is the particle's [mobility](https://en.wikipedia.org/wiki/Einstein_relation_\(kinetic_theory\) "Einstein relation (kinetic theory)") in the fluid. [George Stokes](https://en.wikipedia.org/wiki/Sir_George_Stokes,_1st_Baronet "Sir George Stokes, 1st Baronet") had shown that the mobility for a spherical particle with radius r is ![{\\displaystyle \\mu ={\\tfrac {1}{6\\pi \\eta r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be24ee78d91ecf2f45fba9b087986821463a7858), where η is the [dynamic viscosity](https://en.wikipedia.org/wiki/Dynamic_viscosity "Dynamic viscosity") of the fluid. In a state of dynamic equilibrium, and under the hypothesis of isothermal fluid, the particles are distributed according to the [barometric distribution](https://en.wikipedia.org/wiki/Barometric_formula "Barometric formula") ![{\\displaystyle \\rho =\\rho \_{o}\\,\\exp \\left({-{\\frac {mgh}{k\_{\\text{B}}T}}}\\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de96f9973a262aa9c33869b220ae9f1b76307452) where *ρ* − *ρ*o is the difference in density of particles separated by a height difference, of ![{\\displaystyle h=z-z\_{o}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f968d881d543c1ece3483bb6788b61da38908e4), *k*B is the [Boltzmann constant](https://en.wikipedia.org/wiki/Boltzmann_constant "Boltzmann constant") (the ratio of the [universal gas constant](https://en.wikipedia.org/wiki/Universal_gas_constant "Universal gas constant"), *R*, to the [Avogadro constant](https://en.wikipedia.org/wiki/Avogadro_constant "Avogadro constant"), *N*A), and *T* is the [absolute temperature](https://en.wikipedia.org/wiki/Thermodynamic_temperature "Thermodynamic temperature"). [![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Brownian_motion_gamboge.jpg/250px-Brownian_motion_gamboge.jpg)](https://en.wikipedia.org/wiki/File:Brownian_motion_gamboge.jpg) [Perrin](https://en.wikipedia.org/wiki/Jean_Baptiste_Perrin "Jean Baptiste Perrin") examined the equilibrium ([barometric distribution](https://en.wikipedia.org/wiki/Barometric_formula "Barometric formula")) of granules (0.6 [microns](https://en.wikipedia.org/wiki/Micron "Micron")) of [gamboge](https://en.wikipedia.org/wiki/Gamboge "Gamboge"), a viscous substance, under the microscope. The granules move against gravity to regions of lower concentration. The relative change in density observed in 10 microns of suspension is equivalent to that occurring in 6 km of air. [Dynamic equilibrium](https://en.wikipedia.org/wiki/Dynamic_equilibrium "Dynamic equilibrium") is established because the more that particles are pulled down by [gravity](https://en.wikipedia.org/wiki/Gravity "Gravity"), the greater the tendency for the particles to migrate to regions of lower concentration. The flux is given by [Fick's law](https://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion "Fick's laws of diffusion"), ![{\\displaystyle J=-D{\\frac {d\\rho }{dh}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2729e7641f9c691261fb9ddc635e161f61dd63) where *J* = *ρv*. Introducing the formula for ρ, we find that ![{\\displaystyle v={\\frac {Dmg}{k\_{\\text{B}}T}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d70f9b4b3f064d2b3edf6ab1e39ed33e02c6095) In a state of dynamical equilibrium, this speed must also be equal to *v* = *ÎŒmg*. Both expressions for v are proportional to *mg*, reflecting that the derivation is independent of the type of forces considered. Similarly, one can derive an equivalent formula for identical [charged particles](https://en.wikipedia.org/wiki/Charged_particle "Charged particle") of charge q in a uniform [electric field](https://en.wikipedia.org/wiki/Electric_field "Electric field") of magnitude E, where *mg* is replaced with the [electrostatic force](https://en.wikipedia.org/wiki/Electrostatic_force "Electrostatic force") *qE*. Equating these two expressions yields the [Einstein relation](https://en.wikipedia.org/wiki/Einstein_relation_\(kinetic_theory\) "Einstein relation (kinetic theory)") for the diffusivity, independent of *mg* or *qE* or other such forces: ![{\\displaystyle {\\frac {\\mathbb {E} {\\left\[x^{2}\\right\]}}{2t}}=D=\\mu k\_{\\text{B}}T={\\frac {\\mu RT}{N\_{\\text{A}}}}={\\frac {RT}{6\\pi \\eta rN\_{\\text{A}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d19b78ac840afe66ba1ad694390512cf72bd90d2) Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of the [Boltzmann constant](https://en.wikipedia.org/wiki/Boltzmann_constant "Boltzmann constant") as *k*B = *R* / *N*A, and the fourth equality follows from Stokes's formula for the mobility. By measuring the mean squared displacement over a time interval along with the universal gas constant *R*, the temperature T, the viscosity η, and the particle radius r, the Avogadro constant *N*A can be determined. The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously by [J. J. Thomson](https://en.wikipedia.org/wiki/J._J._Thomson "J. J. Thomson")[\[22\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Thomson-1904-22) in his series of lectures at Yale University in May 1903 that the dynamic equilibrium between the velocity generated by a [concentration gradient](https://en.wikipedia.org/wiki/Concentration_gradient "Concentration gradient") given by Fick's law and the velocity due to the variation of the partial pressure caused when ions are set in motion "gives us a method of determining Avogadro's constant which is independent of any hypothesis as to the shape or size of molecules, or of the way in which they act upon each other".[\[22\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Thomson-1904-22) An identical expression to Einstein's formula for the diffusion coefficient was also found by [Walther Nernst](https://en.wikipedia.org/wiki/Walther_Nernst "Walther Nernst") in 1888[\[23\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-23) in which he expressed the diffusion coefficient as the ratio of the [osmotic pressure](https://en.wikipedia.org/wiki/Osmotic_pressure "Osmotic pressure") to the ratio of the [frictional](https://en.wikipedia.org/wiki/Friction "Friction") force and the velocity to which it gives rise. The former was equated to the law of van 't Hoff while the latter was given by [Stokes's law](https://en.wikipedia.org/wiki/Stokes%27s_law "Stokes's law"). He writes ![{\\displaystyle k'=p\_{o}/k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d61c5206a0db1e99ba184d2a3649cf4ad38db948) for the diffusion coefficient kâ€Č, where ![{\\displaystyle p\_{o}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c058013bfdb87708b775d59bda857fd5e6d53b3) is the osmotic pressure and k is the ratio of the frictional force to the molecular viscosity which he assumes is given by Stokes's formula for the viscosity. Introducing the [ideal gas law](https://en.wikipedia.org/wiki/Ideal_gas_law "Ideal gas law") per unit volume for the osmotic pressure, the formula becomes identical to that of Einstein's.[\[24\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-24) The use of Stokes's law in Nernst's case, as well as in Einstein and Smoluchowski, is not strictly applicable since it does not apply to the case where the radius of the sphere is small in comparison with the [mean free path](https://en.wikipedia.org/wiki/Mean_free_path "Mean free path").[\[25\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-25) Confirming Einstein's formula experimentally proved difficult. Initial attempts by [Theodor Svedberg](https://en.wikipedia.org/wiki/Theodor_Svedberg "Theodor Svedberg") in 1906 and 1907 were critiqued by Einstein and by Perrin as not measuring a quantity directly comparable to the formula. [Victor Henri](https://en.wikipedia.org/wiki/Victor_Henri "Victor Henri") in 1908 took cinematographic shots through a microscope and found quantitative disagreement with the formula but again the analysis was uncertain.[\[26\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-26) Einstein's predictions were finally confirmed in a series of experiments carried out by Chaudesaigues in 1908 and Perrin in 1909.[\[27\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-27)[\[12\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Brush-1968-12) The confirmation of Einstein's theory constituted empirical progress for the [kinetic theory of heat](https://en.wikipedia.org/wiki/Kinetic_theory_of_gases "Kinetic theory of gases"). In essence, Einstein showed that the motion can be predicted directly from the kinetic model of [thermal equilibrium](https://en.wikipedia.org/wiki/Thermal_equilibrium "Thermal equilibrium"). The importance of the theory lay in the fact that it confirmed the kinetic theory's account of the [second law of thermodynamics](https://en.wikipedia.org/wiki/Second_law_of_thermodynamics "Second law of thermodynamics") as being an essentially statistical law.[\[28\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-28) Brownian motion model of the trajectory of a particle of dye in water [Smoluchowski](https://en.wikipedia.org/wiki/Marian_Smoluchowski "Marian Smoluchowski")'s theory of Brownian motion,[\[29\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-29) later contextualized in comprehensive reviews of stochastic physics,[\[30\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-30) starts from the same premise as that of Einstein and derives the same probability distribution *ρ*(*x*, *t*) for the displacement of a Brownian particle along the x axis in time t. He therefore gets the same expression for the mean squared displacement: ![{\\textstyle \\mathbb {E} {\\left\[(\\Delta x)^{2}\\right\]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb176c581545de90d31f1ffea58ef73ed686093d). However, when he relates it to a particle of mass m moving at a velocity u which is the result of a frictional force governed by Stokes's law, he finds ![{\\displaystyle \\mathbb {E} {\\left\[(\\Delta x)^{2}\\right\]}=2Dt=t{\\frac {32}{81}}{\\frac {mu^{2}}{\\pi \\mu a}}=t{\\frac {64}{27}}{\\frac {{\\frac {1}{2}}mu^{2}}{3\\pi \\mu a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86135c60133697844b01ee4c06a9d5fa84470b14) where ÎŒ is the viscosity coefficient, and a is the radius of the particle. Associating the kinetic energy ![{\\textstyle mu^{2}/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27984b6f95bafd1da2add2193f974aec53bc671c) with the thermal energy ![{\\textstyle RT/N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24cac37f97fde56f0713eab5107b57a509c68b50), the expression for the mean squared displacement is 64/27 times that found by Einstein. This discrepancy arises from differing theoretical approaches: Einstein assumed Stokes drag applied directly to the macroscopic drift velocity, while Smoluchowski performed a more detailed kinematic collision analysis but introduced a slight calculation variance when averaging over the Maxwellian velocity distribution. The fraction 27/64 was commented on by [Arnold Sommerfeld](https://en.wikipedia.org/wiki/Arnold_Sommerfeld "Arnold Sommerfeld") in his necrology on Smoluchowski: "The numerical coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt."[\[31\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-31) Smoluchowski[\[32\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-32) attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are exactly equal. To address this paradox, he relies on the inevitability of statistical fluctuations: - If the probability of m gains and ![{\\textstyle n-m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9408cf61fe6c72a387c31a444cfc7ac3137ddd5d) losses follows a [binomial distribution](https://en.wikipedia.org/wiki/Binomial_distribution "Binomial distribution"), ![{\\displaystyle P\_{m,n}={\\binom {n}{m}}2^{-n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98367f0a10e66d5b02b9998d80de5cbe77416e9e) with equal *a priori* probabilities of 1/2, the mean total gain is ![{\\displaystyle \\mathbb {E} {\\left\[2m-n\\right\]}=\\sum \_{m={\\frac {n}{2}}}^{n}(2m-n)P\_{m,n}={\\frac {nn!}{2^{n+1}\\left\[\\left({\\frac {n}{2}}\\right)!\\right\]^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72f3259e6d8b5f8b588ba14dbae36071836cc493) - If n is large enough so that Stirling's approximation can be used in the form ![{\\textstyle n!\\approx \\left({\\frac {n}{e}}\\right)^{n}{\\sqrt {2\\pi n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/487096849d06aea7ffa11b9be260dedd53c3391e), then the expected total absolute gain representing the net drift can be approximated.[\[33\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-33) This expected gain will be: ![{\\displaystyle \\mathbb {E} {\\left\[2m-n\\right\]}\\approx {\\sqrt {\\frac {n}{2\\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e27b8f4165bc2e0c804df161fd4c33d6fd50d3e) showing that the net displacement increases proportionally to the square root of the total population of collision events. Suppose that a Brownian particle of mass M is surrounded by lighter particles of mass m which are traveling at a speed u. Smoluchowski reasons that the mechanics of these interactions produce a macroscopically observable effect: - In any single collision between a surrounding fluid particle and the Brownian particle, the velocity transmitted to the latter will be ![{\\textstyle mu/M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57ce044c79db00202c9dbf9ea6aaaf88336a1d8b). This ratio is exceedingly small, on the order of 10−7 cm/s. - However, in a gas there will be more than 1016 collisions in a second, and even more in a liquid, where there are expected to be roughly 1020 collisions in one second. - Because of the immense scale of collisions, statistical imbalances are inevitable. While some collisions will accelerate the Brownian particle, others will decelerate it. - If there is a mean excess of one kind of collision (e.g., more impacts from the left than the right) on the order of 108 to 1010 collisions in a single second, then the instantaneous velocity of the Brownian particle may be anywhere between 10 and 1000 cm/s. - Thus, even though there are equal probabilities for forward and backward collisions, the sheer volume of events creates a net tendency to keep the Brownian particle in erratic, continuous motion, much like the fluctuations predicted by the ballot theorem. These orders of magnitude do not take into consideration the velocity of the Brownian particle, U, which actively depends on the collisions that tend to accelerate and decelerate it. The larger U is, the greater will be the resistive drag of collisions that will retard it, so that the velocity of a Brownian particle can never increase without limit. If such an unbounded process could occur, it would be tantamount to a perpetual motion machine of the second kind. Since the equipartition of energy applies to this system in thermal equilibrium, the kinetic energy of the Brownian particle, ![{\\textstyle MU^{2}/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e04abbc311be0a74c39f0d76dd428d46979282c6), will be equal, on the average, to the kinetic energy of the surrounding fluid particle, ![{\\textstyle mu^{2}/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27984b6f95bafd1da2add2193f974aec53bc671c). In 1906, Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion.[\[34\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-34) The model assumes collisions with ![{\\textstyle M\\gg m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de9fc63624060a1a77eb908a9a3909dbf706c0ba) where M is the test particle's mass and m the mass of one of the individual particles composing the fluid. To build this theoretical framework, several simplifying assumptions are made: - The particle collisions are confined to one dimension along a single axis. - It is equally probable for the test particle to be hit from the left as from the right. - Every collision always imparts the exact same discrete magnitude of velocity change, ![{\\textstyle \\Delta V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6098f6ae65a71ec75c77595a1ce6c6b844a81972). If ![{\\textstyle N\_{\\text{R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d52c69d029e1517130dc6283d10a4fb12b257f7) is the number of collisions from the right and ![{\\textstyle N\_{\\text{L}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f92bed4f749b91147fadfa9164cba3b03141ff94) the number of collisions from the left then after N collisions the particle's velocity will have changed by ![{\\textstyle \\Delta V(2N\_{\\text{R}}-N)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4944e34eda114f9c1b868debd9b33950c658822). The [multiplicity](https://en.wikipedia.org/wiki/Multiplicity_\(mathematics\) "Multiplicity (mathematics)") is then simply given by: ![{\\displaystyle {\\binom {N}{N\_{\\text{R}}}}={\\frac {N!}{N\_{\\text{R}}!(N-N\_{\\text{R}})!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb98a3f680bae27365c32c7a87b1cf1ef660969f) and the total number of possible states is given by ![{\\textstyle 2^{N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef00a629421e96c78395ff90b9cf4d57b0176782). Therefore, the probability of the particle being hit from the right ![{\\textstyle N\_{\\text{R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d52c69d029e1517130dc6283d10a4fb12b257f7) times is: ![{\\displaystyle P\_{N}(N\_{\\text{R}})={\\frac {N!}{2^{N}N\_{\\text{R}}!(N-N\_{\\text{R}})!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/711bd4ce858b9e6ba3a2f6c6bac3ca4de26d5de2) As a result of its simplicity, Smoluchowski's 1D model can only qualitatively describe Brownian motion. For a realistic particle undergoing Brownian motion in a fluid, many of the assumptions do not apply. For example, the assumption that on average there are an equal number of collisions from the right as from the left falls apart once the particle is in motion, as it will sweep into particles in its path and create a velocity-dependent drag force. Furthermore, there would be a continuous statistical distribution of different possible ![{\\textstyle \\Delta V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6098f6ae65a71ec75c77595a1ce6c6b844a81972)s governed by the Maxwell-Boltzmann distribution of fluid molecule velocities, rather than a single discrete value in a physical liquid or gas. The [diffusion equation](https://en.wikipedia.org/wiki/Diffusion_equation "Diffusion equation") yields an approximation of the time evolution of the [probability density function](https://en.wikipedia.org/wiki/Probability_density_function "Probability density function") associated with the position of the particle going under a Brownian movement under the physical definition. The approximation becomes valid on timescales much larger than the timescale of individual atomic collisions, since it does not include a term to describe the acceleration of particles during collision. The time evolution of the position of the Brownian particle over all time scales described using the [Langevin equation](https://en.wikipedia.org/wiki/Langevin_equation "Langevin equation"), an equation that involves a random force field representing the effect of the [thermal fluctuations](https://en.wikipedia.org/wiki/Thermal_fluctuations "Thermal fluctuations") of the solvent on the particle.[\[17\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Velocity-2010-17) At longer times scales, where acceleration is negligible, individual particle dynamics can be approximated using [Brownian dynamics](https://en.wikipedia.org/wiki/Brownian_dynamics "Brownian dynamics") in place of [Langevin dynamics](https://en.wikipedia.org/wiki/Langevin_dynamics "Langevin dynamics"). ### Astrophysics: star motion within galaxies \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=6 "Edit section: Astrophysics: star motion within galaxies")\] In [stellar dynamics](https://en.wikipedia.org/wiki/Stellar_dynamics "Stellar dynamics"), a massive body (star, [black hole](https://en.wikipedia.org/wiki/Black_hole "Black hole"), etc.) can experience Brownian motion as it responds to [gravitational](https://en.wikipedia.org/wiki/Gravitational "Gravitational") forces from surrounding stars.[\[35\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Merritt-2013-35) The rms velocity V of the massive object, of mass M, is related to the rms velocity ![{\\displaystyle v\_{\\star }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/913c9f2b0add99440a59ec96a769759b57ab05ed) of the background stars by ![{\\displaystyle MV^{2}\\approx mv\_{\\star }^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc109205eeeaa8d536a127bf22ff3e716d6fe29e) where ![{\\displaystyle m\\ll M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9913ddecf5981f7796c73b2b166d3f4e6fb9e39e) is the mass of the background stars. The gravitational force from the massive object causes nearby stars to move faster than they otherwise would, increasing both ![{\\displaystyle v\_{\\star }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/913c9f2b0add99440a59ec96a769759b57ab05ed) and V.[\[35\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Merritt-2013-35) The Brownian velocity of [Sgr A\*](https://en.wikipedia.org/wiki/Sagittarius_A* "Sagittarius A*"), the [supermassive black hole](https://en.wikipedia.org/wiki/Supermassive_black_hole "Supermassive black hole") at the center of the [Milky Way galaxy](https://en.wikipedia.org/wiki/Milky_Way_galaxy "Milky Way galaxy"), is predicted from this formula to be less than 1 km s−1.[\[36\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Reid-2004-36) An animated example of a Brownian motion-like [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk") on a 2D surface with periodic boundary conditions. In the [scaling limit](https://en.wikipedia.org/wiki/Scaling_limit "Scaling limit"), random walk approaches the Wiener process according to [Donsker's theorem](https://en.wikipedia.org/wiki/Donsker%27s_theorem "Donsker's theorem"). In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), Brownian motion is described by the **Wiener process**, a continuous-time [stochastic process](https://en.wikipedia.org/wiki/Stochastic_process "Stochastic process") named in honor of [Norbert Wiener](https://en.wikipedia.org/wiki/Norbert_Wiener "Norbert Wiener"). It is one of the best known [LĂ©vy processes](https://en.wikipedia.org/wiki/L%C3%A9vy_process "LĂ©vy process") ([cĂ dlĂ g](https://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g "CĂ dlĂ g") stochastic processes with [stationary](https://en.wikipedia.org/wiki/Stationary_increments "Stationary increments") [independent increments](https://en.wikipedia.org/wiki/Independent_increments "Independent increments")) and occurs frequently in pure and applied mathematics, [economics](https://en.wikipedia.org/wiki/Economy "Economy") and [physics](https://en.wikipedia.org/wiki/Physics "Physics"). [![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wiener_process_3d.png/250px-Wiener_process_3d.png)](https://en.wikipedia.org/wiki/File:Wiener_process_3d.png) A single realisation of three-dimensional Brownian motion for times 0 ≀ *t* ≀ 2 The Wiener process *Wt* is characterized by four facts:[\[37\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-37) 1. *W*0 = 0 2. *Wt* is [almost surely](https://en.wikipedia.org/wiki/Almost_surely "Almost surely") continuous 3. *Wt* has independent increments 4. ![{\\displaystyle W\_{t}-W\_{s}\\sim {\\mathcal {N}}(0,t-s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c502ae53a6726372cce17efdd787e8a791abf88) (for ![{\\displaystyle 0\\leq s\\leq t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efd3155df471ed05348da5307e4104e085f77d45)). ![{\\displaystyle {\\mathcal {N}}(\\mu ,\\sigma ^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/863304aaa42a945f2f07d79facc3d2eebc845ce7) denotes the [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution "Normal distribution") with [expected value](https://en.wikipedia.org/wiki/Expected_value "Expected value") ÎŒ and [variance](https://en.wikipedia.org/wiki/Variance "Variance") *σ*2. The condition that it has independent increments means that if ![{\\displaystyle 0\\leq s\_{1}\<t\_{1}\\leq s\_{2}\<t\_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/753cdf256564c50198f038805e98885e058e2119) then ![{\\displaystyle W\_{t\_{1}}-W\_{s\_{1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23dcd43cde7eb8ce6ce229da07b72ec2f98d47cb) and ![{\\displaystyle W\_{t\_{2}}-W\_{s\_{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5c045f44a9b78e02815793e35af984c270f729) are independent random variables. In addition, for some [filtration](https://en.wikipedia.org/wiki/Filtration_\(probability_theory\) "Filtration (probability theory)") ![{\\displaystyle {\\mathcal {F}}\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5209aeab53016ae720999b2a50e7f0bc27cb33), ![{\\displaystyle W\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50680c5535c83badfa630dba63b583d2eeaa2977) is ![{\\displaystyle {\\mathcal {F}}\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5209aeab53016ae720999b2a50e7f0bc27cb33) [measurable](https://en.wikipedia.org/wiki/Measurable "Measurable") for all ![{\\displaystyle t\\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/248525429e9cd266f53ab8c52d17bc206c546060). An alternative characterisation of the Wiener process is the so-called *LĂ©vy characterisation* that says that the Wiener process is an almost surely continuous [martingale](https://en.wikipedia.org/wiki/Martingale_\(probability_theory\) "Martingale (probability theory)") with *W*0 = 0 and [quadratic variation](https://en.wikipedia.org/wiki/Quadratic_variation "Quadratic variation") ![{\\displaystyle \[W\_{t},W\_{t}\]=t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b53fb1bd6ab3cb0b4d2732924e5b654454b11171). A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent ![{\\displaystyle {\\mathcal {N}}(0,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3eeb356405c0b33b680b5caa425ada4e9f53e8b) random variables. This representation can be obtained using the [Kosambi–Karhunen–LoĂšve theorem](https://en.wikipedia.org/wiki/Kosambi%E2%80%93Karhunen%E2%80%93Lo%C3%A8ve_theorem "Kosambi–Karhunen–LoĂšve theorem"). The Wiener process can be constructed as the [scaling limit](https://en.wikipedia.org/wiki/Scaling_limit "Scaling limit") of a [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk"), or other discrete-time stochastic processes with stationary independent increments. This is known as [Donsker's theorem](https://en.wikipedia.org/wiki/Donsker%27s_theorem "Donsker's theorem"). Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed [neighborhood](https://en.wikipedia.org/wiki/Neighborhood_\(mathematics\) "Neighborhood (mathematics)") of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is [scale invariant](https://en.wikipedia.org/wiki/Scale_invariance "Scale invariance"). A d-dimensional [Gaussian free field](https://en.wikipedia.org/wiki/Gaussian_free_field "Gaussian free field") has been described as "a d-dimensional-time analog of Brownian motion."[\[38\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-38) The Brownian motion can be modeled by a [random walk](https://en.wikipedia.org/wiki/Random_walk "Random walk").[\[39\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-39) In the general case, Brownian motion is a [Markov process](https://en.wikipedia.org/wiki/Markov_process "Markov process") and described by [stochastic integral equations](https://en.wikipedia.org/wiki/Stochastic_calculus "Stochastic calculus").[\[40\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Morozov-2011-40) ### LĂ©vy characterisation \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=9 "Edit section: LĂ©vy characterisation")\] The French mathematician [Paul LĂ©vy](https://en.wikipedia.org/wiki/Paul_L%C3%A9vy_\(mathematician\) "Paul LĂ©vy (mathematician)") proved the following theorem, which gives a necessary and sufficient condition for a continuous **R***n*\-valued stochastic process *X* to actually be n\-dimensional Brownian motion. Hence, LĂ©vy's condition can actually be used as an alternative definition of Brownian motion. Let *X* = (*X*1, ..., *X**n*) be a continuous stochastic process on a [probability space](https://en.wikipedia.org/wiki/Probability_space "Probability space") (Ω, ÎŁ, **P**) taking values in **R***n*. Then the following are equivalent: 1. *X* is a Brownian motion with respect to **P**, i.e., the law of *X* with respect to **P** is the same as the law of an n\-dimensional Brownian motion, i.e., the [push-forward measure](https://en.wikipedia.org/wiki/Push-forward_measure "Push-forward measure") *X*∗(**P**) is [classical Wiener measure](https://en.wikipedia.org/wiki/Classical_Wiener_measure "Classical Wiener measure") on *C*0(\[0, ∞); **R***n*). 2. both 1. *X* is a [martingale](https://en.wikipedia.org/wiki/Martingale_\(probability_theory\) "Martingale (probability theory)") with respect to **P** (and its own [natural filtration](https://en.wikipedia.org/wiki/Natural_filtration "Natural filtration")); and 2. for all 1 ≀ *i*, *j* ≀ *n*, *X**i*(*t*) *X**j*(*t*) − *ÎŽ**ij* *t* is a martingale with respect to **P** (and its own [natural filtration](https://en.wikipedia.org/wiki/Natural_filtration "Natural filtration")), where *ÎŽ**ij* denotes the [Kronecker delta](https://en.wikipedia.org/wiki/Kronecker_delta "Kronecker delta"). The spectral content of a stochastic process ![{\\displaystyle X\_{t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82120d04dfb3cbadc4912951dd12b5568c9cd8f3) can be found from the [power spectral density](https://en.wikipedia.org/wiki/Power_spectral_density "Power spectral density"), formally defined as ![{\\displaystyle S(\\omega )=\\lim \_{T\\to \\infty }{\\frac {1}{T}}\\mathbb {E} \\left\\{\\left\|\\int \_{0}^{T}e^{i\\omega t}X\_{t}dt\\right\|^{2}\\right\\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a23858585053a9d0db9a71bcafe5970d4e72cd0) where ![{\\displaystyle \\mathbb {E} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad9faf1fd4a61d36d7f8a2f3204f3805a43c0d4a) stands for the [expected value](https://en.wikipedia.org/wiki/Expected_value "Expected value"). The power spectral density of Brownian motion is found to be[\[41\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-41) ![{\\displaystyle S\_{BM}(\\omega )={\\frac {4D}{\\omega ^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6397d3243cdb5b94d63dd1e37b481f61ac8e172) where D is the [diffusion coefficient](https://en.wikipedia.org/wiki/Diffusion_coefficient "Diffusion coefficient") of *Xt*. For naturally occurring signals, the spectral content can be found from the power spectral density of a single realization, with finite available time, i.e., ![{\\displaystyle S^{(1)}(\\omega ,T)={\\frac {1}{T}}\\left\|\\int \_{0}^{T}e^{i\\omega t}X\_{t}dt\\right\|^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5353bd441f2a3f690560a9287276afea554172) which for an individual realization of a Brownian motion trajectory,[\[42\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Krapf-2018-42) it is found to have expected value ![{\\displaystyle \\mu \_{BM}(\\omega ,T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a192047176a2599314e03efc49fc4d99a91db73) ![{\\displaystyle \\mu \_{\\text{BM}}(\\omega ,T)={\\frac {4D}{\\omega ^{2}}}\\left\[1-{\\frac {\\sin \\left(\\omega T\\right)}{\\omega T}}\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1620301f27770202e280584b0bf65d3cad88dc59) and [variance](https://en.wikipedia.org/wiki/Variance "Variance") ![{\\displaystyle \\sigma \_{\\text{BM}}^{2}(\\omega ,T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f093f46c9333d664e26790ad65ce12242fec46d4)[\[42\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Krapf-2018-42) ![{\\displaystyle \\sigma \_{S}^{2}(f,T)=\\mathbb {E} \\left\\{\\left(S\_{T}^{(j)}(f)\\right)^{2}\\right\\}-\\mu \_{S}^{2}(f,T)={\\frac {20D^{2}}{f^{4}}}\\left\[1-{\\Big (}6-\\cos \\left(fT\\right){\\Big )}{\\frac {2\\sin \\left(fT\\right)}{5fT}}+{\\frac {{\\Big (}17-\\cos \\left(2fT\\right)-16\\cos \\left(fT\\right){\\Big )}}{10f^{2}T^{2}}}\\right\].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8390eff4a72b6d313d0800fee1e33fcb978b784) For sufficiently long realization times, the expected value of the power spectrum of a single trajectory converges to the formally defined power spectral density ![{\\displaystyle S(\\omega )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cd569c75ee7dad692b0f20a695130098e18c279), but its coefficient of variation ![{\\displaystyle \\gamma =\\sigma /\\mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/076b60902718422c76d5985bbffa9f9a4ef10b68) tends to ![{\\displaystyle {\\sqrt {5}}/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97f3998e64a3d3ca91193be8e617198eb1a83346). This implies the distribution of ![{\\displaystyle S^{(1)}(\\omega ,T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe2f9e122d38c84f693175abcf1a850a4f2d593) is broad even in the infinite time limit. ### Riemannian manifolds \[[edit](https://en.wikipedia.org/w/index.php?title=Brownian_motion&action=edit&section=11 "Edit section: Riemannian manifolds")\] [![](https://upload.wikimedia.org/wikipedia/commons/4/4e/BMonSphere.jpg)](https://en.wikipedia.org/wiki/File:BMonSphere.jpg) Brownian motion on a sphere Brownian motion is usually considered to take place in [Euclidean space](https://en.wikipedia.org/wiki/Euclidean_space "Euclidean space"). It is natural to consider how such motion generalizes to more complex shapes, such as [surfaces](https://en.wikipedia.org/wiki/Surface "Surface") or higher dimensional [manifolds](https://en.wikipedia.org/wiki/Manifold "Manifold"). The formalization requires the space to possess some form of a [derivative](https://en.wikipedia.org/wiki/Derivative "Derivative"), as well as a [metric](https://en.wikipedia.org/wiki/Metric_space "Metric space"), so that a [Laplacian](https://en.wikipedia.org/wiki/Laplacian "Laplacian") can be defined. Both of these are available on [Riemannian manifolds](https://en.wikipedia.org/wiki/Riemannian_manifold "Riemannian manifold"). Riemannian manifolds have the property that [geodesics](https://en.wikipedia.org/wiki/Geodesic "Geodesic") can be described in [polar coordinates](https://en.wikipedia.org/wiki/Polar_coordinates "Polar coordinates"); that is, displacements are always in a radial direction, at some given angle. Uniform random motion is then described by Gaussians along the radial direction, independent of the angle, the same as in Euclidean space. The [infinitesimal generator](https://en.wikipedia.org/wiki/Infinitesimal_generator_\(stochastic_processes\) "Infinitesimal generator (stochastic processes)") (and hence [characteristic operator](https://en.wikipedia.org/wiki/Characteristic_operator "Characteristic operator")) of Brownian motion on Euclidean **R***n* is ⁠1/2⁠Δ, where Δ denotes the [Laplace operator](https://en.wikipedia.org/wiki/Laplace_operator "Laplace operator"). Brownian motion on an m\-dimensional [Riemannian manifold](https://en.wikipedia.org/wiki/Riemannian_manifold "Riemannian manifold") (*M*, *g*) can be defined as diffusion on M with the characteristic operator given by ⁠1/2⁠ΔLB, half the [Laplace–Beltrami operator](https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator "Laplace–Beltrami operator") ΔLB. One of the topics of study is a characterization of the [PoincarĂ© recurrence time](https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem "PoincarĂ© recurrence theorem") for such systems.[\[16\]](https://en.wikipedia.org/wiki/Brownian_motion#cite_note-Grigoryan-1999-16) The [narrow escape problem](https://en.wikipedia.org/wiki/Narrow_escape_problem "Narrow escape problem") is a ubiquitous problem in biology, biophysics and cellular biology which has the following formulation: a Brownian particle ([ion](https://en.wikipedia.org/wiki/Ion "Ion"), [molecule](https://en.wikipedia.org/wiki/Molecule "Molecule"), or [protein](https://en.wikipedia.org/wiki/Protein "Protein")) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is that of calculating the mean escape time. This time diverges as the window shrinks, thus rendering the calculation a [singular perturbation](https://en.wikipedia.org/wiki/Singular_perturbation "Singular perturbation") problem. - [Brownian bridge](https://en.wikipedia.org/wiki/Brownian_bridge "Brownian bridge") – Stochastic process in physics - [Brownian covariance](https://en.wikipedia.org/wiki/Brownian_covariance "Brownian covariance") – Statistical measure - [Brownian dynamics](https://en.wikipedia.org/wiki/Brownian_dynamics "Brownian dynamics") – Ideal molecular motion where no average acceleration takes place - [Brownian motor](https://en.wikipedia.org/wiki/Brownian_motor "Brownian motor") – Nanoscale machine - [Brownian noise](https://en.wikipedia.org/wiki/Brownian_noise "Brownian noise") – Type of noise produced by Brownian motion - [Brownian ratchet](https://en.wikipedia.org/wiki/Brownian_ratchet "Brownian ratchet") – Perpetual motion device - [Brownian surface](https://en.wikipedia.org/wiki/Brownian_surface "Brownian surface") - [Brownian tree](https://en.wikipedia.org/wiki/Brownian_tree "Brownian tree") – Concept in probability theory - [Brownian web](https://en.wikipedia.org/wiki/Brownian_web "Brownian web") - [Fractional Brownian motion](https://en.wikipedia.org/wiki/Fractional_Brownian_motion "Fractional Brownian motion") – Probability theory concept - [Geometric Brownian motion](https://en.wikipedia.org/wiki/Geometric_Brownian_motion "Geometric Brownian motion") – Continuous stochastic process - [ItĂŽ diffusion](https://en.wikipedia.org/wiki/It%C3%B4_diffusion "ItĂŽ diffusion") – Solution to a specific type of stochastic differential equation - [LĂ©vy arcsine law](https://en.wikipedia.org/wiki/L%C3%A9vy_arcsine_law "LĂ©vy arcsine law") – Collection of results for one-dimensional random walks and Brownian motion - [Marangoni effect](https://en.wikipedia.org/wiki/Marangoni_effect "Marangoni effect") – Physical phenomenon between two fluids - [Nanoparticle tracking analysis](https://en.wikipedia.org/wiki/Nanoparticle_tracking_analysis "Nanoparticle tracking analysis") – Method for visualizing and analyzing particles in liquids - [Reflected Brownian motion](https://en.wikipedia.org/wiki/Reflected_Brownian_motion "Reflected Brownian motion") – Wiener process with reflecting spatial boundaries - [Rotational Brownian motion](https://en.wikipedia.org/wiki/Rotational_Brownian_motion "Rotational Brownian motion") - [Schramm–Loewner evolution](https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution "Schramm–Loewner evolution") – Concept in probability theory - [Single particle tracking](https://en.wikipedia.org/wiki/Single_particle_tracking "Single particle tracking") - [Single particle trajectories](https://en.wikipedia.org/wiki/Single_particle_trajectories "Single particle trajectories") - [Surface diffusion](https://en.wikipedia.org/wiki/Surface_diffusion "Surface diffusion") – Physical Process - [Tyndall effect](https://en.wikipedia.org/wiki/Tyndall_effect "Tyndall effect") – Scattering of light by tiny particles in a colloidal suspension 1. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-1)** Meyburg, Jan Philipp; Diesing, Detlef (2017). "Teaching the Growth, Ripening, and Agglomeration of Nanostructures in Computer Experiments". *Journal of Chemical Education*. **94** (9): 1225–1231\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2017JChEd..94.1225M](https://ui.adsabs.harvard.edu/abs/2017JChEd..94.1225M). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1021/acs.jchemed.6b01008](https://doi.org/10.1021%2Facs.jchemed.6b01008). 2. ^ [***a***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Feynman-1964_2-0) [***b***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Feynman-1964_2-1) Feynman, Richard (1964). ["The Brownian Movement"](https://feynmanlectures.caltech.edu/I_41.html). *The Feynman Lectures of Physics, Volume I*. p. 41. 3. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-3)** Pathria, RK (1972). Statistical Mechanics. Pergamon Press. pp. 43–48, 73–74. ISBN 0-08-016747-0. 4. ^ [***a***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Einstein-1905_4-0) [***b***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Einstein-1905_4-1) Einstein, Albert (1905). ["Über die von der molekularkinetischen Theorie der WĂ€rme geforderte Bewegung von in ruhenden FlĂŒssigkeiten suspendierten Teilchen"](http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_549-560.pdf) \[On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat\] (PDF). *Annalen der Physik* (in German). **322** (8): 549–560\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1905AnP...322..549E](https://ui.adsabs.harvard.edu/abs/1905AnP...322..549E). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/andp.19053220806](https://doi.org/10.1002%2Fandp.19053220806). [Archived](https://ghostarchive.org/archive/20221009/http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_549-560.pdf) (PDF) from the original on 9 October 2022. 5. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-5)** ["The Nobel Prize in Physics 1926"](https://www.nobelprize.org/prizes/physics/1926/perrin/lecture/). *NobelPrize.org*. Retrieved 29 May 2019. 6. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-6)** Tsekov, Roumen (1995). "Brownian motion of molecules: the classical theory". *Ann. Univ. Sofia*. **88**: 57. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1005\.1490](https://arxiv.org/abs/1005.1490). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1995AUSFC..88...57T](https://ui.adsabs.harvard.edu/abs/1995AUSFC..88...57T). "the behavior of a Brownian particle is quite irregular and can be described only in the frames of a statistical approach." 7. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-7)** Knight, Frank B. (1 February 1962). ["On the random walk and Brownian motion"](https://www.ams.org/jourcgi/jour-getitem?pii=S0002-9947-1962-0139211-2). *Transactions of the American Mathematical Society*. **103** (2): 218–228\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1090/S0002-9947-1962-0139211-2](https://doi.org/10.1090%2FS0002-9947-1962-0139211-2). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0002-9947](https://search.worldcat.org/issn/0002-9947). 8. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-8)** ["Donsker invariance principle – Encyclopedia of Mathematics"](https://encyclopediaofmath.org/wiki/Donsker_invariance_principle). *encyclopediaofmath.org*. Retrieved 28 June 2020. 9. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-9)** Perrin, Jean (1914). [*Atoms*](https://archive.org/stream/atomsper00perruoft#page/115/mode/1up). London : Constable. p. 115. `{{cite book}}`: CS1 maint: publisher location ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_publisher_location "Category:CS1 maint: publisher location")) 10. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-10)** [Lucretius](https://en.wikipedia.org/wiki/Lucretius "Lucretius") (1951). *On the Nature of the Universe*. Translated by Latham, R. E. London: Penguin Books. pp. 63–64\. 11. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-11)** Tabor, D. (1991). [*Gases, Liquids and Solids: And Other States of Matter*](https://books.google.com/books?id=bTrJ15y2IksC&pg=PA120) (3rd ed.). Cambridge: Cambridge University Press. p. 120. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-521-40667-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-40667-3 "Special:BookSources/978-0-521-40667-3") . 12. ^ [***a***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Brush-1968_12-0) [***b***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Brush-1968_12-1) Brush, Stephen G. (1968). ["A History of Random Processes: I. Brownian Movement from Brown to Perrin"](https://www.jstor.org/stable/41133279). *Archive for History of Exact Sciences*. **5** (1): 1–36\. [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0003-9519](https://search.worldcat.org/issn/0003-9519). 13. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-13)** Brown, Robert; Bennett, John J.; Hardwicke, Robert (1866). [*The miscellaneous botanical works of Robert Brown*](https://www.biodiversitylibrary.org/page/18531045). Vol. 1. Published for the Ray society by R. 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[CiteSeerX](https://en.wikipedia.org/wiki/CiteSeerX_\(identifier\) "CiteSeerX (identifier)") [10\.1.1.167.8245](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.167.8245). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1126/science.1189403](https://doi.org/10.1126%2Fscience.1189403). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [20488989](https://pubmed.ncbi.nlm.nih.gov/20488989). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [45828908](https://api.semanticscholar.org/CorpusID:45828908). Archived from [the original](http://chaos.utexas.edu/wp-uploads/2010/06/science.1189403v1.pdf) (PDF) on 31 March 2011. 18. ^ [***a***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Einstein-1956_18-0) [***b***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Einstein-1956_18-1) Einstein, Albert (1956) \[1926\]. [*Investigations on the Theory of the Brownian Movement*](http://users.physik.fu-berlin.de/~kleinert/files/eins_brownian.pdf) (PDF). Dover Publications. [Archived](https://ghostarchive.org/archive/20221009/http://users.physik.fu-berlin.de/~kleinert/files/eins_brownian.pdf) (PDF) from the original on 9 October 2022. Retrieved 25 December 2013. 19. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-19)** Stachel, J., ed. (1989). ["Einstein's Dissertation on the Determination of Molecular Dimensions"](http://www.csun.edu/~dchoudhary/Physics-Year_files/ed_diss.pdf) (PDF). *The Collected Papers of Albert Einstein, Volume 2*. Princeton University Press. [Archived](https://ghostarchive.org/archive/20221009/http://www.csun.edu/~dchoudhary/Physics-Year_files/ed_diss.pdf) (PDF) from the original on 9 October 2022. 20. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-20)** Kozdron, Michael J. (2008). ["Brownian Motion and the Heat Equation – chpt. 3 Albert Einstein's proof of the existence of Brownian motion"](https://web.archive.org/web/20241126184903/https://uregina.ca/~kozdron/Research/UgradTalks/BM_and_Heat/heat_and_BM.pdf) (PDF). *University of Regina*. Archived from [the original](https://uregina.ca/~kozdron/Research/UgradTalks/BM_and_Heat/heat_and_BM.pdf) (PDF) on 26 November 2024. Retrieved 3 November 2025. 21. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-21)** Lavenda, Bernard H. (1985). [*Nonequilibrium Statistical Thermodynamics*](https://archive.org/details/nonequilibriumst00lave). John Wiley & Sons. p. [20](https://archive.org/details/nonequilibriumst00lave/page/n29). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-471-90670-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-90670-4 "Special:BookSources/978-0-471-90670-4") . 22. ^ [***a***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Thomson-1904_22-0) [***b***](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-Thomson-1904_22-1) Thomson, J. J. (1904). [*Electricity and Matter*](https://archive.org/details/electricitymatte00thomuoft). Yale University Press. pp. [80](https://archive.org/details/electricitymatte00thomuoft/page/80)–83. 23. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-23)** Nernst, Walther (1888). "Zur Kinetik der in Lösung befindlichen Körper". *[Zeitschrift fĂŒr Physikalische Chemie](https://en.wikipedia.org/wiki/Zeitschrift_f%C3%BCr_Physikalische_Chemie "Zeitschrift fĂŒr Physikalische Chemie")* (in German). **9**: 613–637\. 24. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-24)** Leveugle, J. (2004). *La RelativitĂ©, PoincarĂ© et Einstein, Planck, Hilbert*. Harmattan. p. 181. 25. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-25)** Townsend, J.E.S. (1915). [*Electricity in Gases*](https://archive.org/details/electricityinga00towngoog). Clarendon Press. p. [254](https://archive.org/details/electricityinga00towngoog/page/n282). 26. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-26)** Maiocchi, Roberto (September 1990). ["The case of Brownian motion"](https://www.cambridge.org/core/journals/british-journal-for-the-history-of-science/article/abs/case-of-brownian-motion/7E6FCB8188956D072CC83581B5645099). *The British Journal for the History of Science*. **23** (3): 257–283\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1017/S0007087400043983](https://doi.org/10.1017%2FS0007087400043983). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1474-001X](https://search.worldcat.org/issn/1474-001X). 27. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-27)** Haw, M D (26 August 2002). ["Colloidal suspensions, Brownian motion, molecular reality: a short history"](https://iopscience.iop.org/article/10.1088/0953-8984/14/33/315). *Journal of Physics: Condensed Matter*. **14** (33): 7769–7779\. 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["Sur le chemin moyen parcouru par les molĂ©cules d'un gaz et sur son rapport avec la thĂ©orie de la diffusion"](https://archive.org/stream/bulletininternat1906pols#page/202/mode/2up) \[On the average path taken by gas molecules and its relation with the theory of diffusion\]. *[Bulletin International de l'AcadĂ©mie des Sciences de Cracovie](https://en.wikipedia.org/w/index.php?title=Bulletin_International_de_l%27Acad%C3%A9mie_des_Sciences_de_Cracovie&action=edit&redlink=1 "Bulletin International de l'AcadĂ©mie des Sciences de Cracovie (page does not exist)")* (in French): 202. 30. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-30)** Chandrasekhar, Subrahmanyan (1943). "Stochastic Problems in Physics and Astronomy". *Reviews of Modern Physics*. **15** (1): 1–89\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1103/RevModPhys.15.1](https://doi.org/10.1103%2FRevModPhys.15.1). 31. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-31)** See p. 535 in Sommerfeld, A. (1917). "Zum Andenken an Marian von Smoluchowski" \[In Memory of Marian von Smoluchowski\]. *[Physikalische Zeitschrift](https://en.wikipedia.org/wiki/Physikalische_Zeitschrift "Physikalische Zeitschrift")* (in German). **18** (22): 533–539\. 32. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-32)** Smoluchowski, M. M. (1906). 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[ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0471257080](https://en.wikipedia.org/wiki/Special:BookSources/978-0471257080 "Special:BookSources/978-0471257080") . 34. **[^](https://en.wikipedia.org/wiki/Brownian_motion#cite_ref-34)** von Smoluchowski, M. (1906). ["Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen"](https://zenodo.org/record/1424073). *[Annalen der Physik](https://en.wikipedia.org/wiki/Annalen_der_Physik "Annalen der Physik")* (in German). **326** (14): 756–780\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1906AnP...326..756V](https://ui.adsabs.harvard.edu/abs/1906AnP...326..756V). 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["A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies"](http://sciweb.nybg.org/science2/pdfs/dws/Brownian.pdf) (PDF). *[Philosophical Magazine](https://en.wikipedia.org/wiki/Philosophical_Magazine "Philosophical Magazine")*. **4** (21): 161–173\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1080/14786442808674769](https://doi.org/10.1080%2F14786442808674769). [Archived](https://ghostarchive.org/archive/20221009/http://sciweb.nybg.org/science2/pdfs/dws/Brownian.pdf) (PDF) from the original on 9 October 2022. Also includes a subsequent defense by Brown of his original observations, *Additional remarks on active molecules*. - Chaudesaigues, M. (1908). 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[Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1965PNAS...53..913D](https://ui.adsabs.harvard.edu/abs/1965PNAS...53..913D). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1073/pnas.53.5.913](https://doi.org/10.1073%2Fpnas.53.5.913). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [72837](https://www.jstor.org/stable/72837). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [301348](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC301348). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [16591279](https://pubmed.ncbi.nlm.nih.gov/16591279). - Einstein, A. (1956). [*Investigations on the Theory of Brownian Movement*](https://archive.org/details/investigationson00eins). New York: Dover. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-486-60304-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-60304-9 "Special:BookSources/978-0-486-60304-9") . Retrieved 6 January 2014. - Henri, V. (1908). "Études cinĂ©matographique du mouvement brownien" \[Cinematographic studies of Brownian motion\]. *[Comptes Rendus](https://en.wikipedia.org/wiki/Comptes_Rendus "Comptes Rendus")* (in French) (146): 1024–6\. - [Lucretius](https://en.wikipedia.org/wiki/Lucretius "Lucretius"), *On The Nature of Things*, translated by [William Ellery Leonard](https://en.wikipedia.org/wiki/William_Ellery_Leonard "William Ellery Leonard"). (*[on-line version](http://onlinebooks.library.upenn.edu/webbin/gutbook/lookup?num=785)*, from [Project Gutenberg](https://en.wikipedia.org/wiki/Project_Gutenberg "Project Gutenberg"). See the heading 'Atomic Motions'; this translation differs slightly from the one quoted). - [Nelson, Edward](https://en.wikipedia.org/wiki/Edward_Nelson "Edward Nelson"), (1967). *Dynamical Theories of Brownian Motion*. [(PDF version of this out-of-print book, from the author's webpage.)](https://web.math.princeton.edu/~nelson/books/bmotion.pdf) This is primarily a mathematical work, but the first four chapters discuss the history of the topic, in the era from Brown to Einstein. - Pearle, P.; Collett, B.; Bart, K.; Bilderback, D.; Newman, D.; Samuels, S. (2010). "What Brown saw and you can too". *[American Journal of Physics](https://en.wikipedia.org/wiki/American_Journal_of_Physics "American Journal of Physics")*. **78** (12): 1278–1289\. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[1008\.0039](https://arxiv.org/abs/1008.0039). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2010AmJPh..78.1278P](https://ui.adsabs.harvard.edu/abs/2010AmJPh..78.1278P). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1119/1.3475685](https://doi.org/10.1119%2F1.3475685). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [12342287](https://api.semanticscholar.org/CorpusID:12342287). - Perrin, J. (1909). "Mouvement brownien et rĂ©alitĂ© molĂ©culaire" \[Brownian movement and molecular reality\]. *[Annales de chimie et de physique](https://en.wikipedia.org/wiki/Annales_de_chimie_et_de_physique "Annales de chimie et de physique")*. 8th series. **18**: 5–114\. - See also Perrin's book "Les Atomes" (1914). - von Smoluchowski, M. (1906). ["Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen"](http://gallica.bnf.fr/ark:/12148/bpt6k15328k/f770.chemindefer). *[Annalen der Physik](https://en.wikipedia.org/wiki/Annalen_der_Physik "Annalen der Physik")*. **21** (14): 756–780\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[1906AnP...326..756V](https://ui.adsabs.harvard.edu/abs/1906AnP...326..756V). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1002/andp.19063261405](https://doi.org/10.1002%2Fandp.19063261405). - Svedberg, T. (1907). *Studien zur Lehre von den kolloiden Losungen*. - [Theile, T. N](https://en.wikipedia.org/wiki/Thorvald_N._Thiele "Thorvald N. Thiele"). - Danish version: "Om Anvendelse af mindste Kvadraters Methode i nogle TilfĂŠlde, hvor en Komplikation af visse Slags uensartede tilfĂŠldige Fejlkilder giver Fejlene en 'systematisk' Karakter". - French version: "Sur la compensation de quelques erreurs quasi-systĂ©matiques par la mĂ©thodes de moindre carrĂ©s" published simultaneously in *Vidensk. Selsk. Skr. 5. Rk., naturvid. og mat. Afd.*, 12:381–408, 1880. [![Wikisource logo](https://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/40px-Wikisource-logo.svg.png)](https://en.wikipedia.org/wiki/File:Wikisource-logo.svg) English [Wikisource](https://en.wikipedia.org/wiki/Wikisource "Wikisource") has original text related to this article: - [Einstein on Brownian Motion](https://web.archive.org/web/20010222031055/http://www.bun.kyoto-u.ac.jp/~suchii/einsteinBM.html) - [Discusses history, botany and physics of Brown's original observations, with videos](http://physerver.hamilton.edu/Research/Brownian/index.html) - ["Einstein's prediction finally witnessed one century later"](http://www.gizmag.com/einsteins-prediction-finally-witnessed/16212/) : a test to observe the velocity of Brownian motion - [Large-Scale Brownian Motion Demonstration](https://web.archive.org/web/20220331054344/https://demos.smu.ca/demos/thermo/90-brownian-motion)
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