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4 Hydrodynamic model Although the VACF of a Brownian particle was never explicitly measured in the first half of the twentieth century due to experimental limitation, it was widely believed to decay exponentially. When a new era of computational science began in the 1950s, this belief was put to the test and it marked the failure of the molecular chaos assumption. 72 4.1 Observation of algebraic decay in VACFs Using molecular dynamics (MD) simulations, some pioneers started to realize that the VACF of molecules does not follow strictly an exponential decay, but has a slowly decreasing characteristic. This long persistence was found in fluids described by both the Lennard-Jones potential 73 , 74 and the hard-core potential. 75 , 76 A milestone took place in 1970 when Alder and Wainwright 77 delivered a definite answer for the long persistence of the VACF as an algebraic decay, that is, C ( t ) ~ t − d /2 for t → ∞. Here d is the dimension of the problem. Meanwhile this scaling was confirmed by independent numerical simulations of Navier–Stokes equations, which indicate that a (transient) vortex flow pattern forms around a tagged particle. 76 , 77 These observations from computer simulations led to many intriguing questions as to what is missing in the Langevin model. The most suspicious assumption of the Langevin model (and also of the Einstein model) is probably that the friction coefficient ξ is taken as the solution of the steady Stokes flow, whereas a Brownian particle undergoes erratic movements constantly. Therefore, the steady friction may be valid only if the surrounding fluid becomes quasi-steady immediately after each movement, or less strictly, before the relaxation time τ B = m / ξ of the Brownian particle. This deficiency was already pointed out in the early lectures of Hendrik Lorentz: ** 78 ξ = 6 πηa is a good approximation only when the mass density ratio ρ / ρ B of the fluid and the Brownian particle is so small that the fluid inertia is negligible. We shall discuss later why this is true. Since the seminal work of Alder and Wainwright, it was very soon widely acknowledged that unsteady hydrodynamics plays a significant role in the dynamics of the Brownian particle. This motivated many theoretical physicists to work on this subject from various perspectives, and so the algebraic decay was understood by several approaches: a purely hydrodynamic approach based on the linearized Navier–Stokes equations, 80 , 81 a generalized Langevin equation approach based on the fluctuating hydrodynamics, 82 – 84 the mode-coupling theory, 85 – 87 and the kinetic theory. 88 Although these methodologies have different perspectives and mathematical sophistication, all of them respect the inertia of the surrounding fluid and corroborated the same scaling of the asymptotic decay on the VACF. 89 The bold assumption of quasi-steady state in the Langevin model can be examined only if we consider the unsteady solution of the hydrodynamics, which has been available for more than a century from the independent works of Basset and Boussinesq. 4.2 Boussinesq–Basset force For a spherical particle undergoing unsteady motion influenced by the inertia of the surrounding fluid, its resistant force was known to Boussinesq and Basset: 90 – 93 where M = 4 3 π a 3 ρ is the mass of the fluid displaced by the particle. Note that eqn (28) is obtained by linearizing (dropping the v ·∇ v term) the incompressible Navier–Stokes equations together with the no-slip boundary condition on the particle. For a stationary motion v ˙ ( t ) = 0 , only the first term on the right-hand side remains, which is just the Stokes friction in eqn (11) . The second term is due to the added mass of an inviscid incompressible origin, while the third term is the memory effect of the viscous force from the retarding fluid, which is referred to as the Bousinesq–Basset force. Now let us discuss when the Bousinesq–Basset force becomes as important as the Stokes friction. Since the former is expressed as a convolution integral, we may understand it better in the frequency domain. By taking the Laplace transform of eqn (28) , that is, F ( ω ) = ∫ 0 ∞ e − ω t F ( t ) d t , we obtain F ( ω ) = − ξ ( ω ) v ( ω ) with 82 From the transformation, we note that any model with only the steady friction should be considered to be a zero-frequency theory. 80 If we compare the first and third terms on the right-hand side of eqn (29) , the latter becomes larger than the former for frequency ω > η / ρa 2 , or equivalently for time t < ρa 2 / η . Since the relaxation time in Langevin eqn (11) is τ B = m / ξ = 2 ρ B a 2 /9 η , the fluid inertia has non-negligible effects on the dynamics of the Brownian particle for t < (9 ρ /2 ρ B ) τ B . Hence, if 9 ρ /2 ρ B ≪ 1, the fluid inertia is negligible, which also confirms the insightful remark made earlier by Lorentz. Alternatively, we may realize the significance of the fluid inertia more directly by considering the vorticity ω = ∇ × u , which satisfies the diffusion equation ∂ ω /∂ t = v ∇ 2 ω , 94 where the kinematic viscosity v = η /ρ. The time scale for the vorticity to travel a distance of the radius of the Brownian particle is τ v = a 2 / v . For the Langevin model to be valid, it must be τ v ≪ τ B or 9 ρ /2 ρ B ≪ 1 so that the transient behavior of the fluid plays a negligible role in the particle dynamics. This hydrodynamic argument is also in agreement with the analysis of the molecular theory. 70 In summary, while the Langevin equation provides a fair approximation for 9 ρ /2 ρ B ≪ 1, e.g. , a dense particle in gas, it does not apply well to the case of 9 ρ /2 ρ B ~ 1, for example, a pollen particle in water, that is the historic observation recorded by Robert Brown. 4.3 Generalized Langevin equation Now that the importance of the fluid inertia is recognized, we may discuss the equation of motion for the Brownian particle. For a rigid particle suspended in a continuum fluid described by the fluctuating hydrodynamics, 93 the following generalized Langevin equation can be formulated: 83 , 84 Compared to the original Langevin eqn (20) , eqn (30) is non-Markovian as the friction force is history-dependent. The memory kernel ξ ( t ) is the inverse Laplace transform of eqn (29) . In addition, the random force F ∼ ( t ) is non-white or colored, which can be observed via the fluctuation-dissipation relation 57 At first glance, eqn (30) seems to be simple. We note, however, that the form is quite general and all the complicated information is hidden in the memory kernel ξ ( t ) or in the statistics of the random force F ∼ ( t ) . Although theoretically well known, the colored power spectral density of the thermal noise, which is the Fourier transform of eqn (31) , has been confirmed by experiments only recently. 95 , 96 We also note that the same form of equation as eqn (30) can be obtained from microscopic equations of motion for a Hamiltonian fluid through the Mori–Zwanzig formalism. 97 – 102 In fact, the emergence of a non-Markovian process is a typical scenario when insignificant variables (fast fluid variables in our case) are eliminated in a Markovian process under coarse-graining. 54 4.4 Heuristic derivations of the algebraic decay Here, we discuss how the algebraic decay appears in the generalized Langevin eqn (30) , and how it can be explained from a hydrodynamic perspective. The first question can be answered by deriving a differential equation that the VACF C ( t ) = 〈 v (0)· v ( t )〉 satisfies. After multiplying eqn (30) by v (0) and taking averages, we obtain the Volterra equation (also known as the memory function equation 103 ) It is known that if either C ( t ) or ξ ( t ) decays algebraically, then the other also decays algebraically with the same power law and the opposite sign. 104 From the ω term of ξ (ω) in eqn (29) , we know that ξ ( t ) decays like t −3/2 with negative values at large time t . Therefore, it is expected that C ( t ) also decays like t −3/2 but with positive values at large time t . This mathematical argument shows that no matter how small ρ / ρ B is, the asymptotic decay of the VACF is always algebraic rather than exponential. However, for smaller ρ / ρ B , the exponential decay yields to algebraic decay later in time and the Langevin model becomes a better approximation. The persistent scaling of the VACF can also be easily understood by a heuristic hydrodynamic argument. Suppose a particle has initial velocity v 0 , due to viscous diffusion, after time t , a vortex ring ( d = 2) or shell ( d = 3) with radius r ~ v t develops. The total mass within the influenced zone is M * ~ ρr d . If the surrounding fluid is entrained and moves with the particle at time t , by momentum conservation we have v ( t ) = m v 0 M ∗ ~ m v 0 ρ ( v t ) − d / 2 . Then, it is simple to see that C ( t ) ~ ( vt ) − d /2 . The argument above assumes that the particle does not move when the vortex forms. If the particle moves evidently as the vortex develops, we may still extend this hydrodynamic argument by adding in the self-diffusion constant D of the tagged particle into the scaling so that we have C ( t ) ~ [( v + D ) t ] − d /2 . In fact, by introducing the evolution of the probability distribution function of the tagged particle, the following expression was derived rigorously (one-dimensional case): 87 This power law scaling is demonstrated by dissipative particle dynamics simulations in Fig. 2 . Fig. 2. Asymptotic limit of the velocity autocorrelation function for a diffusive particle. Eqn (33) with or without diffusion coefficient D is compared with the results of tagged fluid particles in dissipative particle dynamics (DPD) simulations. The inset shows the long-time limit in the logarithmic scale. Input parameters of DPD are taken from a previous work, 105 , 106 which correspond to a fluid with k B T = 1, ρ = 3, v = 0.54, and D = 0.15 in DPD units. If the momentum diffusion is much stronger than the mass diffusion or if the Schmidt number Sc = v / D is very large ( e.g. , a solid particle suspended in a liquid), we can ignore the contribution of D . Under this condition, which is favored by the linearized hydrodynamics, the full expression of C ( t ) was derived from the fluctuating hydrodynamics of an incompressible fluid for a neutrally buoyant particle: 83 , 107 Other than the integral form of eqn (34) , an alternative closed form of C ( t ) is also available. 82 , 108 , 109 We compare the VACF from the hydrodynamics theory with that of Langevin’s model in Fig. 3(a) . We observe that the Langevin model underestimates the decay rate of the VACF at short time ( t ≲ τ v ) while overestimates it at long time ( t ≳ τ v ). 4.5 Diffusion coefficient and mean-squared displacement The time-dependent diffusion coefficient D ( t ) of a Brownian particle can be obtained directly by integrating eqn (34) as shown in eqn (27) . Furthermore, the MSD may also be obtained by further integrating D ( t ) or directly from the VACF as 110 , 111 The non-diffusive signatures of the MSD and the time-dependent diffusion coefficient due to hydrodynamic memory have been validated for Brownian particles in a suspension probed by dynamic light scattering. †† 107 , 111 More recently, to avoid any (weak) hydrodynamic interactions between particles, optical trapping interferometry has been applied to a single micrometer particle 112 which is trapped in a weakly harmonic potential. 113 Consequently, the hydrodynamic theory for the non-diffusive regime has been explicitly confirmed with excellent accuracy. 112 We compare the time-dependent diffusion coefficients and MSDs from different theoretical models in Fig. 3(b) and (c) . We observe that the D ( t ) from Langevin’s model approaches exponentially fast to Einstein’s diffusion coefficient, whereas it takes a substantially longer time for the hydrodynamic model to reach a plateau value. It is worth noting that even when the fluid inertia is important for the dynamics such as the asymptotic decay of C ( t ) of the Brownian particle, the equation for the diffusion coefficient D ∞ = ∫ 0 ∞ C ( t ) d t = k B T / ξ always holds. This means that the steady motion or the zero-frequency mobility component provides the largest displacement and dominates the diffusive process. 83 , 109 Therefore, the Stokes–Einstein–Sutherland formula in eqn (9) is still correct for a diffusive process, which is universally captured by Einstein’s model, Langevin’s model and the hydrodynamic model. 4.6 Limitations and underlying assumptions The heuristic approach above assumes that the long-time decay of the VACF for the particle is solely affected by the dynamics of vortex formation driven by the transversal component of the hydrodynamic equations. 89 , 116 The longitudinal component drives compressibility effects, which vanish in a sonic time scale, and therefore, they do not contribute to the long-time behavior of the dynamics. 87 If the short-time dynamics is of interest, the compressibility should be reconsidered. When the fluid is considered mathematically to be incompressible, the particle mass m is augmented by an induced mass M /2, where M is the mass of the fluid displaced by the particle. 93 Due to this mathematical treatment, for any infinitesimal time δt , C ( δt ) = k B T /( m + M /2). However, the equipartition theorem requires that C ( t ) starts with C (0) = k B T / m . Therefore, the incompressible assumption generates a discontinuity of C ( t ) at short time and violates the equipartition theorem of statistical physics. ‡‡ 117 , 118 A similar paradox was recognized when inverse-transforming eqn (29) to get ξ ( t ), which is singular at t = 0 and leads to a substantial difference between v (0) and v ( δt ) in the case of impulsive particle motion. 89 , 93 The unphysical consequences at short time may be alleviated by realizing that every fluid is (slightly) compressible. Therefore, we may find a reconciliation of the dynamics from short to long time by considering the propagation of sound waves and incorporating a frequency dependent friction at a frequency similar to the inverse of the sound speed c s . 80 , 117 , 118 For a neutrally buoyant particle, the sound wave dissipates 1/3 of the total energy and the contribution on the VACF from the compressibility effects reads 109 , 117 , 118 We may see in Fig. 3(a) that adding the compressible correction of eqn (36) to the incompressible VACF of eqn (34) indeed respects the equipartition theorem at short time. The effects of the compressibility are not so apparent for the diffusion coefficient or MSD, as indicated in Fig. 3(b) and (c) . Another interesting phenomenon at the short-time scale due to sound propagation is the “backtracking”, which may contribute negatively to the overall friction experienced by the particle. 120 , 121 From the ratio of the added mass and the particle mass M 2 m = ρ 2 ρ B , it is simple to see that for a lighter fluid the compressibility becomes less important for the particle dynamics. Similarly any viscoelasticity effects may be incorporated into the generalized friction at a different frequency after introducing a new relaxation time scale. 80 Moreover, one would need to select a suitable viscoelastic model and also determine its relaxation time by other means. The problem is that viscoelasticity includes a vast range of time scales, but most models do not. The hydrodynamic theory is based on continuum-fluid mechanics, which necessarily cannot resolve the ballistic motion over δt > 0 accurately. This fact is indicated in the inset of Fig. 3(c) , where the Langevin model shows a finite period for the ballistic regime, whereas the hydrodynamic model deviates from it quickly. In the hydrodynamic model (also in Langevin and Einstein models), we consider only the continuous friction such as the Stokes or Bousinesq–Basset drag on the particle, but ignore the Enskog friction on the Brownian particle due to molecular collisions with the solvent. 122 , 123 Here we focused on the translational motion of a single spherical particle with the no-slip boundary condition. There are various extensions based on this simple scenario. For example, for a sphere with the slip or partial-slip boundary condition, the magnitude but not the scaling of the asymptotic decay changes. 80 The dynamics for a particle with an arbitrary shape can be formulated as a similar problem. 83 , 108 , 124 The VACF of the angular velocity for a rotating particle may also be calculated with an asymptotic behavior as C R ( t ) ∝ t −5/2 , §§ 83 , 125 , 126 and the non-spherical shape alters only its magnitude but not the power law. 127 For a test particle immersed in a suspension of particles, the asymptotic power law does not change and its magnitude is obtained by replacing the fluid viscosity with the suspension viscosity. 128 , 129 The unsteady equation of motion for a sphere in a nonuniform flow is also available. 130 For a Brownian particle of molecular size, the value of its radius or slip length on the surface is always conceptually subtle in a continuum description 131 and needs extra care.