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[Learn more about Collectives](https://stackoverflow.com/collectives) **Teams** Q\&A for work Connect and share knowledge within a single location that is structured and easy to search. [Learn more about Teams](https://stackoverflow.co/teams/) # [Random walk- Brownian Motion](https://stackoverflow.com/questions/10067325/random-walk-brownian-motion) [Ask Question](https://stackoverflow.com/questions/ask) Asked 13 years, 4 months ago Modified [13 years, 4 months ago](https://stackoverflow.com/questions/10067325/random-walk-brownian-motion?lastactivity "2012-04-09 05:11:17Z") Viewed 9k times 2 I've been assigned the project of creating a simple console app. that models brownian motion in a 2D plane. I wasn't given much information on how to do so (and I'm hoping that it's a pretty popular assignment so that I could get some insight) just that it relies on random number generation. I researched brownian motion for a little bit and saw some formulas that looked complicated, but by the description is just seems to have to move randomly within a certain number interval. Can anyone clarify? Am I to create a program that continually creates a random number in an interval and then modify the particles "x" and "y" coordinate or is there more to it? Thanks for any help. - [c++](https://stackoverflow.com/questions/tagged/c%2B%2B "show questions tagged 'c++'") - [random](https://stackoverflow.com/questions/tagged/random "show questions tagged 'random'") [Share](https://stackoverflow.com/q/10067325 "Short permalink to this question") [Improve this question](https://stackoverflow.com/posts/10067325/edit) Follow [edited Apr 8, 2012 at 23:14](https://stackoverflow.com/posts/10067325/revisions "show all edits to this post") [](https://stackoverflow.com/users/198244/jamessan) [jamessan](https://stackoverflow.com/users/198244/jamessan) 42\.8k88 gold badges8888 silver badges8585 bronze badges asked Apr 8, 2012 at 23:12 [](https://stackoverflow.com/users/699969/sam) [Sam](https://stackoverflow.com/users/699969/sam)Sam 2,34799 gold badges3939 silver badges5454 bronze badges 3 - What, specifically, have you tried or are you having a problem with? – [Chad](https://stackoverflow.com/users/185915/chad "19,152 reputation") Commented Apr 8, 2012 at 23:16 - 2 This sounds like something to ask your TA. – [Michael Petrotta](https://stackoverflow.com/users/23897/michael-petrotta "61,068 reputation") Commented Apr 8, 2012 at 23:16 - Is it full-blown [brownian motion](http://en.wikipedia.org/wiki/Brownian_motion) or the far-simpler [random walk](http://en.wikipedia.org/wiki/Random_walk)? Will you have to migrate the program from random walk to Brownian motion in the next iteration? – [sarnold](https://stackoverflow.com/users/377270/sarnold "104,382 reputation") Commented Apr 8, 2012 at 23:18 [Add a comment](https://stackoverflow.com/questions/10067325/random-walk-brownian-motion "Use comments to ask for more information or suggest improvements. Avoid answering questions in comments.") \| ## 4 Answers 4 Sorted by: [Reset to default](https://stackoverflow.com/questions/10067325/random-walk-brownian-motion?answertab=scoredesc#tab-top) 3 Brownian motion is the result of random air molecules hitting a small particle. Since the sum of a bunch of random forces is unlikely to be exactly 0, and the mass of the particle is so small, it appears to jiggle around, hence Brownian motion. So you get a motion that appears random, but is not uniformly so. The dumb way to model it would be to get a uniform distribution for the direction and Gaussian distribution for the momentum of hundreds of air molecules, apply collisions to a particle, and get the sum. Do this many times and you'll get Brownian-type motion. (The individual air molecules have an average momentum dependent on the temperature, and the number of air molecules is dependent on the pressure.) Note that the resulting motion is not Gaussian, but rather the sum of many samples from the Gaussian distribution. Not sure what it's called. [Share](https://stackoverflow.com/a/10067356 "Short permalink to this answer") [Improve this answer](https://stackoverflow.com/posts/10067356/edit) Follow [edited Apr 8, 2012 at 23:28](https://stackoverflow.com/posts/10067356/revisions "show all edits to this post") answered Apr 8, 2012 at 23:19 user684934user684934 1 - Shouldn't the sum of many samples from the Gaussian (normal) distribution be Gaussian itself? By the central limit theorem, the sampling distribution of the sample mean of *any* distribution approaches the normal distribution as the number of samples grows without bound. – [wchargin](https://stackoverflow.com/users/732016/wchargin "16,097 reputation") Commented Oct 2, 2014 at 3:24 [Add a comment](https://stackoverflow.com/questions/10067325/random-walk-brownian-motion "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| 1 Keep in mind that the random motion is not "uniform", but rather if you plot the frequency of movements vs movement distance you'll see that most are short, some are longer, and a few are very long, creating something resembling an exponential decline. I can't remember what statistical curve the motion observes, but you can probably figure that out, and then you need to craft a random number generator that generates values to fit that curve. What I would do is calculate distance using this RNG, then use a uniform RNG to calculate angle, from zero to 2\*pi, and make the motion polar. You could calculate a random X and a random Y separately, but I'm not sure you'd get the same distribution. [Share](https://stackoverflow.com/a/10067364 "Short permalink to this answer") [Improve this answer](https://stackoverflow.com/posts/10067364/edit) Follow answered Apr 8, 2012 at 23:19 [](https://stackoverflow.com/users/581994/hot-licks) [Hot Licks](https://stackoverflow.com/users/581994/hot-licks)Hot Licks 47\.8k1919 gold badges9696 silver badges155155 bronze badges 2 - Yep, bdares had the distribution -- Gaussian. (Haven't thought about this stuff since 1972.) There are RNGs that will compute Gaussian random numbers directly (or at least there were in 1972). – [Hot Licks](https://stackoverflow.com/users/581994/hot-licks "47,809 reputation") Commented Apr 8, 2012 at 23:21 - 1 C++ has a gaussian distribution for use with its random number library. It's called `std::normal_distribution`. – [bames53](https://stackoverflow.com/users/365496/bames53 "88,566 reputation") Commented Apr 8, 2012 at 23:55 [Add a comment](https://stackoverflow.com/questions/10067325/random-walk-brownian-motion "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| 1 Your question is horribly ill-posed. This almost certainly isn't your fault since your instructor should have pointed out to you that a proper implementation of Brownian motion requires lots and lots of pretty sophisticated specification and analysis of the problem domain even before you get to coding. The precise definition of Brownian motion is probably going to be opaque to you unless you've taken the relevant courses in measure theory. However, there are plenty of resources on the net that give adequate descriptions of Ito processes (of which Brownian motion is an example). If you're interested in coding such a process up, here's a decent tip. At some stage you're going to need to generate random numbers. Almost certainly, you're going to be interested in generating draws from a normal distribution. Thankfully, there are some great ways of doing this available to a C++ programmer. My favourite is to use the Boost.Random library (or the relevant libraries in C++11). The smartest strategy is to use a function object to generate the random variates, probably by using a variate\_generator: ``` #include <iostream> #include <vector> using namespace std; #include <boost/random/mersenne_twister.hpp> #include <boost/random/normal_distribution.hpp> #include <boost/random/variate_generator.hpp> int main() { // Some typedefs to help keep the code clean // Always a good idea when using Boost! typedef boost::mt19937 T_base_prng; typedef boost::normal_distribution<> T_norm_varg; typedef boost::variate_generator<T_base_prng&, T_norm_dist> T_norm_varg; unsigned int base_seed = 42; // Seed for the base pseudo-random number generator double mean = 0.0; // Mean of the normal distribution double stdev = 1.0; // Standard deviation of the normal distribution T_base_prng base_prng(base_seed); // Base PRNG T_norm_dist norm_dist(mean, stdev); // Normal distribution T_norm_varg norm_varg(base_prng, norm_dist); // Variate generator // Generate 1000 draws from a standard normal distribution vector<double> drawVec(1000); for (vector<double>::iterator iter = drawVec.begin(); iter != drawVec.end(); ++iter) { *iter = norm_varg(); } // More stuff... return 0; } ``` Once you get a handle on what a Brownian motion is, it should then be trivial to construct some examples using the functionality in Boost.Random. [Share](https://stackoverflow.com/a/10069154 "Short permalink to this answer") [Improve this answer](https://stackoverflow.com/posts/10069154/edit) Follow answered Apr 9, 2012 at 5:11 [](https://stackoverflow.com/users/834712/pmcs) [pmcs](https://stackoverflow.com/users/834712/pmcs)pmcs 1,1231111 silver badges1616 bronze badges 1 - Probably better for the example to use the standard library rather than boost. – [bames53](https://stackoverflow.com/users/365496/bames53 "88,566 reputation") Commented Apr 9, 2012 at 6:25 [Add a comment](https://stackoverflow.com/questions/10067325/random-walk-brownian-motion "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| 0 Yes, you only need to add the random number to the `x` and `y` coordinates at each time step as follow: ``` int x=0, y=0; for (int t=0; t<N; t++) { x += distribution(gen); y += distribution(gen); display(x, y); } ``` where the distribution can be simple as `{0,1}` , a interval, or a Guassian distribution. **Edit:** For very large N, you can measure whether the mean distance `R = d(x,y)` and check whether it is scaled like `t ~ R^2`. Surely, the above code only generate one Brownian motion, in order for the relationship to hold, you have to repeat many time. Do the experiment yourself. [Share](https://stackoverflow.com/a/10067366 "Short permalink to this answer") [Improve this answer](https://stackoverflow.com/posts/10067366/edit) Follow [edited Apr 8, 2012 at 23:25](https://stackoverflow.com/posts/10067366/revisions "show all edits to this post") answered Apr 8, 2012 at 23:20 [](https://stackoverflow.com/users/535615/unsym) [unsym](https://stackoverflow.com/users/535615/unsym)unsym 2,20011 gold badge2020 silver badges1919 bronze badges 3 - Do note that we like to avoid giving pre-written code directly in response to [homework](https://stackoverflow.com/questions/tagged/homework) questions. (Not that this is drop-in ready, but at least bdares and Hot Licks have further raised the issue of which distribution specifically to pick that I feel they are superior answers.) – [sarnold](https://stackoverflow.com/users/377270/sarnold "104,382 reputation") Commented Apr 8, 2012 at 23:22 - This is not true. How far a Brownian process moves within an interval follows a Gaussian distribution. – [Apprentice Queue](https://stackoverflow.com/users/230436/apprentice-queue "2,036 reputation") Commented Apr 8, 2012 at 23:24 - @ApprenticeQueue: It is statistically true only when the N is very large. For small `N`, you can tell whether it is a Brownian motion – [unsym](https://stackoverflow.com/users/535615/unsym "2,200 reputation") Commented Apr 8, 2012 at 23:26 [Add a comment](https://stackoverflow.com/questions/10067325/random-walk-brownian-motion "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \| Start asking to get answers Find the answer to your question by asking. 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