# Definition of Problem

Consider 1D Brownian motion of a system with

nparticles. Formulate the profile of particle density wherexis the position andtis time.

# Annotation

*n* = Total number of particles in system

*τ* = Infinitesimal time interval between each change of profile *f(x, t)*

*D* = = A set of random variables denoting the change in x-coordinates of each particle

= Probability density function of each random variables , note that this is an even function

# Derivation

Let *v = f(x, t) *be the particle density. Consider that the change in profile *f(x, t) *to *f(x, t + τ) *is equivalent to taking all the changes of x-position of particles into account. We therefore have [Eq.1]:

And because *τ* is infinitesimal, we can write the derivative [Eq.2]:

In the next step, expand *f(x + Δ, t) *about the point *x + Δ *= *x* [Eq.3]* :*

Combining the three equations, and lose the insignificant terms of [Eq.3], obtain [Eq.4]:

Because *φ(Δ) *is even, the even terms of [Eq.4] vanish and the first integral sums to 1 by definition (normalization constrain of probability), we have the partial deferential equation [Eq.5]:

where:

Solving this equation [Eq.5] for the initial conditions where infinite particles diffuse from single point (*x* = 0) with the constrain of particle conservation, which are expressed as:

and

gives the solution [Eq.6]:

# Side Notes

[Eq.5] is actually the famous **equation of diffusion** for continuous time and continuous space, also known as *Fokker-Planck equation *or *Kolmogorov’s equation*.

We can also know from [Eq.6] that the mean square velocity of the particles is

# Reference

Gardiner, Crispin W.

Stochastic methods. Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1985.Einstein, Albert.

Investigations on the Theory of the Brownian Movement. Courier Corporation, 1956.

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