Definition of Problem
Consider 1D Brownian motion of a system with n particles. Formulate the profile of particle density where x is the position and t is time.
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
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]:
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:
gives the solution [Eq.6]:
[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
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.