Learning Notes

Stochastic – Brownian Motion Einstein Derivation

Definition of Problem

Consider 1D Brownian motion of a system with n particles. Formulate the profile of particle density f(x, t) where x is the position and t is time.

Annotation

n = Total number of particles in system

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

D = \{D_1, D_2, \dots, D_N\} = A set of random variables denoting the change in x-coordinates of each particle

\phi (\Delta) = Probability density function of each random variables D_i, 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 D_i = \Delta into account. We therefore have [Eq.1]:

\displaystyle f(x, t+\tau) = \int_{-\infty}^{\infty} f(x + \Delta, t) \cdot \phi(\Delta)d\Delta

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

\displaystyle f(x, t+\tau) = f(x, t) + \tau \cdot \frac{\partial f}{\partial t}

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

\displaystyle f(x + \Delta, t) \\ \indent = f(x, t) + [x + \Delta - x]\cdot \frac{\partial f}{\partial x} + \frac{[x + \Delta - x]^2}{2!}\cdot\frac{\partial^2 f}{\partial x^2} + \cdots \\ \indent = f(x, t) + \Delta\cdot \frac{\partial f}{\partial x} + \frac{\Delta^2}{2!}\cdot\frac{\partial^2 f}{\partial x^2} + \cdots

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

\displaystyle f + \tau\frac{\partial f}{\partial t} = \\ \indent \indent f \int_{-\infty}^{\infty}\phi (\Delta)d\Delta +\frac{\partial f}{\partial x} \int_{-\infty}^{\infty}\Delta\cdot\phi (\Delta)d\Delta + \frac{\partial^2 f}{\partial x^2}\int_{-\infty}^{\infty}\frac{\Delta^2}{2!}\cdot\phi (\Delta)d\Delta

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]:

\displaystyle \frac{\partial f}{\partial t}= \zeta \cdot \frac{\partial^2 f}{\partial x^2}

where:

\displaystyle \zeta = \frac{1}{\tau} \int_{-\infty}^{\infty}\frac{\Delta^2}{2!}\cdot\phi (\Delta)d\Delta

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:

\displaystyle f(x, t)=0,  \{\forall x \neq 0 \text{ and } t=0\} and \displaystyle \int_{-\infty}^{\infty} f(x, t) dx = n

gives the solution [Eq.6]:

\displaystyle f(x, t) = \frac{\exp ({-x^2/4\zeta t})}{\sqrt{t}}

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 \sqrt{2\zeta t}

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.

See also

Stochastic Process

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