# Definition of Problem

Consider 1D Brownian motion of a system with infinite amount of particles in equilibrium state. Derive the profile of particle density where

xis the position andtis time.

# Annotation

*k *= Boltzmann’s constant

*T* = Temperature

*η *= Viscosity constant of fluid

*a* = Diameter of particles considered (Assumed spherical shape)

*X *= Random variable which denotes the a fluctuating force

*v* = First time derivative of *x , *namely velocity

# Derivation

Consider two force acting on the particles:

1) A varying force *X*.

2) A drag force caused by viscosity dependent on particle velocity *v*:

Through statistical mechanics, it is well established that the mean K.E. in equilibrium is proportional to the temperature of the system [Eq.1].

Then we can write Newton’s second law of motion as follow:

Inspired by [Eq.1], multiply both sides by *x *and get [Eq.2]:

consider the following equations:

and taking time average of both sides (reducing *X* term to zeros) of [Eq.2], we have:

This equation is called **stochastic differential equation**. The general solution of the equation is:

Observations by Langevin suggest the exponential term of the equation approaches zeros rapidly with a time constant of order 10^-8, so it is insignificant if we are considering time average. To recover Einstein’s result, integrate this one more time:

# Reference

Gardiner, Crispin W.

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

# See Also

Stochastic – Brownian Motion Einstein Derivation

Stochastic – Python Example of a Random Walk Implementation