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 x is the position and t is time.
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
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:
Gardiner, Crispin W. Stochastic methods. Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1985.