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Subsections

# Diffusion Approach. Focker-Planck Equation

## Markov Systems

### Time-dependent Distribution Function

In equilibrium thermodynamics we introduced distribution function : the probability to have momenta around and coordinates around is

• We will study one Brownian particle--one coordinate and one velocity
• We want a time-dependent solution--we must introduce t
We introduce a time-dependent distribution function --the probability to be in certain place with certain velocity at certain time!

### Transfer Probabilities

If the Brownian motion is the slowest mode in the system, there exists such time that:

1.
is large enough, so there are many molecular collisions during this time
2.
is small enough, so and do not change much''
We know the state of our particle at time t: and . What is its state at the time ?

Assumption:
The state of the system at the moment depends only on the state at the moment t. The system has short memory.
Definition:
A system satisfying this assumption is called Markov system. We assume Brownian motion to be Markovian.

We introduce --the probability to go from state 1 to state 2:

or, introducing the variable

The function is called transfer function. The evolution of Markov system depends only on transfer function and initial state.

### Property of the Transfer Function

Whatever is the state you started from, you will finish at some state with probability 1:
 (8)

## Chapman-Kolmogorov Equation

What is the probability to find particle at the point u at the time ? If it was at the point at time t, the probability is . The probability to be at was . We obtain:
 (9)
This is called Chapman-Kolmogorov (or master) equation.

## Expansion

Now we will use the fact that is small. In the left hand side of (9) we have:

In the right hand side (do not forget that u is a vector!):

Integration:

1.
First term gives

2.
Second term gives:

3.
Third term gives:

## Focker-Planck Equation

We obtained:
 (10)
with drift coefficients

and diffusion coefficients

This is called Focker-Planck equation. It describes diffusion in phase space.

The evolution is described by the moments . In the Focker-Planck equation we use only the first and second moments. In principle, we can construct an approximation based on higher moments (Kramers, 1940).

## Calculation of Coefficients

For Brownian motion we have:

1.
Linear terms are

2.
Cross terms are zero:

3.

4.

Result (summation over & is implied):

with

## Large Viscosity. Smoluchowski Equation

If viscosity is large, we can assume that quickly relaxes to equilibrium value:

Integrating Focker-Planck equation, we can obtain diffusion equation

Up: Brownian Motion and Focker-Planck Previous: Langevin Equation

© 1997 Boris Veytsman and Michael Kotelyanskii
Sun Nov 2 18:50:28 EST 1997