- Markov Systems
- Chapman-Kolmogorov Equation
- Expansion
- Focker-Planck Equation
- Calculation of Coefficients
- Large Viscosity. Smoluchowski Equation

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*

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''

**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
Whatever is the state you started from, you will finish at some state
with probability 1^{}:

(8) |

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) |

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 thatIntegration:

- 1.
- First term gives
- 2.
- Second term gives:
- 3.
- Third term gives:

(10) |

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).*

For Brownian motion we have:

- 1.
- Linear terms are
- 2.
- Cross terms are zero:
- 3.
- Quadratic in term is:
- 4.
- Quadratic in term is

Result (summation over & is implied):

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

Integrating Focker-Planck equation, we can obtain diffusion equation© 1997

Sun Nov 2 18:50:28 EST 1997