In equilibrium thermodynamics we introduced distribution function : the probability to have momenta around and coordinates around is
If the Brownian motion is the slowest mode in the system, there exists such time that:
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.
Whatever is the state you started from, you will finish at some state with probability 1:
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:
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!):
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).
For Brownian motion we have:
Result (summation over & is implied):
If viscosity is large, we can assume that quickly relaxes to equilibrium value:
Integrating Focker-Planck equation, we can obtain diffusion equation
© 1997 Boris Veytsman and Michael Kotelyanskii