Subsections

Langevin Equation

Problem

We have one particle surrounded by small molecules. At t=0 it had . We want to calculate its position at the time t.

Assumptions:

• The velocity is the slowest mode
• Friction coefficient is

Equation

By definition,
 (1)
We already obtained equation for :
 (2)
with random force :
 (3)

This is called Langevin equation. Langevin equation describes system with white noise acting on .

Solution

Let . In equilibrium
 (4)

Solution of Langevin equation for is
 (5)

Solution for is

1.
Integration of the first term in (5):

2.
Second term--by parts:

We obtained:

Averages: we know and . The cross-average

1.
First term gives

2.
Second term gives

Interpretation

We obtained:
 (6)
At small times

At large t

The large time limit could be obtained from the Wiener equation
 (7)
with white noise :

In the limit of small t Wiener equation does not work!

White noise in Wiener equation is the consequence of the averaging out a fast process--in our case, velocity relaxation!

Next: Diffusion Approach. Focker-Planck Equation Up: Brownian Motion and Focker-Planck Previous: Brownian Motion and Focker-Planck

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