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