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We have one particle surrounded by small molecules. At t=0 it had . We want to calculate its position at the time t.
This is called Langevin equation. Langevin equation describes system with white noise acting on .
Let . In equilibrium
Solution of Langevin equation for is
Solution for is
Averages: we know and . The cross-average
At large t
The large time limit could be obtained from the Wiener equation
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!
© 1997 Boris Veytsman and Michael Kotelyanskii