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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.
Assumptions:
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(1) |
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(2) |
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(3) |
This is called Langevin equation. Langevin equation describes
system with white noise acting on .
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(4) |
Solution of Langevin equation for is
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(5) |
Solution for is
We obtained:
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(6) |
At large t
The large time limit could be obtained from the Wiener equation
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(7) |
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