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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:

- The velocity is the slowest mode
- Friction coefficient is

(1) |

(2) |

(3) |

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

(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

(6) |

At large *t*

The large time limit could be obtained from the *Wiener equation*

(7) |

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

© 1997

Sun Nov 2 18:50:28 EST 1997