Noise term η(t) in Langevin equation

[xiaoding]

Langevin equation describes the brown motion. But I don't understand the nose term η(t) in the equation. What's the relationship between η(t) and the force proportional to the velocity due to stoke's law? I mean they both belong to the force between the collisions with the molecules of the fluid. So η(t) is just the stochastic part of the force due to stoke's law?

 

[vanhees71]

The Langevin equation is a stochstic differential equation. Physically it applies to the motion of a heavy object (like a pollen in water) which interacts with light particles such that a single collision of the heavy object with a light matter particle (here water molecules) has little impact on it. You need a lot of collisions to make a macroscopically noticable effect.

The important point is that this situation has a clear separation of time scales. On the one hand you have the time between two collisions of the medium particles with the heavy object which is pretty short compared to the time scales over which the heavy object moves a noticable macroscopic distance. The microscopic time scale is small compared to this macroscopic time scale.

On the average momentum transfer from the heavy particle to the medium is described by friction. Without any external forces you have
\frac{\mathrm{d}}{\mathrm{d t}} \langle p \rangle=-\gamma \langle p \rangle.
This is a usual differential equation for the average momentum of the heavy particle.

The Langevin equation takes into account also fluctuations of the force from the many random collisions per macroscopic time step. It reads
\mathrm{d} p=-\gamma p \mathrm{d} t+\sqrt{\mathrm{d t} D} \eta.
Here \eta is a Gaussian-normal distributed uncorrelated random variable with the properties
\langle{\eta(t)} \rangle=0, \quad \langle{\eta(t) \eta(t')} \rangle=\delta(t-t').
You can show that this equation is equivalent to the Focker-Planck equation for the phase-space distribution function. You find more details about the derivation and properties of the Langevin equation in one of my papers on heavy quarks in the Quark Gluon Plasma. There it's derived for the relativistic motion, but the general arguments are the same for relativistic and non-relativistic situations:

http://fias.uni-frankfurt.de/~hees/publ/hq-qgp4-bibtex.pdf