A Gas of Brownian Particles in Statistical Dynamics

Preprint prepared: 27/5/1996;

by R. F. Streater, J. Stat. Phys., 88, 447-469, 1997.

ABSTRACT

We consider a gas of particles on a one-dimensional lattice in interaction with a heat-particle; the latter is located on the bond linking the position of the particle to the point to which it jumps. The energy of the particle is given by a potential V(x), x in Z. In the continuum limit, the classical version leads to Brownian motion with drift. A quantum version leads to a drift velocity which is independent of the applied force (but is in the same direction as the force). Both these models obey Einstein's relation between drift, diffusion and applied force. The system obeys a pair of coupled non-linear heat equations, one for the density of the Brownian particle and one for the heat occupation number; together they obey the first and second laws of thermodynamics. The equation for the Brownian particle can be expressed as a non-linear stochastic differential equation, whose terms depend on the initial distribution.

The paper was reviewed by Lorenza Viola.

REMARKS

In a subsequent paper I show that for one of these models there exist solutions for small time if the initial conditions are suitable. Further work has been done by Piotr Biler, Greg Karch, and Tad Nadzieja. See the paper by Biler and Nadzieja listed here.