A GAS OF BROWNIAN PARTICLES IN STATISTICAL DYNAMICS

We consider a large number of particles on a one-dimensional lattice l Z in interaction with a heat particle; the latter is located on the bo nd linking the position of the particle to the point to which it jumps . The energy of a single particle is given by a potential V(x), x is a n element of Z. In the continuum limit, the classical version leads to Brownian motion with drift. A quantum version leads to a local drift velocity which is independent of the applied force. Both these models obey Einstein's relation between drift, diffusion, and applied force. The system obeys the first and second laws of thermodynamics, with the time evolution given by a pair of coupled non linear heat equations, one for the density of the Brownian particles and one for the heat occ upation number; the equation for a tagged Brownian particle can be wri tten as a stochastic differential equation.