Exact Conditions for the Preservation of a Canonical Distribution in Markovian Relaxation Processes

Abstract

Necessary and sufficient conditions have been determined for the exact preservation of a canonical distribution characterized by a time‐dependent temperature (canonical invariance) in Markovian relaxation processes governed by a master equation. These conditions, while physically realizable, are quite restrictive so that canonical invariance is the exception rather than the rule. For processes with a continuous energy variable, canonical invariance requires that the integral master equation is exactly equivalent to a Fokker‐Planck equation with linear transition moments of a special form. For processes with a discrete energy variable, canonical invariance requires, in addition to a special form of the level degeneracy, equal spacing of the energy levels and transitions between nearest‐neighbor levels only. Physically, these conditions imply that canonical invariance is maintained only for weak interactions of a special type between the relaxing subsystem and the reservoir. It is also shown that canonical invariance is a sufficient condition for the exponential relaxation of the mean energy. A number of systems (hard‐sphere Rayleigh gas, Brownian motion, harmonic oscillators, nuclear spins) are discussed in the framework of the above theory. Conditions for approximate canonical invariance valid up to a certain order in the energy are also developed and then applied to nuclear spins in a magnetic field.