On the Microscopic Conditions for Linear Macroscopic Laws

Abstract

We have investigated the conditions which must be imposed on the microscopic equations of motion to obtain exact linear laws for macroscopic (phase averaged) variables. The starting point in this study has been the lowest order master equation (Pauli equation) which is a linear microscopic equation in the state probabilities with a time‐independent transition matrix. Discrete and continuous variable master equations as well as their multivariate generalizations have been considered. In the case of continuum state variables, we have used various Fokker‐Planck equations and their corresponding Langevin equations as our starting microscopic equation of motion. In each case the conditions which must be imposed to obtain linear macroscopic transport equations have been derived and discussed. The problem of the derivation of linear macroscopic laws from microscopic laws which are nonlinear in the dynamical variables has been discussed in the context of our results. We find that exact linear macroscopic laws can be derived from microscopic laws only when stringent conditions are imposed on the form of microscopic transition rates.