Information Theory and Statistical Mechanics. II

TLDR

Irreversible processes cannot be fully described by differential rate equations based on time‑proportional transition probabilities. The study extends statistical inference in statistical mechanics to the density‑matrix formalism, applying it to irreversibility and information loss, and examines.

Abstract

Treatment of the predictive aspect of statistical mechanics as a form of statistical inference is extended to the density-matrix formalism and applied to a discussion of the relation between irreversibility and information loss. A principle of "statistical complementarity" is pointed out, according to which the empirically verifiable probabilities of statistical mechanics necessarily correspond to incomplete predictions. A preliminary discussion is given of the second law of thermodynamics and of a certain class of irreversible processes, in an approximation equivalent to that of the semiclassical theory of radiation.It is shown that a density matrix does not in general contain all the information about a system that is relevant for predicting its behavior. In the case of a system perturbed by random fluctuating fields, the density matrix cannot satisfy any differential equation because $\stackrel{\ifmmode \dot{}\else \'{} {}.\fi{}}{\ensuremath{\rho}}(t)$ does not depend only on $\ensuremath{\rho}(t)$, but also on past conditions The rigorous theory involves stochastic equations in the type $\ensuremath{\rho}(t)=\mathcal{G}(t, 0)\ensuremath{\rho}(0)$, where the operator $\mathcal{G}$ is a functional of conditions during the entire interval ($0\ensuremath{\rightarrow}t$). Therefore a general theory of irreversible processes cannot be based on differential rate equations corresponding to time-proportional transition probabilities. However, such equations often represent useful approximations.