#### 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 $\stackrel{\u0307}{\rho}\mathrm{}\left(t\right)\mathrm{}$ does not depend only on $\rho \mathrm{}\left(t\right)\mathrm{}$, but also on past conditions The rigorous theory involves stochastic equations in the type $\rho \mathrm{}\left(t\right)=\mathcal{G}(t,0)\rho \mathrm{}\left(0\right)\mathrm{}$, where the operator $\mathcal{G}$ is a functional of conditions during the entire interval ($0\to t$). 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.

- Received 15 March 1957

DOI:https://doi.org/10.1103/PhysRev.108.171

©1957 American Physical Society