Generalizations of statistical mechanics on the basis of an incomplete information theory

Qiuping Alexandre Wang

Although Gödel's incompleteness theorem made mathematician recognize that no axiomatic system could completely prove its correctness and that there is an eternal hole between our knowledge and the world, physicists so far continue to work on the approaches based on the hypothesis to completely or approximately know the systems of interest. In this paper, however, I review the recent development of a different approach, a statistical theory based upon the notion of incomplete information. This consideration leads to generalized statistical mechanics characterized by an incompleteness parameter which equals unity when information is complete. The mathematical and physical bases of the information incompleteness are discussed. The application of the concomitant incomplete quantum distribution to correlated electron systems is reviewed.

A. Katz

E. T. Jaynes

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.

Marco Baldovin, Giacomo Gradenigo, Angelo Vulpiani +1 more

R. Blankenbecler, M. Hossein Partovi

Noting that quantum measurements are in general incomplete, we develop, starting from a recent entropic formulation of uncertainty, a maximum uncertainty principle to define the statistical mechanics of microscopic systems. The resulting ensemble entropy coincides with the expression of von Neumann, thus providing a unified, quantum basis for statistical physics of all systems. Examples involving momentum and position measurements are discussed.

Paul A. Benioff