Differential entropy and time

Abstract

We give a detailed analysis of the Gibbs-type entropy notion and its dynamical behavior in case of time-dependent continuous probability distributions of varied origins: related to classical and quantum systems. The purpose-dependent usage of conditional Kullback-Leibler and Gibbs (Shannon) entropies is explained in case of non-equilibrium Smoluchowski processes. A very different temporal behavior of Gibbs and Kullback entropies is confronted. A specific conceptual niche is addressed, where quantum von Neumann, classical Kullback-Leibler and Gibbs entropies can be consistently introduced as information measures for the same physical system. If the dynamics of probability densities is driven by the Schrödinger picture wave-packet evolution, Gibbs-type and related Fisher information functionals appear to quantify nontrivial power transfer processes in the mean. This observation is found to extend to classical dissipative processes and supports the view that the Shannon entropy dynamics provides an insight into physically relevant non-equilibrium phenomena, which are inaccessible in terms of the Kullback-Leibler entropy and typically ignored in the literature.

Final, unabridged version; http://www.mdpi.org/entropy/ Dedicated to Professor Rafael Sorkin on his 60th birthday

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Cited by in corpus (6)

• On applications of Orlicz Spaces to Statistical Physics
• Heat and work distributions for mixed Gauss-Cauchy process
• Comment on "Time-dependent entropy of simple quantum model systems"
• Delocalization and the semiclassical description of molecular rotation
• The role of interparticle interaction and environmental coupling in a two-particle open quantum system
• Remarks on osmosis, quantum mechanics, and gravity