**Review:**

**Differential entropy and time**

**Piotr Garbaczewski**

Institute of Physics, University of Zielona Góra,
ul. Szafrana 4a, 65-516 Zielona Góra, Poland

E-mail: p.garbaczewski@if.uz.zgora.pl

*Received: 22 August 2005 / Accepted: 17 October 2005 / Published: 18 October 2005*

**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.

**Keywords:**
differential entropy; von Neumann entropy; Shannon
entropy; Kullback-Leibler entropy; Gibbs entropy; entropy methods;
entropy functionals; Fisher information; Ornstein-Uhlenbeck
process; Schrödinger picture evolution; dynamics of
probability densities; invariant density.

**PACS codes:** 05.45.+b, 02.50.-r, 03.65.Ta, 03.67.-a

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