  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 Schr\"{o}dinger 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.
