Entropic Functionals in Quantum Statistical Mechanics

Uncertainty relations have become the trademark of quantum theory since they were formulated by Bohr and Heisenberg. This review covers various generalizations and extensions of the uncertainty relations in quantum theory that involve the R\'enyi and the Shannon entropies. The advantages of these entropic uncertainty relations are pointed out and their more direct connection to the observed phenomena is emphasized. Several remaining open problems are mentioned

A microscopic understanding of the thermodynamic entropy in quantum systems has been a mystery ever since the invention of quantum mechanics. In classical physics, this entropy is believed to be the logarithm of the volume of phase space accessible to an isolated system [1]. There is no quantum mechanical analog to this. Instead, Von Neumann's hypothesis for the entropy [2] is most widely used. However this gives zero for systems with a known wave function, that is a pure sta...

These notes provide a brief primer on the basic aspects of "observational entropy" (also known as "quantum coarse-grained entropy"), a general framework for applying the concept of coarse-graining to quantum systems. We review the basic formalism, survey applications to thermodynamics, make a connection to quantum correlations and entanglement entropy, compare to the corresponding classical theory, and discuss a generalization based on POVM measurements.

The formulation of quantum mechanics within the framework of entropic dynamics includes several new elements. In this paper we concentrate on one of them: the implications for the theory of time. Entropic time is introduced as a book-keeping device to keep track of the accumulation of changes. One new feature is that, unlike other concepts of time appearing in the so-called fundamental laws of physics, entropic time incorporates a natural distinction between past and future.

We propose an approach to the realization of many-body quantum state distributions inspired by combined principles of thermodynamics and mesoscopic physics. Its essence is a maximum entropy principle conditioned by conservation laws. We go beyond traditional thermodynamics and condition on the full distribution of the conserved quantities. The result are quantum state distributions whose deviations from `thermal states' get more pronounced in the limit of wide input distribut...

This work is concerned with the minimization of quantum entropies under local constraints of density, current, and energy. The problem arises in the work of Degond and Ringhofer about the derivation of quantum hydrodynamical models from first principles, and is an adaptation to the quantum setting of the moment closure strategy by entropy minimization encountered in kinetic equations. The main mathematical difficulty is the lack of compactness needed to recover the energy con...

We present a particular approach to the non-equilibrium dynamics of quantum field theory. This approach is based on the Jaynes-Gibbs principle of the maximal entropy and its implementation, throghout the initial value data, into the dynamical equations for Green's functions.

What is the major difference between large and small systems? At small length-scales the dynamics is dominated by fluctuations, whereas at large scales fluctuations are irrelevant. Therefore, any thermodynamically consistent description of quantum systems necessitates a thorough understanding of the nature and consequences of fluctuations. In this chapter, we outline two closely related fields of research that are commonly considered separately -- fluctuation forces and fluct...

We study a version of the generalized (h, {\phi})-entropies, introduced by Salicr\'u et al, for a wide family of probabilistic models that includes quantum and classical statistical theories as particular cases. We extend previous works by exploring how to define (h, {\phi})-entropies in infinite dimensional models.

We gave a simple derivation of density operator with the quantum analysis. We dealt with the functional of a density operator, and applied maximum entropy principle. We obtained easily the density operators for the Tsallis entropy and R\'enyi entropy with the $q$-expectation value (escort average), and also obtained easily the density operators for the Boltzmann-Gibbs entropy and the Burg entropy with the conventional expectation value. The quantum analysis works effectively ...