Statistical systems with well-defined entropy

Tian Qiu, Zhaoyu Fei, Rui Pan +1 more

Entropy is one of the most basic concepts in thermodynamics and statistical mechanics. The most widely used definition of statistical mechanical entropy for a quantum system is introduced by von Neumann. While in classical systems, the statistical mechanical entropy is defined by Gibbs. The relation between these two definitions of entropy is still not fully explored. In this work, we study this problem by employing the phase-space formulation of quantum mechanics. For those quantum states having well-defined classical counterparts, we study the quantum-classical correspondence and quantum corrections of the entropy. We expand the von Neumann entropy in powers of ℏ by using the phase-space formulation, and the zeroth-order term reproduces the Gibbs entropy. We also obtain the explicit expression of the quantum corrections of the entropy. Moreover, we find that for the thermodynamic equilibrium state, all terms odd in ℏ are exactly zero. As an application, we derive quantum corrections for the net work extraction during a quantum Carnot cycle. Our results bring important insights into the understanding of quantum entropy and may have potential applications in the study of quantum heat engines.

Roger Balian

Alfred Wehrl

Jagdish Rai, Suranjana Rai

The entropy of an ensemble of systems is defined in classical statistical mechanics by the relation where k is the Boltzmann constant and p n is the probability of finding the system in state n. It is possible to generalize this definition of entropy to a quantum state ψ(x) with Fourier transform ϕ(p) by defining p n as1 The corresponding entropies for the two quadratures x ^ and p ^ are defined as For the coherent state S ( x ) = S ( p ) and has the minimum value. For the x quadrature squeezed state, S ( x ) decreases at the expense of S ( p ) and vice versa. This definition of entropy has applications in information theory and optical communication using squeezed states.

Yu. L. Klimontovich

Harrison Crecraft

The thermocontextual interpretation (TCI) establishes a system’s exergy, entropic energy, and thermal entropy as thermocontextual properties of state, defined with respect to its positive temperature surroundings. This work extends previous applications of the TCI to irreversible and statistical transitions. The TCI defines statistical entropy as a transactional process of derandomization and transition to a negative-entropy state. Statistical measurements of a confined quantum particle’s position are detailed in terms of reversible processes of instantiation and actualization. The TCI then formalizes the MaxEnt as a fundamental physical principle. We apply MaxEnt and statistical entropy measurements to the double-slit experiment. Particles passing through parallel slits record a wave interference pattern, but a “which-slit” detector eliminates wave interference. Richard Feynman called the double-slit experiment the only mystery, at the heart of quantum mechanics. The TCI offers a simple explanation. The which-slit detector breaks the system’s symmetry, enabling particles to pass through one slit or the other, and MaxEnt selects the asymmetrical transition, which has no wave interference and a higher statistical entropy.