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Entropy

On Quantum Superstatistics and the Critical Behavior of Nonextensive Ideal Bose Gases

Octavio Obregón, José Luis López, Marco Ortega-Cruz

Published Oct 9, 2018• 6 citations

Abstract

We explore some important consequences of the quantum ideal Bose gas, the properties of which are described by a non-extensive entropy. We consider in particular two entropies that depend only on the probability. These entropies are defined in the framework of superstatistics, and in this context, such entropies arise when a system is exposed to non-equilibrium conditions, whose general effects can be described by a generalized Boltzmann factor and correspondingly by a generalized probability distribution defining a different statistics. We generalize the usual statistics to their quantum counterparts, and we will focus on the properties of the corresponding generalized quantum ideal Bose gas. The most important consequence of the generalized Bose gas is that the critical temperature predicted for the condensation changes in comparison with the usual quantum Bose gas. Conceptual differences arise when comparing our results with the ones previously reported regarding the q-generalized Bose–Einstein condensation. As the entropies analyzed here only depend on the probability, our results cannot be adjusted by any parameter. Even though these results are close to those of non-extensive statistical mechanics for q ∼ 1 , they differ and cannot be matched for any q.

DOI: 10.3390/e20100773

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We discuss the Bose–Einstein condensation (BEC) for an ideal gas of bosons in the framework of Tsallis's nonextensive statistical mechanics. We study the corrections to the st and ard BEC formulas due to a weak nonextensivity of the system. In particular, we consider three cases in the D-dimensional space: the homogeneous gas, the gas in a harmonic trap and the relativistic homogenous gas. The results show that small deviations from the extensive Bose statistics produce remarkably large changes in the BEC transition temperature.

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I discuss ideal and interacting quantum gases obeying general fractional exclusion statistics. For systems with constant density of single-particle states, described in the mean field approximation, the entropy depends neither on the microscopic exclusion statistics, nor on the interaction. Such systems are called thermodynamically equivalent and I show that the microscopic reason for this equivalence is a one-to-one correspondence between the excited states of these systems. This provides a method, different from the bosonization technique, of transforming between systems of different exclusion statistics. In the last section the macroscopic aspects of this method are discussed. In appendix A, I calculate the fluctuation of the ground-state population of a condensed Bose gas in a grand-canonical ensemble and mean field approximation, while in appendix B I show a situation where although the system exhibits fractional exclusion properties on microscopic energy intervals, a rigorous calculation of the population of single-particle states reveals a condensation phenomenon. This also implies a malfunction of the usual and simplified calculation technique of the most probable statistical distributions.