Thermostatistics of aq-deformed boson gas

Abstract

We show that a natural realization of the thermostatistics of q bosons can be built on the formalism of q calculus, and that the entire structure of thermodynamics is preserved if we use an appropriate Jackson derivative in place of the ordinary thermodynamics derivative. This framework allows us to obtain a generalized q boson entropy which depends on the q basic number. We study the ideal q boson gas in the thermodynamic limit which is shown to exhibit Bose-Einstein condensation with a higher critical temperature and a discontinuous specific heat.

Bose–Einstein condensation in the framework of κ-statistics

“…The liquid 4 He behaviour at low temperature and pressure has been studied in the approximation of the standard ideal boson gas, and described by the Bose-Einstein (BE) statistical distribution:…”

“…Moreover the heat capacity behavior presents in the two cases some qualitative and quantitative differences. Some models depending on a free parameter based on the group theory, describing the liquid 4 He behavior, have been proposed in Ref.s [1,2,3,4]. Recently the liquid 4 He is considered in the framework of the nonextensive statistical mechanics [5,6,7].…”

“…In Ref. [8], a model is developed for the liquid 4 He, starting from the nonextensive quantum statistical distribution of Ref.s [9,10,11], given by:…”

Deformed quantum mechanics andq-Hermitian operators

“…In Ref. [12,13] it was shown that a natural realization of quantum thermostatistics of q-deformed bosons and fermions can be built on the formalism of q-calculus. In Ref.…”

Self-similarity degree of deformed statistical ensembles

“…From a mathematical point of view, the q-calculus represents the most suited formalism to investigate (multi)fractal sets whose generation is provided with manifold action of the dilatation operator related to the Jackson derivative [13]. It appears [21,22] for q-deformed bosons and fermions, a natural generalization of thermostatistics is based on the formalism of the q-calculus, whereas the stationary solution of a classical deformed Fokker-Planck equation represents a q-analogue of the exponential function in the framework of the basic hypergeometric series [11,23]. Related systems exhibit a discrete scale invariance whose description [24] is achieved by using the Jackson derivative and integral, generalizing the regular derivative and integral for discretely self-similar systems (for example, the free energy of spin systems on a hierarchical lattice is related to a homogeneous function being the q-integral [25]).…”