Thermodynamic geometry of deformed bosons and fermions

Abstract

We construct the thermodynamic geometry of an ideal q-deformed boson and fermion gas. We investigate some thermodynamic properties such as the stability and statistical interaction. It will be shown that the statistical interaction of qdeformed boson gas is attractive, while it is repulsive for the q-deformed fermion one. Also, we will consider the singular point of the thermodynamic curvature to obtain some new results about the condensation of q-deformed bosons and show that there exist a finite critical phase transition temperature even in low dimensions. It is shown that the thermodynamic curvature of q-deformed boson and fermion quantum gases diverges as a power-law function with respect to temperature at zero temperature limit.PACS number(s): 05.70.Fh

Effects of bosonic and fermionic q-deformation on the entropic gravity

“…Let us now derive a modified version of the Poisson equation based on the entropic gravity approach by using the q-deformed temperature obtained in Eq. (21). According to Bekenstein [4], if we suppose that there is a test particle near the black hole horizon which is distant from one Compton wavelength, it increases the black hole mass and horizon area.…”

“…Before calculating the Einstein equations we give some information about the deformed temperature function in Eq. (21). The Unruh temperature on the holographic screen is given as…”

Condensation of an ideal gas with intermediate statistics on the horizon

“…A great deal of effort has been directed toward obtaining the thermodynamic and statistical properties of a q-deformed boson gas by Jackson derivative. These efforts have resulted in deriving both the expressions of the physical quantities such as Bose-Einstein condensation temperature and the heat capacity of the system [20]. In this section, we will first introduce the q-deformed algebra defined by the following relations [34][35][36]:…”

“…We will consider a gas of particles with intermediate statistics as an (approximate) effective theory for the horizon degrees of freedom. For this purpose, two different intermediate statistics [19,20] will be investigated in which a condensed state of particles is possible. It is interesting that we may obtain entropy by determining the correct parameter in the statistics.…”

Stability and anyonic behavior of systems with M-statistics

“…The answer to these questions can be given by performing a calculation of the thermodynamic curvature R, which in our case, is nothing else than the scalar curvature in the two-dimensional space defined by the parameters β and γ = −βµ. The idea of using geometry to study some properties of thermodynamic systems [2]- [6] opened the way to the basic formalism of defining a metric in a two dimensional parameter space and calculate the corresponding scalar curvature as a measure of the correlations strength of the system [8]- [17], with applications to classical and quantum gases [7][13] [18] [19], magnetic systems [20]- [23], non-extensive statistical mechanics [24]- [26], anyon gas [27]- [28], fractional statistics [29], deformed boson and fermion systems [30], systems with fractal distribution functions [31] and quantum group invariance [32]. Some of the basic results of this formalism is the relationship between the departure of the scalar curvature R from the zero value and the stability of the system, and the fact that R vanishes for a classical gas, R > 0 (R < 0) for a boson (fermion) gas and becomes singular at a critical point.…”