On the thermodynamics of relativistic ideal gases in the presence of a maximal length

Highlights

• The thermostatistics of relativistic ideal gases is studied in the presence of a maximal length.

• The ultrarelativistic and nonrelativistic regimes are discussed.

• The effects of the maximal length do not depend on the considered regime.

• A modified equation of state emerges naturally in the presence of a maximal length.

Abstract

We investigate the thermostatistics of relativistic ideal gases within the recently proposed deformed Heisenberg algebra (Perivolaropoulos, 2017), which includes a maximal length. By using the semiclassical method, the generalized canonical partition function is established and the modified thermodynamic functions are then calculated. It is explicitly shown that the maximal length, unlike the minimal length, modifies the equation of state of relativistic ideal gases. Furthermore, the ultrarelativistic and nonrelativistic regimes are discussed. In contrast to the minimal length corrections, the maximal length corrections do not depend on the considered regime. Such a result confirms that the effects of the maximal length and those of the minimal length are fundamentally different.

Introduction

In Refs. [1], [2], the thermostatistics of an ideal gas and harmonic oscillators has been studied in connection with a Generalized Uncertainty Principle (GUP), which implies the existence of a maximal length [3]. These studies showed that this GUP could lead to different effects compared to those induced by the minimal length GUPs. Especially, for the ideal gas, an equation of state similar to that of real gases emerges naturally in the presence of a maximal length.

In order to investigate further implications of the maximal length assumption, and to make a comparative analysis of the effects induced by the maximal and minimal length scales, we consider here relativistic ideal gases, which have been considered previously in the context of the minimal length formalisms [4], [5].

The modification of the Heisenberg Uncertainty Principle (HUP) to the so-called GUP is an unavoidable idea, which serves to account for the existence of limit values for physical observables [6], [7]. In this context, numerous theories suggest the existence of optimum quantities; for instance, a minimum length appears in quantum gravity [8], a maximum momentum is predicted in doubly special relativity [9], and a maximum length is hypothesized in cosmology due to the presence of the particle horizon or from possible nontrivial cosmic topology [3], [10], [11], [12]. Thus, in order to reconcile the HUP with these fundamental proposals, various modifications were suggested in the literature, which gave rise to diverse models of GUP.

The well known GUP is that incorporating a minimal length, which has been discussed by Kempf and coworkers in their seminal work [6]. In the framework of this deformed version of Heisenberg algebra, diverse topics have been addressed in the literature, see for instance Refs. [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], and Refs. [5], [24], [25] devoted to statistical physics.

Other GUPs have been suggested to include both a minimal length and a maximal momentum [26], [27], [28], [29], [30], [31]. In this context, several works have been performed to investigate the implications of such GUPs in statistical mechanics [4], [32], [33] and other physical problems [26], [27], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43].

Recently, to include a maximal length, Perivolaropoulos proposed a new form of GUP in 1D, and studied numerically the one-dimensional Schrödinger equation for the harmonic oscillator [3]. Moreover, it is claimed that the formalism of quantum mechanics, which follows from this new GUP, may reflect quantum effects on the early Universe and their observational imprints at the present time [3]. An extended discussion of cosmological effects of maximum length quantum mechanics is reported in Ref. [44], in which the authors addressed the primordial power spectra of cosmological fluctuations with the GUP incorporating a maximum length. Furthermore, the relativistic gas models are widely used for studying the physics of the early Universe, for instance, quark-gluon plasma system [45], the standard big-bang model [46] and cosmological plasma physics [47]. This suggests that investigations of relativistic statistical systems, in the presence of such a hypothetical maximal length, would be of special importance for eventual cosmological applications.

In this work, we consider the GUP of Ref. [3] and investigate in 3D statistical mechanics a relativistic ideal gas, as well as the special case of an extreme relativistic gas. The effects of this GUP are examined and compared with those of minimal length GUPs, studied previously in the literature.

The rest of this Letter is structured as follows: in the next section, we give a brief review of quantum mechanics with a maximum length; we present in particular an extension of the formalism of Ref. [3] to three dimensions. In the third section, by using the semiclassical approach of Ref. [48], we study the thermodynamics of relativistic ideal gases within the maximal length formalism. The Letter is fenced in the fourth section by summarizing the main obtained results.