Thermostatistics of the polymeric ideal gas

Abstract

In this paper, we formulate statistical mechanics of the polymerized systems in the semiclassical regime. On the corresponding polymeric symplectic manifold, we set up a noncanonical coordinate system in which all of the polymeric effects are summarized in the density of states. Since we show that the polymeric effects only change the number of microstates of a statistical system, working in this coordinate is quite reasonable from the statistical point of view. The results show that the number of microstates decreases due to existence of an upper bound for the momentum of the test particles in the polymer framework. We obtain a corresponding canonical partition function by means of the deformed density of states. By using the partition function, we study thermodynamics of the ideal gas in the polymer framework and show that our results are in good agreement with those that arise from the full quantum consideration at high temperature, and they coincide with their usual counterpart in the limit of low temperature.

Noncommutative spaces and covariant formulation of statistical mechanics

“…However, quantum mechanics provides the semiclassical approximation in the spirit of the uncertainty principle. Indeed, a phase space with finite resolution and well-defined number of classical microstates can be achieved through the approximation [17] Tr…”

“…Thermodynamical properties of the ideal gases and harmonic oscillator are also studied in Refs. [17,37]. To see the effects of minimal length on the thermodynamics of the black holes in noncommutative space, the generalized uncertainty principle framework, and the polymer quantization scheme, see Refs.…”

“…The direct consequence of space-time with noncommutative coordinates is the modification to the dispersion relation and the associated density of states [11][12][13][14][15][16]. In symplectic formulation, the density of states is properly determined by the Liouville measure which will be constructed from the symplectic structure [17]. In noncommutative space-time, however, the corresponding symplectic structure seems to be noncanonical, which induces a nonuniform probability distribution over the set of microstates through a deformed Liouville volume element.…”

Photon gas thermodynamics in dS and AdS momentum spaces

“…In the Maxwell-Boltzmann statistics, the form of equation of state in all phenomenological approaches to the minimal length is also preserved and it seems that this is a general feature (see for instance Refs. [21,22,23,24]). An interesting result in this setup is that there is a maximal pressure for the photon gas for the case of dS momentum space.…”

“…The nontrivial effects may occur when the measure dµ X or/and the constraint C is deformed. In theories with minimal length that formulated on the reduced (non-relativistic) phase space such as generalized uncertainty principle [21], noncommutative reduced phase spaces [22] and polymerized phase spaces [23], always one of the phase space measure or the dispersion relation is modified (it depends on what one prefers to work, in canonical (Darboux) or noncanonical charts on the reduced phase space [16,24,25]). In deformed special relativity theories such as the DSR models, depending on the coordinate that one implements, both the measure and the dispersion relation can be simultaneously modified [26,27].…”