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Abstract

Motivated by the doubly special relativity theories and noncommutative spacetime structures, thermodynamical properties of the photon gas in a phase space with compact spatial momentum space is studied. At the high temperature limit, the upper bounds for the internal energy and entropy are obtained which are determined by the size of the compact spatial momentum space. The maximum internal energy turns out to be of the order of the Planck energy and the entropy bound is then determined by the factor View the MathML source through the relevant identification of the size of the momentum space with Planck scale. The entropy bound is very similar to the case of Bekenstein–Hawking entropy of black holes and suggests that thermodynamics of black holes may be deduced from a saturated state in the framework of a full quantum gravitational statistical mechanics.

5. Summary and conclusions

Existence of a minimal length, preferably of the order of Planck length, is suggested by quantum gravity candidates such as string theory and loop quantum gravity. Although a full quantum theory of gravity is not formulated yet, it is widely believed that a non-gravitational theory that admits a minimal length scale would be emerged at the flat limit of quantum gravity. Evidently, such a theory could be achieved through the deformation of the algebraic structure of the standard relativistic quantum mechanics in such a way that spacetime coordinates become non-commutating operators. Recently, in the context of doubly special relativity theories, it was shown that this issue could be also realized from a curved four-momentum space with constant curvature such as the de Sitter geometry with topology R × S3. Identifying R with the space of Fig. 3. Specific heat versus the temperature. The solid line represents the heat capacity of the photon gas in noncommutative space and the dashed line corresponds to the non-deformed case. While the specific heat is constant (for a fixed N) for the standard photon gas, it becomes temperature-dependent at high temperature regime in noncommutative spacetime. Indeed, the photon gas saturates at the high energy regime and then the specific heat tends to zero in this regime. In other words, the system cannot access a higher energy scale by increasing the temperature. energy and S3 with the space of spatial momenta, a universal maximal momentum (corresponds to a minimal observer-independent length scale) naturally arises which is completely determined by the radius of three-sphere or equivalently with curvature of the de Sitter four-momentum space. The deformation to the dispersion relation and density of states are the direct consequence of these setups. Since the number of microstates is determined by the density of states, the significant effects on the thermodynamical properties of the physical systems will be arisen in these setups. In this paper, after obtaining the modified partition function by means of the deformed density of states in spacetime with stable noncommutative algebra, we have studied the thermodynamical properties of the photon gas in this setup. The results show that the entropy of the photon gas increases with a smaller rate at the high temperature regime in noncommutative setup in comparison with the standard non-deformed case. Also, the entropy approaches to a maximum entropy bound around the Planck temperature. The number of accessible microstates associated to the resultant entropy bound is totally determined by the factor V/l 3 Pl which is qualitatively very similar to the black holes’ Bekenstein– Hawking entropy SBH = A/l 2 Pl in which the number of accessible microstates for a black hole is precisely determined by the factor A/l 2 Pl , where A is the black hole horizon area. In other words, similar to the case of black holes, the entropy for the photon gas in high temperature regime is precisely determined by the accessible spatial volume and an ultraviolet cutoff in noncommutative spacetime. This result suggests that thermodynamics of black holes may be obtained from a saturated state in the framework of a full quantum gravitational statistical mechanics. Furthermore, the internal energy of the photon gas also gets a finite maximum value, of the order of the Planck energy, at very high temperature regime. The