Quantum gravity modifications of the relativistic ideal gas thermodynamics

In the frame work of the generalized uncertainty principle, we study the theoretical modifications induced by the quantum gravity on the thermodynamics of the relativistic ideal gas. The merit of this work is that we have applied two different research methods while the results of both methods point to a complete consistency between them. These approaches emerge from different viewpoints of the incorporation of the underlying theory. Here, we call them as the modified Hamiltonian method and the modified density of states method. In the first method, we have an active viewpoint and suppose that the Hamiltonian is modified via the momentum transformation induced by the theory. However, in the second method, we adopted rather a passive interpretation and consider a transformation of the coordinates as a result of the theory. In order to have a time invariant volume element of phase space, this method leads to a redefinition of the density of states. We show that these approaches expectedly lead to the same partition functions and hence are equivalent. Since, there are two models of the generalized uncertainty principle, with quadratic and linear momentum terms, we show these equivalencies are established for both models. We obtain the modifications of the thermodynamics of the relativistic ideal gas induced by the quadratic model and estimate the consequences for the limiting case the nonrelativistic domains. All the results are in agreement with the previous issues. Also the results for the extreme relativistic and asymptotic ultrarelativistic domains are obtained which shows novel properties. In addition, we indicate that the black body energy spectrum will be changed, due to the quantum gravity corrections, but this effect can be seen at high temperature limits.