Thermostatistics with minimal length uncertainty relation

Abstract

Existence of minimal length is suggested in any quantum theory of gravity such as string theory, double special relativity and black hole physics. One way to impose minimal length is deforming Heisenberg algebra in phase space which is called Generalized Uncertainty Principle (GUP). In this paper, we develop statistical mechanics in GUP framework. Our method is quite general and does not need to fix the generalized coordinates and momenta. We define general transformation in phase space which transforms usual Heisenberg algebra to a deformed one. In this method, quantum gravity effects only acts on the structure of phase space and we relate these effects to the density of states. We find an interesting phenomenon in Maxwell-Boltzmann statistics which has not a classical analogy. We show that there is an upper bound for the number of excited particles in the limit of high temperature which implies to the condensation. Also we study modification of Bose-Einstein condensation and the completely degenerate gas.