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Various frameworks of quantum gravity predict a modification in the Heisenberg uncertainty principle to a so-called generalized uncertainty principle (GUP). Introducing quantum gravity effect makes a considerable change in the density of states inside the volume of the phase space which changes the statistical and thermodynamical properties of any physical system. In this paper we investigate the modification in thermodynamic properties of ideal gases and photon gas. The partition function is calculated and using it we calculated a considerable growth in the thermodynamical functions for these considered systems. The growth may happen due to an additional repulsive force between constitutes of gases which may be due to the existence of GUP, hence predicting a considerable increase in the entropy of the system. Besides, by applying GUP on an ideal gas in a trapped potential, it is found that GUP assumes a minimum measurable value of thermal wavelength of particles which agrees with discrete nature of the space that has been derived in previous studies from the GUP.

One of the intriguing predictions of various frameworks of quantum gravity such as string theory and black hole physics is the existence of a minimum measurable length. This has given a rise to the so-called generalized uncertainty principle (GUP) or, equivalently, modified commutation relations between position coordinates and momenta [

This definitely would affect a host of quantum phenomena. In a series of earlier papers, various applications of the new model of GUP were investigated on atomic physics, condensed matter physics, preheating phase of the universe, and black holes at LHC [

For more details about proving the above equation, (

It appears that the total number of microstates is increased due to GUP correction. It is worth mentioning that there are other theories that predict other forms of modified dispersion relations. These models were introduced in [

In this paper we continue investigating the modification of thermodynamics but this time in the presence of GUP, proposed by Ali et al. [

Consider

We have considered classical Maxwell-Boltzmann statistics along with Gibb's factor and

As one might expect, the approximation used so far, that is,

The dashed curves represent the unmodified internal energy and its non- and ultrarelativistic limits. The solid curve represents the modified internal energy.

The dashed curves represent the unmodified specific heat and its non- and ultrarelativistic limits. The solid curve represents the modified specific heat.

Let us consider an ideal gas contained by some potential well acting as a trap. This can take place by wall of a container, such that the wall represents positions of a classical turning points. Also the trap can be represented by surrounding particle in a very high densities and pressure case. Quantum mechanically, the particle has a small probability to tunnel past the turning point. So the linear size of the potential trap

In this section, we aim to determine the thermal properties of the radiation field in the presence of GUP. With quantum gravity effects, the photon obeys the dispersion relation:

Various approaches to quantum gravity such as string theory, black hole physics, and doubly special relativity predict a considerable modification in Heisenberg uncertainty principle to be a generalized uncertainty principle. This modification leads to a change in the energy-momentum dispersion relation and in the physical phase space. These lead to a considerable enhancement in the number of accessible microscopic states of the phase space volume with GUP effects. In this paper we investigate the effect of the GUP in thermodynamic properties of ideal and photon gases. An analytical expression of the partition function for massive ideal gas and photon gas is derived. Using the modified partition function, we determined the thermodynamic functions such as free energy, entropy, pressure, internal energy, and specific heat. We found that there is a considerable increase in these quantities in comparison with the corresponding relativistic quantities. This is due to an increase in accessible microscopic states in the phase space which leads to an increase in entropy that carries a physical properties of the system. This in turn leads to an increase in thermodynamic properties of the system. The pressure of the ideal gas does not change, but the pressure of the photon gas is increased. Another problem is considered and we study the effect of GUP in a gas contained in a potential trap. When the size of the potential trap is comparable with the thermal wavelength of the particles, the pressure and the volume cannot be treated as commuting observable. Quantum gravity puts some restrictions on the particle dynamics, where we found that the thermal wavelength of the particles should be greater than, or equal to, a minimum length and this definitely agrees with the discrete nature of the space that has been derived based on GUP in previous studies. Besides, it is found that the thermal energy of the particle of the ideal gas should not exceed a maximum energy. The results we obtained in this paper could be useful to study the effect of the GUP with astrophysical objects such as standard model of stars (photon plus nonrelativistic ideal gas) [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research of Ahmed Farag Ali is supported by Benha University (