Statistical aspects of the massive photon gases in the presence of a minimal length

Abstract

This work attempts to investigate the influence of the generalized uncertainty principle on the statistical parameters of the massive photon gases. The modified energy-momentum relations for the de Broglie Proca electrodynamics are obtained. Based on modified energy-momentum relations, we find thermodynamical characteristics such as partition function, mean energy, pressure, and entropy of the massive photon gases in the presence of a minimal length scale. Also, the upper bound on the isotropic minimal length which is close to the electroweak length scale is derived.

Appendix A

In this section, we show that how energy, pressure, and entropy of the massive photon gases in the presence of a minimal length satisfy the thermodynamics laws.

First law: $Q=ST$ is the heat supplied to the system and $W=PV$ is the work done by the system on its surrounding, the first law of thermodynamics can be defined as follows: [58]

$$\begin{aligned} \Delta U=Q-W. \end{aligned}$$

(35)

Now, according to Eq. (30), we have

$$\begin{aligned} Q_{ML}= & {} S_{ML}T=\frac{gV}{(2\pi ^{2}\hbar ^{3})c\beta ^{2}}\left[ m^{2}-\frac{8}{\beta ^{2}c^{2}}+\frac{a^{2}}{\hbar ^{2}}\left( 12\frac{m^{2}}{\beta ^{2}}\right. \right. \nonumber \\{} & {} \quad \left. \left. -\frac{12}{\beta ^{2}c^{2}}+2m^{2}c^{2}(3-m^{2}c^{2})\right) \right] ,\nonumber \\ W_{ML}= & {} (P_{Pressure})_{ML}V=-\frac{g}{(2\pi ^{2}\hbar ^{3})c\beta ^{2}}\left[ \frac{2}{\beta ^{2}c^{2}}-\frac{m^{2}}{2}\right. \nonumber \\{} & {} \quad \left. +\frac{a^{2}}{\hbar ^{2}}\left( \frac{3}{\beta ^{2}c^{2}}-3\frac{m^{2}}{\beta ^{2}}-m^{2}c^{2}(3-m^{2}c^{2})\right) \right] . \end{aligned}$$

(36)

Based on Eqs. (35) and  (36), we can find the modified energy system as follows:

$$\begin{aligned} Q_{ML}-W_{ML}= & {} \frac{gV}{(2\pi ^{2}\hbar ^{3})c\beta ^{2}}\left[ \frac{m^{2}}{2}-\frac{6}{\beta ^{2}c^{2}}+\frac{a^{2}}{\hbar ^{2}}\left( 9\frac{m^{2}}{\beta ^{2}}\right. \right. \nonumber \\{} & {} \quad \left. \left. -9\frac{1}{\beta ^{2}c^{2}}+m^{2}c^{2}(3-m^{2}c^{2})\right) \right] . \end{aligned}$$

(37)

Second law: In a natural thermodynamics process, the total entropy of the interacting thermodynamics systems increases [49], so we can write

$$\begin{aligned} \Delta S\ge 0. \end{aligned}$$

(38)

Now, let us assume the entropy of massive photon gases, change from an initial state $S_{1}$ with temperature $T_{1}$ to a final state $S_{2}$ with temperature $T_{2}$. According to Eq. (30) and considering $T_{2}>T_{1}$, the following nonequation can be found

$$\begin{aligned} S_{2\mathrm ML}-S_{1\mathrm ML}\ge 0,\nonumber \\ \Delta S_{ML}\ge 0. \end{aligned}$$

(39)

Third law: when temperature of the system drops to absolute zero, the entropy of the system tends to a universal constant(which can be taken to be zero). Based on Eq. (30), the modified entropy ($S_{ML}$) tends to zero in the limit $T\rightarrow 0$.