Photon Gas Thermodynamics with $\kappa$-generalized statistics at Planck-Scale

Abstract

Kaniadakis, or $\kappa$-generalized, statistical mechanics provides a flexible framework for describing complex systems in the relativistic regime and predicts a richer phenomenology compared to its standard Maxwell-Boltzmann counterpart. In the present work, we extend the investigation of photon gas thermodynamics at the Planck scale within doubly special relativity to the framework of $\kappa$-generalized statistical mechanics. By adopting the $\kappa$-generalized statistical approach, we derive the principal thermodynamic quantities of a photon gas at the Planck scale, including the internal energy $U$, Helmholtz free energy $F$, pressure $P$, entropy $S$, and heat capacity $C_{\rm V}$. We find that these $\kappa$-deformed thermodynamic quantities exhibit nontrivial dependence on the deformation parameter $\kappa$. We then numerically evaluate these quantities as functions of temperature $T$, and reveal that, as $T$ approaches the Planck scale, the $\kappa$-deformed thermodynamic quantities are significantly suppressed relative to those in special relativity, while in the low-temperature regime they are enhanced and exceed their counterparts in special relativity, as a consequence of the $\kappa$-generalized statistics.