Thermodynamic and statistical properties of a multifractional modified dispersion relation via the grand-canonical ensemble

TL;DR

This paper investigates the thermodynamic and statistical properties of a gas influenced by a multifractional modified dispersion relation, revealing nonstandard thermal scalings, altered density of states, and modified thermodynamic laws in different regimes.

Contribution

It introduces a detailed analysis of a gas with a multifractional dispersion relation, deriving new thermodynamic behaviors and scaling laws, including in the ultraviolet limit.

Findings

Density of states scales as ω^{7/5} in the ultraviolet regime.

Stefan-Boltzmann law is deformed to u∝E_{*}^{3/5}T^{17/5}.

Equation-of-state parameter approaches w=5/12 instead of 1/3.

Abstract

We study the thermodynamic and statistical properties of a gas governed by a multifractional modified dispersion relation of the form $\omega^{2}=k^{2}+4E_{*}^{-1/2}k^{5/2}$, where $E_{*}$ sets the characteristic scale of the multifractional correction. Working within the grand-canonical ensemble, we derive the modified density of states, the grand potential, the partition function, and the main thermodynamic quantities for both bosonic and fermionic sectors. The deformation changes the available phase-space distribution and produces nonstandard thermal scalings controlled by the ratio $T/E_{*}$. In the infrared regime, the usual relativistic gas behavior is recovered with leading corrections proportional to powers of $(T/E_{*})^{1/2}$. In the ultraviolet regime, the density of states scales as $\varrho(\omega)\propto \omega^{7/5}$, corresponding to an effective density-of-states dimension $d_{\mathrm{eff}}=12/5$. As a consequence, the Stefan-Boltzmann law is deformed from $u\propto T^{4}$ to $u\propto E_{*}^{3/5}T^{17/5}$, while the equation-of-state parameter approaches $w=5/12$ instead of the standard radiation value $w=1/3$. We also analyze thermal stability, particle number and energy fluctuations, Bose-Einstein condensation, and the degenerate Fermi gas limit. The multifractional correction increases the critical temperature of a conserved bosonic gas and modifies the Fermi energy, pressure, sound speed, and low-temperature heat capacity of degenerate fermions.