Renyi statistics in equilibrium statistical mechanics

TL;DR

This paper investigates the Renyi statistics within equilibrium statistical mechanics, demonstrating its equivalence to Boltzmann-Gibbs statistics in the microcanonical and canonical ensembles for the ideal gas, especially in the thermodynamic limit.

Contribution

It provides an analytical comparison showing Renyi statistics aligns with Boltzmann-Gibbs statistics in key ensembles and satisfies thermodynamic principles.

Findings

Renyi statistics is equivalent to Boltzmann-Gibbs in microcanonical ensemble.

In the canonical ensemble, Renyi statistics matches Boltzmann-Gibbs in the thermodynamic limit.

Renyi statistics satisfies thermodynamic homogeneity and equilibrium conditions.

Abstract

The Renyi statistics in the canonical and microcanonical ensembles is examined in the general case and in particular for the ideal gas. In the microcanonical ensemble the Renyi statistics is equivalent with the Boltzmann-Gibbs statistics. By the exact analytical results for the ideal gas, it is shown that in the canonical ensemble in the thermodynamic limit the Renyi statistics is also equivalent with the Boltzmann-Gibbs statistics. Furthermore it satisfies the requirements of the equilibrium thermodynamics, i.e. the thermodynamical potential of the statistical ensemble is a homogeneous function of degree 1 of its extensive variables of state. We conclude that the Renyi statistics duplicates the thermodynamical relations stemming from the Boltzmann-Gibbs statistics in the thermodynamical limit.