Momentum distribution and correlation function of free particles in the Tsallis statistics using conventional expectation value and equilibrium temperature

TL;DR

This paper derives the momentum distribution and correlation functions for free particles within Tsallis statistics using conventional expectation values, highlighting differences from Boltzmann-Gibbs results and dependence on the entropic parameter q.

Contribution

It provides explicit formulas for momentum distribution and correlations in Tsallis statistics with conventional expectation, emphasizing the role of q and particle number N.

Findings

Momentum distribution differs from Boltzmann-Gibbs case for q<1.

Correlation exists among free particles in Tsallis framework.

q is constrained between 1 - 1/(3N/2+1) and 1.

Abstract

We applied the Tsallis statistics with the conventional expectation value to a system of free particles, adopting the equilibrium temperature which is often called the physical temperature. The entropic parameter $q$ in the Tsallis statistics is less than one for power-law-like distribution. The well-known relation between the energy and the temperature in the Boltzmann--Gibbs statistics holds in the Tsallis statistics, when the equilibrium temperature is adopted. We derived the momentum distribution and the correlation in the Tsallis statistics. The momentum distribution and the correlation in the Tsallis statistics are different from those in the Boltzmann--Gibbs statistics, even when the equilibrium temperature is adopted. These quantities depend on $q$ and $N$, where $N$ is the number of particles. The correlation exists even for free particles. The parameter $q$ satisfies the inequality $1-1/(3N/2+1) < q < 1$.