Relativistic Entropy and Related Boltzmann Kinetics

TL;DR

This paper proposes a relativistic generalization of the Boltzmann equation by modifying the two-particle correlation function, leading to a new entropy form and stationary distribution with power-law tails, consistent with special relativity.

Contribution

It introduces a fully relativistic Boltzmann equation based on a generalized correlation function and entropy, extending classical kinetic theory to high-energy regimes.

Findings

The new relativistic Boltzmann equation obeys the H-theorem.

Predicts stationary distributions with power-law tails.

Recovers classical results in the non-relativistic limit.

Abstract

It is well known that the particular form of the two-particle correlation function, in the collisional integral of the classical Boltzmman equation, fix univocally the entropy of the system, which turn out to be the Boltzmann-Gibbs-Shannon entropy. In the ordinary relativistic Boltzmann equation, some standard generalizations, with respect its classical version, imposed by the special relativity, are customarily performed. The only ingredient of the equation, which tacitly remains in its original classical form, is the two-particle correlation function, and this fact imposes that also the relativistic kinetics is governed by the Boltzmann-Gibbs-Shannon entropy. Indeed the ordinary relativistic Boltzmann equation admits as stationary stable distribution, the exponential Juttner distribution. Here, we show that the special relativity laws and the maximum entropy principle, suggest a relativistic generalization also of the two-particle correlation function and then of the entropy. The so obtained, fully relativistic Boltzmann equation, obeys the H-theorem and predicts a stationary stable distribution, presenting power-law tails in the high energy region. The ensued relativistic kinetic theory preserves the main features of the classical kinetics, which recovers in the $c \to \infty$ limit.