Two generalizations of the Boltzmann equation

TL;DR

This paper unifies two generalizations of Boltzmann's kinetic theory by showing their equivalence in producing non-extensive statistics through different but related modifications.

Contribution

It establishes a direct connection between two approaches to generalizing the Boltzmann equation, linking nonlinear collision rates and nontrivial energy composition rules.

Findings

Demonstrates the equivalence of two generalizations in stationary solutions

Provides transformation formulas between key functions of the approaches

Shows how non-extensive statistics can be derived from either method

Abstract

We connect two different generalizations of Boltzmann's kinetic theory by requiring the same stationary solution. Non-extensive statistics can be produced by either using corresponding collision rates nonlinear in the one-particle densities or equivalently by using nontrivial energy composition rules in the energy conservation constraint. Direct transformation formulas between key functions of the two approaches are given.