On a Boltzmann equation for Haldane statistics

TL;DR

This paper develops a nonlinear Boltzmann equation model for quantum quasi-particles obeying Haldane statistics, establishing existence, uniqueness, and stability of solutions under various conditions.

Contribution

It introduces a novel kinetic model for Haldane statistics and proves strong L1 solutions for the associated Cauchy problem with large initial data.

Findings

Existence of strong L1 solutions for the Boltzmann equation with Haldane statistics.

Uniqueness and stability of solutions under certain conditions.

Results vary between local and global in time depending on parameters.

Abstract

The study of quantum quasi-particles at low temperatures including their statistics, is a frontier area in modern physics. In a seminal paper F.D. Haldane proposed a definition based on a generalization of the Pauli exclusion principle for fractional quantum statistics. The present paper is a study of quantum quasi-particles obeying Haldane statistics in a fully non-linear kinetic Boltzmann equation model with large initial data on a torus. Strong L1 solutions are obtained for the Cauchy problem. The main results concern existence, uniqueness and stability. Depending on the space dimension and the collision kernel, the results obtained are local or global in time.