A Kac Model for Fermions

TL;DR

This paper introduces a stochastic particle system modeling fermions, demonstrating that as the number of particles grows, the system converges to a fermionic Uehling-Uhlenbeck equation, incorporating quantum exclusion effects.

Contribution

It extends the Kac model to include fermionic exclusion, deriving a new particle system that converges to the fermionic Uehling-Uhlenbeck equation in the large particle limit.

Findings

Convergence of the particle system to the fermionic Uehling-Uhlenbeck equation as N→∞

Inclusion of Pauli exclusion principle in the Kac model

Validation of the model through mathematical analysis

Abstract

We introduce a stochastic $N$-particle system and show that, as $N\to \infty$, an effective description ruled by the homogeneous fermionic Uehling-Uhlenbeck equation is recovered. The particle model we consider is the same as the Kac model for the homogeneous Boltzmann equation with an additional exclusion constraint taking into account the Pauli Exclusion Principle.