Discrete Boltzmann Equation for Anyons
Niclas Bernhoff
TL;DR
This work develops and analyzes a discrete Boltzmann equation for anyons (Haldane statistics) with a finite velocity set, introducing a filling factor $\Psi_{\alpha}$ and a collision operator $Q_i^{\alpha}$ that recovers bosonic and fermionic limits at $\alpha=0$ and $\alpha=1$. Equilibria are characterized by a transcendental condition $F/\Psi_{\alpha}(F)=M$, yielding Maxwellian-like forms for the limiting cases and nontrivial equilibria for intermediate statistics. The authors establish trend-to-equilibrium results in both planar stationary and spatially homogeneous settings via monotone entropy-like functionals ($\widetilde{\mathcal{H}}$ and $\mathcal{H}$) and prove that the linearized collision operator $L$ is symmetric and positive semidefinite with a precisely described kernel tied to collision invariants. These results enable the application of general discrete-velocity half-space theory to the anyon Boltzmann equation and point to extensions to mixtures and multiple internal energy states, broadening the kinetic theory of fractional statistics in two dimensions.
Abstract
A semi-classical approach to the study of the evolution of anyonic excitations--elementary particles with fractional statistics, complementing bosons and fermions--is through the Boltzmann equation for anyons. This work reviews a discretized version--a system of partial differential equations--of such a quantum equation. Trend to equilibrium is studied for a planar stationary system, as well as the spatially homogeneous system. Essential properties of the linearized operator are proven, implying that results for general steady half-space problems for the discrete Boltzmann equation in a slab geometry can be applied.