Beltrami operators

TL;DR

This paper explores Beltrami operators as generators of particle dynamics, demonstrating that they satisfy an H-theorem and lead to a generalized Boltzmann distribution, with implications for topologically constrained diffusion.

Contribution

It extends previous work by analyzing Beltrami operators as dynamical generators and establishing conditions for equilibrium distributions, including cases where the Beltrami condition is violated.

Findings

Random motion with Beltrami operators satisfies an H-theorem.

Beltrami operators lead to a generalized Boltzmann distribution.

Violation of the Beltrami condition results in heterogeneous distributions.

Abstract

Beltrami fields occur as stationary solutions of the Euler equations of fluid flow and as force free magnetic fields in magnetohydrodynamics. In this paper we discuss the role of Beltrami fields when considered as operators acting on a Hamiltonian function to generate particle dynamics. Beltrami operators, which include Poisson operators as a special subclass, arise in the description of topologically constrained diffusion in non-Hamiltonian systems. Extending previous results, we show that random motion generated by a Beltrami operator satisfies an H-theorem, leading to a generalized Boltzmann distribution on the coordinate system where the Beltrami condition holds. When the Beltrami condition is violated, random fluctuations do not work anymore to homogenize the particle distribution in the coordinate system where they are applied. The resulting distribution becomes heterogeneous. The heterogeneity depends on the `field charge' measuring the departure of the operator from a Beltrami field. Examples of both Beltrami and non-Beltrami operators in three real dimensions together with the corresponding equilibrium distribution functions are given.