Two-dimensional pseudo-gravity model

TL;DR

This paper introduces and analyzes a novel 2D pseudo-gravity model for particle clouds in magneto-optical traps, highlighting its unique force field properties and establishing solution existence and convergence results.

Contribution

It presents a new macroscopic model with non-potential, singular forces, proving solution existence under certain conditions and demonstrating mean field convergence.

Findings

Existence of solutions under small mass or high diffusion.

Solutions exhibit exponential moments under symmetry.

Convergence of N-particle system to PDE in mean field limit.

Abstract

We analyze a simple macroscopic model describing the evolution of a cloud of particles confined in a magneto-optical trap. The behavior of the particles is mainly driven by self--consistent attractive forces. In contrast to the standard model of gravitational forces, the force field does not result from a potential; moreover, the non linear coupling is more singular than the coupling based on the Poisson equation. We establish the existence of solutions, under a suitable smallness condition on the total mass, or, equivalently, for a sufficiently large diffusion coefficient. When a symmetry assumption is fulfilled, the solutions satisfy strengthened estimates (exponential moments). We also investigate the convergence of the $N$-particles description towards the PDE system in the mean field regime.