Dynamics of point-vortex type systems near thermal equilibrium: relaxation or not?

Abstract

This article is devoted to the long-time dynamics of point-vortex type systems near thermal equilibrium and to the possible emergence of collisional relaxation. More precisely, we consider a tagged particle coupled to a large number of background particles that are initially at equilibrium, and we analyze its resulting slow dynamics. On the one hand, in the spirit of the Lenard-Balescu relaxation for plasmas, we establish in a generic setting the outset of the slow thermalization of the tagged particle. On the other hand, we show that a completely different phenomenology is also possible in some degenerate regime: the slow dynamics of the tagged particle then remains conservative and the thermalization no longer holds in a strict sense. We provide the first detailed description of this degenerate regime and of its mixing properties. Note that it is particularly delicate to handle due to statistical closure problems, which manifest themselves as a lack of self-adjointness of the effective Hamiltonian.

63 pages

References in corpus (10)

• Resonant Relaxation in Stellar Systems
• Kinetic theory of point vortices: diffusion coefficient and systematic drift
• Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations
• A numerical study of vector resonant relaxation
• On the size of chaos via Glauber calculus in the classical mean-field dynamics
• Vector Resonant Relaxation of Stars around a Massive Black Hole
• Kinetic theory of Onsager's vortices in two-dimensional hydrodynamics
• Kinetic theory of two-dimensional point vortices with collective effects
• Lenard-Balescu correction to mean-field theory
• Orbital alignment and mass segregation in galactic nuclei via vector resonant relaxation