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

Mitia Duerinckx, Pierre-Emmanuel Jabin

TL;DR

This work rigorously analyzes the slow dynamics of a tagged particle in a large background of thermally equilibrated particles for 2D point-vortex type systems. By a detailed BBGKY/ cumulant framework and spectral analysis of the linearized mean-field operators, it establishes a dual phenomenology: (i) in non-degenerate, non-Gaussian equilibria the tagged particle thermalizes on the slow scale $t=O(N)$, described by a Fokker-Planck type limit with explicit diffusion coefficient $a_β(r)$; (ii) in Gaussian equilibria the dynamics remains conservative on the critical scale $t=O(N^{1/2})$, leading to an infinite, unitary-like hierarchical system with a RAGE theorem ensuring weak relaxation for $t o o o\

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.

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

TL;DR

This work rigorously analyzes the slow dynamics of a tagged particle in a large background of thermally equilibrated particles for 2D point-vortex type systems. By a detailed BBGKY/ cumulant framework and spectral analysis of the linearized mean-field operators, it establishes a dual phenomenology: (i) in non-degenerate, non-Gaussian equilibria the tagged particle thermalizes on the slow scale , described by a Fokker-Planck type limit with explicit diffusion coefficient ; (ii) in Gaussian equilibria the dynamics remains conservative on the critical scale , leading to an infinite, unitary-like hierarchical system with a RAGE theorem ensuring weak relaxation for $t o o o\

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.
Paper Structure (25 sections, 23 theorems, 348 equations)

This paper contains 25 sections, 23 theorems, 348 equations.

Table of Contents

  1. Introduction
  2. General overview
  3. Formal BBGKY analysis
  4. Point-vortex type systems
  5. Main results
  6. Non-degenerate (non-Gaussian) setting
  7. Degenerate (Gaussian) setting
  8. Comparison to Lenard--Balescu thermalization for plasmas
  9. Rigorous BBGKY analysis
  10. BBGKY hierarchy
  11. Correlation functions or cumulants
  12. A priori correlation estimates
  13. Weighted Sobolev spaces
  14. Non-degenerate case: non-Gaussian equilibrium
  15. Preliminary BBGKY analysis
  16. ...and 10 more sections

Key Result

Theorem 1.1

Assume that the external potential $V$ further satisfies $\nabla(V'/r)\in C^\infty_b(\mathbb R^2)$, and that $\beta\ll1$ is small enough so that the mean-field equilibrium $\mu_\beta$ is well-defined, cf. eq:mubeta. Let the angular velocity $\Omega_\beta$ be the smooth radial function given by Consider the non-Gaussian setting when $\Omega_\beta$ is nowhere constant: more precisely, we assume for

Theorems & Definitions (46)

  • Conjecture 1
  • Theorem 1.1: Non-Gaussian setting
  • Theorem 1.2: Gaussian case
  • Theorem 1.3
  • Lemma 2.1: BBGKY hierarchy for correlations
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: Interpolation
  • proof
  • ...and 36 more