Self containment radius for rotating planar flows, single-signed vortex gas and electron pl

2. Auxillary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 2.1. Monte-Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 2.2. Heuristic derivation from mean field theory . . . . . . . . . . . . . . . . . . . . . . . . . . 244 3. Variational Problems for E [q ] : Derivation of central formula . . . . . . . . . . . . . . . . . 245 3.1. Problem 1: Unconstrained extremization of E . . . . . . . . . . . . . . . . . . . . . . . . . 246 3.2. Problem 2: The variable radius variational problem . . . . . . . . . . . . . . . . . . . . . 248 4. Monte-Carlo simulations and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
REGULAR AND CHAOTIC DYNAMICS, V. 10, №3, 2005
239
C. C. LIM, S. M. ASSAD
1.rives an elegant formula (2.1) for 2D Euler flows in the unbounded plane, which holds between inverse temperature β, chemical potential µ, total circulation Ω and the radius R of the compact support of steady-state vorticity when β = kµ → ∞ with fixed k > 0. It will be clear from the results in this paper that vortex dynamics on the unbounded plane differs significantly from any bounded domain formulation. The key to self-confinement in the unbounded plane lies in the notion of an angular momentum which is conserved by the dynamics because of SO(2) invariance of the vortex gas. Both in electron plasma [32] and geophysical flows [8] on unbounded domains, the confinement of the vorticity or charge can be expressed in terms of a self-containment radius, which is clearly a very useful quantity for which it will be desirable to have a closed form expression. The approach we take is based on the observation that the minimizers of the internal energy are compactly supported when the angular momentum is present. At this point the reader could well ask why there is an inverse temperature β in the formulation since we are merely minimizing an augmented energy functional in the framework of the continuum Euler model on the unbounded plane. The reason for this lie in the dual origin of the problem at hand, namely, the above dynamical question of steady-states of the Euler equations in the unbounded plane, and the mean field continuum limit of the Onsager Vortex Gas. The latter is an equilibrium statistical mechanics model that consists of N point vortices of like sign in thermal equilibrium with infinite reservoirs of energy and angular momentum. In other words, Onsager’s Vortex Gas is based on a Gibbs ensemble that is canonical in both the flow kinetic energy HN and the angular momentum or moment of vorticity ΓN . For rotating 2D flows, it is arguable that the first point of view is primary. In this first point of view, there is no need for thermodynamics and only the ratio σ = µ/β plays a physical role as the rate of rotation. On the other hand, the second point of view is perhaps more natural in the setting of cylindrical electron plasmas where the discrete vortices correspond in cross section to the long lines of charge in the plasma. Since the main aim of this paper is to rigorously derive the self-containment formula from 2D Euler dynamics, we will focus on the first point of view. The same result could also be obtained from the recent technical results on the mean field limit for the Onsager Vortex Gas discussed next. In view of this paper’s secondary aim to provide a detailed summary of the recent work on the Onsager Vortex Gas, we will include such a review of the known results, a heuristic derivation of the radius formula in section 2 and the statement of an important Open Problem on the Onsager Vortex Gas. The existence and asymptotic exactness of this mean field limit was accomplished in the 1990s by Caglioti et al [3] and Kiessling [18]. Subsequently, the convergence of the finite temperature mean field theory to the zero temperature ground state of the single-sign vortex gas in the unbounded plane was established by Kiessling and Spohn [19]. This zero temperature ground state is the minimizer of the augmented energy functional in the first point of view and is related to Ginibre’s exact expressions from random matrix theory. This paper is organized as follows: In section (1.1)-(1.3) we give an overview of known results for both the bounded and unbounded vortex gas problem. Section 2 discusses some numerical as well as heuristic results. The main theorem is proved in section 3. Section 4 presents the results of Monte Carlo simulations which can be explained by our theorem. The proof that the global minimizers of the energy functional (with angular momentum) must be compactly supported and radially symmetric will be reported in another paper.