February  2005, 5(1): 1-14. doi: 10.3934/dcdsb.2005.5.1

Statistical equilibrium of the Coulomb/vortex gas on the unbounded 2-dimensional plane

1. 

Department of Physics, National University of Singapore

2. 

Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, United States

Received  September 2003 Revised  October 2003 Published  November 2004

This paper presents the statistical equilibrium distributions of single-species vortex gas and cylindrical electron plasmas on the unbounded plane obtained by Monte Carlo simulations. We present detailed numerical evidence that at high values of $\beta >0$ and $\mu >0$, where $\beta $ is the inverse temperature and $\mu $ is the Lagrange multiplier associated with the conservation of the moment of vorticity, the equilibrium vortex gas distribution is centered about a regular crystalline distribution with very low variance. This equilibrium crystalline structure has the form of several concentric nearly regular polygons within a bounding circle of radius $R.$ When $\beta$ ~ $O(1)$, the mean vortex distributions have nearly uniform vortex density inside a circular disk of radius $R.$ In all the simulations, the radius $R=\sqrt{\beta \Omega /(2\mu )}$ where $\Omega $ is the total vorticity of the point vortex gas or number of identical point charges. Using a continuous vorticity density model and assuming that the equilibrium distribution is a uniform one within a bounding circle of radius $R$, we show that the most probable value of $R$ scales with inverse temperature $\beta >0$ and chemical potential $\mu >0$ as in $R=\sqrt{\beta \Omega /(2\mu )}.$
Citation: Syed M. Assad, Chjan C. Lim. Statistical equilibrium of the Coulomb/vortex gas on the unbounded 2-dimensional plane. Discrete and Continuous Dynamical Systems - B, 2005, 5 (1) : 1-14. doi: 10.3934/dcdsb.2005.5.1
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