# American Institute of Mathematical Sciences

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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