Statistical mechanics of twodimensional vortices in a bounded container
Abstract
The equilibrium statistical properties of a system consisting of a large number of twodimensional vortices of arbitrary sign, evolving in an arbitrary domain closed by a bounded curve, are investigated under the fundamental assumptions of the equiprobability of any state compatible with a given value of the energy. The Salzberg and Prager equation of state is proved to be also valid in this general case. An expansion in inverse powers of the density closes the hierarchy of equations deduced from the fundamental assumption. The resulting nonlinear partial differential equation for the onepoint probability density function, is solved numerically in various cases. A numerical method of simulation using a regular grid is tested for a small number of vortices. Some very accurate numerical simulations of vortex motion in a circular domain confirm the theoretical conclusions.
 Publication:

Ph.D. Thesis
 Pub Date:
 February 1976
 Bibcode:
 1976PhDT........61P
 Keywords:

 Statistical Mechanics;
 Two Dimensional Flow;
 Vortices;
 Boundaries;
 Equations Of State;
 Nonlinear Equations;
 Partial Differential Equations;
 Probability Density Functions;
 Fluid Mechanics and Heat Transfer