Dissertation

Monte Carlo simulations, statistical mechanics, and ground states of the logarithmic potential

Nebus, Joseph Francis

2002

Overview

The search for statistical equilibria at very low positive temperatures, using a Monte Carlo algorithm, can locate dynamical equilibria of the N-vortex problem on a sphere. Numerical results show this algorithm accurately and efficiently locates the lowest energy equilibrium. Using an essential tool called the radial distribution function, superficially similar configurations can be easily distinguished and classified. The problem of N bodies on the surface of the sphere interacting by a logarithmic potential is examined for selected N from 4 to 40,962, comparing the energies found by placing points at vertices of polyhedrons to the lowest energies found by a Monte Carlo algorithm. The polyhedron families are generated from simple polyhedrons through two triangular face splitting operations applied iteratively. The closest energy of these polyhedron configurations to the Monte Carlo minimum energy is identified and the energies agree well. The energy per particle pair asymptotically approaches a mean field theory limit of −½(log(2) − 1), approximately 0.153426, for both the polyhedron and the Monte Carlo-generated energies. On a mesh of N points distributed over the plane through methods like those employed above, the logarithmic potential is studied with the circulation is set to zero and the enstrophy held to a fixed value. Using a Monte Carlo Metropolis Rule-based algorithm to maximize the Gibbs factor exp(−β H), with β an inverse temperature and H the Hamiltonian, the arrangement of site vorticities is found to depend on the inverse temperature and number of points. From examining the effects the mesh size has by using the mean nearest neighbor parity introduced here, conclusions about the continuum limit can be drawn, and match analytical predictions that there are no phase transitions except at β = 0. Finally, meshes of N points are distributed over the unit sphere, and the logarithmic potential for this problem studied again with zero and fixed enstrophy. Through a similar Monte Carlo Metropolis algorithm the best arrangements of vorticities for mesh sizes and inverse temperatures are studied. Based on these results conclusions are drawn about the continuum limit, specifically that there are no phase transitions except at β = 0.

• Publisher: ProQuest Dissertations & Theses
• Subject: Mathematics
• ISBN: 0493920900, 9780493920900