Analytic and numerical investigations of lattice-based statistical mechanical models

Abstract

December 2024School of ScienceWe study the canonical ensembles of two lattice models which work with vorticity. In a 2D setting vorticity can be treated as a scalar quantity. Certain functions of vorticity, most notably its first moment, are conserved. By choosing a set of conserved quantities appropriate to the problem being studied, and an inverse temperature which allows one to specify whether a high energy or low energy regime is of interest, one can construct a statistical ensemble. An ensemble encodes certain long-term behaviors of the system, but does not require solving the underlying differential equations which govern the dynamics. The first system is studied is based off the Helmholtz-Onsager point vortex gas, and studies the low positive temperature/low energy regime in a multiply-connected domain. In this regime vorticity particles have high probability to be near the domain walls, as the system energy has a self-interaction term for each vortex which is negative in this neighborhood. This behavior was observed in the canonical ensemble. Due to the simplicity of the point vortex dynamics, the results were supplemented with a microcanonical analysis based on simulating the system and analytically solving a mean field equation which gives its long-term density average. The second system studied is the Kac-Berlin spherical model, which conserves the second moment of the site strengths. The low negative temperature/high energy regime of the spherical model has been used to depict wave systems which undergo inverse energy cascade. We approach the model with new analytical techniques and find evidence for a phase transition, as well as an equation for the expectation of energy after the phase transition. These predictions are then examined and verified with numerical simulations on several lattices which encode a particular instance of the spherical model.Ph