Equilibrium Energy and Entropy of Vortex Filaments on a Cubic Lattice: A Localized Transformations Algorithm

Abstract

In this work we propose a new algorithm for the computation of statistical equilibrium quantities on a cubic lattice when both an energy and a statistical temperature are involved. We demonstrate that the pivot algorithm used in situations such as protein folding works well for a small range of temperatures near the polymeric case, but it fails in other situations. The new algorithm, using localized transformations, seems to perform well for all possible temperature values. Having reliably approximated the values of equilibrium energy, we also propose an efficient way to compute equilibrium entropy for all temperature values. We apply the algorithms in the context of suction or supercritical vortices in a tornadic flow, which are approximated by vortex filaments on a cubic lattice. We confirm that supercritical (smooth, "straight") vortices have the highest energy and correspond to negative temperatures in this model. The lowest-energy configurations are folded up and "balled up" to a great extent. The results support A. Chorin's findings that, in the context of supercritical vortices in a tornadic flow, when such high-energy vortices stretch, they need to fold.