Looking through wavelets to view the Ising problem

Abstract

In the theoretical study of phase transitions and critical phenomena, renormalization groups (RG) provide a course-grain representation of a many particle system and aid in the determination of the critical exponents associated with phase transitions. Monte Carlo and Molecular Dynamic simulations are often the simulation tools used for verification. However, the rich structure and multiscale properties of wavelets have found limited use in these applications. To promote exploration here, we apply and investigate wavelets to the solution of the simplest of molecular models, the Ising model and evaluate the most fundamental of objects in statistical mechanics, the partition function, now based on a wavelet-modifed Hamiltonian. As in [1] we iteratively compute the waveletbased coarse grain 2-D finite-size Ising model. Thermodynamic properties are calculated to predict, given the finite size effect, potential for phase transition. Exact calculations are confirmed using Monte Carlo simulations. Based on this investigation and corresponding insight gained, we propose a wavelet-based investigation in three open areas: Wavelet Bases, Critical Exponents and Phase Transitions, Renormalization Groups and Continuous Wavelets.