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The application of wavelets in field theory

Somchai Ruangpermpool, Ahpisit Ungkitchanukit

Published Jan 1, 1996• 0 citations
Publisher

Abstract

In this thesis, it has been studied and repeated C. Best and A. Schafer’s work on the application of wavelet to the Gaussian model and the Landau-Ginzburg model. The fields of the Gaussian model are expanded in wavelet representation and with the fluctuation strengths of wavelet coefficients as variational parameter, the free energy of the model in Gaussian ensemble is minimized analytically. We then numerically recalculate the fluctuation strengths of wavelet coefficients and thus the correlation function. The result confirms that wavelets provide a reasonable description of the critical phenomena with only a small number of variational parameters, which is the same as the application of wavelet in other fields. In Landau-Ginzburg model, by analytic approximation, it is shown how the phase-transition is represented in wavelet space and how critical exponents can be computed from the wavelet expansion.

DOI
10.58837/chula.the.1996.2340

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In this thesis, it has been studied and repeated C. Best and A. Schafer’s work on the application of wavelet to the Gaussian model and the Landau-Ginzburg model. The fields of the Gaussian model are expanded in wavelet representation and with the fluctuation strengths of wavelet coefficients as variational parameter, the free energy of the model in Gaussian ensemble is minimized analytically. We then numerically recalculate the fluctuation strengths of wavelet coefficients and thus the correlation function. The result confirms that wavelets provide a reasonable description of the critical phenomena with only a small number of variational parameters, which is the same as the application of wavelet in other fields. In Landau-Ginzburg model, by analytic approximation, it is shown how the phase-transition is represented in wavelet space and how critical exponents can be computed from the wavelet expansion.
We investigate the description of statistical field theories using Daubechies' orthonormal compact wavelets on a lattice. A simple variational approach is used to extend mean field theory and make predictions for the fluctuation strengths of wavelet coefficients and thus for the correlation function. The results are compared to Monte Carlo simulations. We find that wavelets provide a reasonable description of critical phenomena with only a small number of variational parameters. This lets us hope for an implementation of the renormalization group in wavelet space.
The sophistication of modern tools used in the study of statistical mechanics and field theory is often an obstacle to the easy understanding of new important current results reported in journals. The main purpose of this book is to introduce the reader to the methods of the fluctuation (field) theory of phase transitions and critical phenomena so as to provide a good source for research. The introductory contents are concerned with ideas of description, thermodynamic stability theory related to phase transitions, major experimental facts, basic models and their relationships. Special attention is paid to the mean field approximation and to the Landau expansion for simple and complex models of critical and multicritical phenomena. An instructive representation of the modern perturbation theory and the method of the renormalization group is developed for field models of phase transitions. The essential influence of the fluctuations on the critical behaviour is established together with the theory of correlation functions, Gaussian approximation, the Ginzburg criterion, [symbol]- and 1/n- expansions as practical realizations of the renormalization group ideas. Applications of the theory to concrete aspects of condensed matter physics are considered: quantum effects, Bose condensation, crystal anisotropy, superconductors and liquid crystals, effects of disorder of type randomly distributed quenched impurities and random fields. This volume can be used as an advanced University course book for students with a basic knowledge of statistical physics and quantum mechanics. It could be considered as a complementary text to a standard University course on statistical physics
In the previous chapter, we developed the Landau theory of phase transitions, emphasizing how macroscopic order parameters and symmetry considerations determine the general form of the free energy near criticality. There, spatial variations of the order parameter were neglected, corresponding to the mean-field approximation . However, as we have seen, close to the critical temperature T c the correlation length ξ diverges, and fluctuations occur over arbitrarily large scales. In this regime, the assumption of spatial uniformity breaks down, and a local field description becomes necessary.
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 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.
Abstract The successful calculation of critical exponents for continuous phase transitions is one of the main achievements of theoretical physics over the last quarter-century. This was achieved through the use of scaling and field-theoretic techniques which have since become standard equipment in many areas of physics, especially quantum field theory. This book provides a thorough introduction to these techniques. Continuous phase transitions are introduced, then the necessary statistical mechanics is summarized, followed by standard models, some exact solutions and techniques for numerical simulations. The real-space renormalization group and mean-field theory are then explained and illustrated. The final chapters cover the Landau-Ginzburg model, from physical motivation, through diagrammatic perturbation theory and renormalization to the renormalization group and the calculation of critical exponents above and below the critical temperature.
In the previous chapter, we developed the perturbative formulation of the Landau–Ginzburg theory and introduced Feynman diagrams as a systematic way to compute corrections to the Gaussian (mean-field) results. Mean-field theory provides a qualitatively correct picture of phase transitions but fails quantitatively: it neglects fluctuations, leading to incorrect values of the critical exponents, especially in low dimensions ( https://www.w3.org/1998/Math/MathML" display="inline"> D < 4 ). To obtain accurate exponents, one must incorporate fluctuations perturbatively and study how couplings evolve with scale.