[Paper Review] Variational description of statistical field theories using Daubechies' wavelets

Christoph Best, Andreas Schäfer|ArXiv.org|Feb 16, 1994

Image and Signal Denoising Methods6 references7 citations

TL;DR

This paper proposes a variational approach using Daubechies' orthonormal compactly supported wavelets to describe statistical field theories on a lattice, extending mean field theory to capture fluctuation strengths and correlation functions. It demonstrates that only a small number of variational parameters are needed to accurately describe critical phenomena, suggesting wavelets as a promising basis for implementing the renormalization group in wavelet space.

ABSTRACT

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.

Motivation & Objective

• To develop a variational method for statistical field theories using Daubechies' wavelets on a lattice.
• To extend mean field theory by incorporating wavelet-based fluctuations to describe correlation functions.
• To assess the accuracy of the wavelet-based approach against Monte Carlo simulations for critical phenomena.
• To explore the potential of wavelets as a basis for implementing the renormalization group in wavelet space.
• To evaluate the efficiency of the method in terms of the number of variational parameters required for convergence.

Proposed method

• Employ Daubechies' orthonormal compactly supported wavelets as a basis for representing field configurations on a lattice.
• Formulate a variational principle using wavelet coefficients as variational parameters to minimize the free energy.
• Derive expressions for fluctuation strengths of wavelet coefficients, which directly relate to correlation functions.
• Use the variational wavelet representation to compute critical exponents and correlation lengths.
• Compare the variational predictions with results from Monte Carlo simulations on the same lattice models.
• Optimize the variational parameters to achieve the best agreement with simulation data.

Experimental results

Research questions

• RQ1 Can Daubechies wavelets provide an efficient and accurate representation of statistical field theories on a lattice?
• RQ2 How well does the variational wavelet method describe critical phenomena compared to Monte Carlo simulations?
• RQ3 What is the minimal number of variational parameters required for a good description of correlation functions?
• RQ4 Can wavelets serve as a viable basis for implementing the renormalization group in wavelet space?
• RQ5 How do the fluctuation strengths of wavelet coefficients relate to the underlying correlation structure of the field theory?

Key findings

• The wavelet-based variational method accurately captures critical behavior in statistical field theories with only a small number of variational parameters.
• The predicted correlation functions from the variational approach show good agreement with Monte Carlo simulation results.
• Fluctuation strengths of wavelet coefficients are directly linked to the correlation function, enabling a physical interpretation of the variational parameters.
• The method extends mean field theory by incorporating non-trivial correlations through wavelet decomposition.
• The compact support and orthonormality of Daubechies wavelets enable efficient numerical implementation and localization in both position and scale.
• The results suggest that wavelet space may offer a natural framework for a wavelet-based renormalization group approach.