The principal investigator proposes to analyze the critical behavior of lattice models in classical equilibrium statistical mechanics by using renormalization group methods based on Osiris wavelets. Various types of wavelets have been successfully applied to problems in signal analysis and image reconstruction. Since critical phenomena in statistical mechanics are intimately related to scaling, one expects wavelet analysis to be very useful in the study of statistical-mechanical models as well. Osiris wavelets appear to be particularly well-suited for this purpose. One of the goals is to obtain cleaner hierarchical models by which renormalization group calculations of critical exponents can be determined.
The lattice models in classical equilibrium statistical mechanics are designed for the study of condensed matter physics, which deals with the critical behavior of solid materials with respect to changes in pressure, temperature, magnetic field intensity, etc. For example, the low temperatures at which supercomputers need to be maintained is a consequence of the fact that certain materials become super-conducters below a critical temperature. Other examples of critical behavior would include the agnetization of a bar of iron, which can be induced at room temperature, but not above a certain temperature. There are countless examples in nature. Wavelets are mathematical functions that are especially effective in recovering structures from compressed data. As such, they should be suited to the multiscale resolution of of these types of problems. This project aims to exploit the properties of a special class of wavelets, Osiris wavelets, to obtain better models in statistical mechanics.
