(cont.) time, while obtaining useful information about the thermodynamic behavior of the system. We show how statistical mechanics can be formulated using the wavelet transform as a coarse-graining technique. For small systems in which exact enumerations of all states is possible, we illustrate how the method recovers reasonably good estimates for physical properties (errors no more than 10%) with several orders of magnitude fewer operations than are required for an exact enumeration. In addition, we illustrate that errors introduced by the wavelet transform vanish in the neighborhood of fixed points of systems as determined by RG theory. Using scaling results from simulations at different length scales, we estimate the thermodynamic behavior of the original system without performing simulations on the full original system. In addition, we make the method adaptive by using fluctuation properties of the system to set criteria under which further coarse graining or refinement of the system is required. We demonstrate our method for the Ising universality class of problems. We also examine the applicability of the WAMC framework to polymer chains. Polymers are quintessential examples of the need for simulations at multiple scales: at one end, we can study short chains using quantum chemistry methods; yet polymers can have relaxation times on the order of seconds or longer, and molecular weights of 10⁶ or more. Even with modern computational resources, simulating behavior at long times or for long chains is still prohibitively expensive ...
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