### Multiresolution analysis in statistical mechanics: Wavelet-accelerated Monte Carlo simulations

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Ahmed Ismail, Massachusetts Institute of Technology
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Multiscale modeling of physical systems often requires the use of multiple types of simulations to bridge the various length scales that need to be considered: for example, a density-functional theory at the electronic scale will be combined with a molecular-dynamics simulation at the atomistic level, and with a finite-element method at the macroscopic level. An improvement to this scheme would be able to simulate consistently a system at multiple levels of resolution without passing from one simulation type to another, so that different simulations can be studied at a common length scale by appropriate coarse-graining or refinement of a given model.

One numerical method for coarse-graining is the wavelet transform, which converts a data set into sets of local averages and differences. As a hierarchical method, it can be used to rescale a Hamiltonian to a desired length scale, and at the same time also rescales the particles of the system by creating “blocked” particles (in the spirit of renormalization group calculations). The wavelet-accelerated Monte Carlo algorithm performs a Monte Carlo simulation on a small system which will be transformed into a block particle to obtain the probability distribution of the blocked particle; a Monte Carlo simulation is then performed on the resulting system of blocked particles. This method, which can be repeated as needed, can achieve significant speedups in computational time, while obtaining useful information about the thermodynamic behavior of the system.

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.

**Abstract Author(s): **Ahmed E. Ismail, Gregory C. Rutledge, and George Stephanopoulos<br />Department of Chemical Engineering<br />Massachusetts Institute of Technology<br />Cambridge, MA