A New Wavelet-Based Paradigm for Hierarchical Coarse-Graining applied to Materials Modeling* Ahmed E. Ismail, Gregory C. Rutledge, and George Stephanopoulos Department of Chemical Engineering, MIT, Cambridge, MA 02139 Abstract— We outline how the wavelet transform, a hi- erarchical averaging scheme, can be used to perform both structural and topological coarse-graining in systems with multiscale physical behavior, such as Ising lattices and polymer models. We illustrate how to create sampling mechanisms, which we call wavelet-accelerated Monte Carlo (WAMC), to study these systems and obtain qualitative and quantitatively accurate results in orders of magnitude less time than using atomistic simulations. I. I NTRODUCTION While there have been impressive computational gains afforded in recent years through advances both in computer hardware and in the expected gains predicted from Moore’s law, it is also clear that relying on improvements in com- putational performance will ultimately be insuf ficient for advancing the state-of-the-art in molecular simulation. In re- sponse, many researchers have turned their attention toward the development of a variety of algorithms for advancing the time and length scales accessible to molecular simulations. One limitation common to many of these algorithms is their limited focus—they are generally designed to study only a specific set of molecular chemistries. However, it is well known that many physical systems possess common structural properties, including self-similarity. As a result, we have created a new simulation paradigm for studying systems with structured behavior over various length scales which exploits these links. Our method is based upon the wavelet transform [1–3, among others]. Although most commonly used for signal processing and image analysis, we use it for its data com- pression properties. The basic principle in our application of the wavelet transform is that we use a characteristic property of our objects—such as the spin of a magnetic particle, or the position of an atom along the backbone of a chain molecule—as the basis for our sampling. The wavelet transform is then used to develop coarse-grained representations of “effective” or “block” variables, as well as potentials, describing the behavior of the system over successively larger length scales. This approach has several advantages, including generality with respect to the range of systems to which it can be applied; extensibility, as it can be used as the basis for a hierarchical simulation scheme; and ef ficiency, as the resulting algorithm is capable of yielding qualitatively accurate predictions of behavior over a wide range of parameters orders of magnitude faster than is capable with traditional atomistic simulations. We discuss our development of this paradigm, in the form of a new simulation technique, wavelet-accelerated Monte Carlo, through its application to lattice systems and to polymer random walks. To date, the wavelet framework transform has not been extensively applied to models in statistical mechanics. Huang uses wavelet analysis to observe the statistical dis- tribution of multiplicity fluctuations in a lattice gas [4], while Gamero et al. employ wavelets to introduce their notion of multiresolution entropy, but for dynamic signal analysis rather than statistical mechanics simulations [5]. O’Carroll attempts to establish a theoretical foundation connecting wavelets to the block renormalization group [6], [7]. The most extensive discussion of the relationship between wavelets and renormalization group theory is a recent monograph by Battle [8]. II. USING THE WAVELET TRANSFORM AS A COARSE- GRAINING SCHEME There are two principal means for carrying out coarse- graining of a physical system: we can call these approaches structural and topological. Structural approaches operate on the simulation space, dividing it into regions, and then combining the regions from one scale to another. Topolog- ical approaches operate on the particles inserted into the simulation space, so that the rules for coarse-graining the system do not depend upon any specific physical structure of the simulation space, such as a lattice, but instead upon the structure of the particles. Ising lattices and polymer chains typify each of these individual approaches. We show how to develop coarse-grained simulations for each of these models below. A. The Ising model The Ising model is the standard model for studying the thermodynamic behavior of lattice systems, such as spin magnets, lattice gases, and binary alloys [9]. The Hamil- tonian for the Ising model, which contains both nearest- neighbor pairwise interactions as well as interactions be- tween lattice sites and an external field, can be generally written in the form −βH = X i h i σ i + X i X j J ij σ i σ j , (1) where σ i is either the occupation number or the spin of lattice site i, h i is the strength of the external field in the Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30 - July 2, 2004 0-7803-8335-4/04/$17.00 ©2004 AACC WeM09.3 920