Statistical mechanical modeling: Computer simulations, analysis and applications.

摘要

This thesis describes the applications of statistical mechanical models and tools, especially computational techniques to the study of several problems in science.; We study in chapter 2, various properties of a non-equilibrium cellular automaton model, the Toom model. We obtain numerically the exponents describing the fluctuations of the interface between the two stable phases of the model.; In chapter 3, we introduce a binary alloy model with three-body potentials. Unlike the usual Ising-type models with two-body interactions, this model is not symmetric in its components. We calculate the exact low temperature phase diagram using Pirogov-Sinai theory and also find the mean-field equilibrium properties of this model. We then study the kinetics of phase segregation following a quenching in this model. We find that the results are very similar to those obtained for Ising-type models with pair interactions, indicating universality.; In chapter 4, we discuss the statistical properties of “Contact Maps”. These maps, are used to represent three-dimensional structures of proteins in modeling problems. We find that this representation space has particular properties that make it a convenient choice. The maps representing native folds of proteins correspond to compact structures which in turn correspond to maps with low degeneracy, making it easier to translate the map into the detailed 3-dimensional structure.; The early stage of formation of a river network is described in Chapter 5 using quasi-random spanning trees on a square lattice. We observe that the statistical properties generated by these models are quite similar (better than some of the earlier models) to the empirical laws and results presented by geologists for real river networks.; Finally, in chapter 6 we present a brief note on our study of the problem of progression of heterogeneous breast tumors. We investigate some of the possible pathways of progression based on the traditional notions of DCIS (Ductal Carcinoma in Situ) being a precursor of IDC (Invasive Ductal Carcinoma). We provide quantitative evidence about the fact that the traditional notion of progression might not be correct.