Statistical Mechanics and Related Probability Theory

University Of Southern California, Los Angeles CA

Investigators

Abstract

Probability models from statistical mechanics are a framework for studying how small-scale randomness produces global-scale phenomena, such as phase transitions, which are essentially nonrandom. Alexander proposes to investigate the following aspects of the subject. (1) Pinning and depinning of lattice polymers and interfaces due to potentials existing in lower-dimensional subspaces of the lattice. Particular emphasis will be placed on the disordered case, in which the potential varies randomly with location. Competition between a potential in a subspace and random potentials in the bulk will also be considered. (2) Layering transitions in the three-dimensional Ising model without an external field, caused solely by the attraction of an interface to a wall in competition with entropic repulsion. (3) Convergence to equilibrium, in systems modeling the phase separation that occurs in the freezing of solutions. (4) Moving interfaces, e.g. between phases of the Ising model, which may "hang up" at locations where a weakened interaction reduces the energetic cost of the interface, resulting in an energy barier which must be crossed.(5) Eigenvalues of the covariance matrix of the two-dimensional Potts model and their relation to decay rates of certain probabilities in the FK model. (6) Potts models in which the external field(s) and the boundary condition are opposed to each other. The research can help provide a theoretical underpinning for applied work in areas such as polymers, adhesion, and superconductivity. The work may lead to predictions which can be tested by experimentalists, leading to new practical discoveries. Introducing a mathematician's perspective to theoretical problems that have previously been considered principally by physicists may in general alter the physicists' perspective,leading to further advances. US-French scientiÞc partnership is enhanced by Alexander's collaboration with Francois Dunlop and Salvador Miracle-Sole. This work is part of an ongoing effort by mathematicians and physicists to understand various systems in the natural world in which nonrandom global-scale phenomena reþect aspects of small-scale randomness. Examples include (i) magnetic properties of materials; (ii) waves traveling through irregular materials, such as seismic waves through the earth's crust; (iii) impurities in semiconductors;(iv) denaturation of DNA; and (v) percolation of liquid through a porous material, such as water or oil through underground rock. It has long been understood that many qualitative aspects of the relation between small-scall randomness and macroscopic properties,including critical phenomena, do not depend too closely on the particular system being studied. One can therefore gain insight into real-world phenomena by studying abstract systems not intended to model specifically magnets, or porous rock, or any other particular part of the physical world. The systems need only exhibit parallel features, such as clustering and critical phenomena. Some of the systems which Alexander will investigate-percolation, Ising and Potts models, and other spin systems-are examples of such abstract systems. Other systems which Alexander will investigate are somewhat more closely based on specific physical systems, such as polymers, DNA molecules, and solutions.