The Lattice and Ising Models

Abstract

As introduced in Chap. 4, the lattice model is a highly coarse-grained model of statistical mechanics for particle systems, with built-in excluded-volume interaction . The model can address the structural and thermodynamic properties on length scales much larger than molecular size. To incorporate the configurational degrees of freedom of many-particle systems, the system is decomposed into identical cells over which the particles are distributed. With the short-range interaction between the adjacent particles included, this seemingly simple model can be usefully extended to a variety of problems such as gas-to liquid transitions, molecular binding on substrates, and mixing and phase separation of binary mixtures. For the particles that are mutually interacting in two and three dimensions, we will introduce the mean field approximations. The lattice model is isomorphic to the Ising model that describes magnetism and paramagnet-to-ferromagnetic transitions. We study the exact solution for the Ising model in one dimension, which is applied to a host of biopolymer properties and the two-state transitions.