Discrete, amorphous physical models

Physical modelling is the process of finding a mathematical description that is consistent with observations of the physical world. Many models, from the heat equation to the Schrödinger equation, are formulated in the continuous language of differential equations. Space and time are thought of as a continuum, hence the models potentially specify detail down to arbitrarily fine scales. Discrete physical models are an attractive alternative to continuous models such as partial differential equations. In discrete models, space is treated as a lattice, and time is discrete. Physical processes are modelled by rules that typically depend on a small number of nearby locations. From a theoretical standpoint, such models have the advantage that they do not have infinitely many locations per unit volume. From a practical standpoint, they correspond to the discrete structure of digital computing machines, and this makes them natural for simulation. Formulations in which space and time are discrete are widely used. Most often, this is done in order to approximate continuous models: space and time are partitioned in order to integrate a model in the form of a differential equation on a computer. By contrast, some models have been proposed as alternatives to, rather than an approximations of, differential equations[16]. Cellular automata (CA’s)[9] and lattice gases[11] are widely-used classes of discrete models in which space is modelled as a regular lattice, with a state associated with each lattice site. Periodically, all sites on the lattice simultaneously update their states. The rule that determines the new state is local, depending only on nearby sites, and is a function of the state of the neighboring sites in the previous time step. CA’s and lattice gases have been used with success to study many different phenomena such as fluid dynamics, polymers, and