Discovering Noncritical Organization: Statistical Mechanical, Information Theoretical, and Computational Views of Patterns in One-Dimensional Spin Systems

We compare and contrast three different, but complementary views of "structure" and "pattern" in spatial processes. For definiteness and analytical clarity we apply all three approaches to the simplest class of spatial processes: one-dimensional Ising spin systems with finite-range interactions. These noncritical systems are well suited for this study since the change in structure as a function of system parameters is more subtle than that found in critical systems where, at a phase transition, many observables diverge thereby making the detection of change in structure obvious. This survey demonstrates that the measures of pattern from information theory and computational mechanics differ from known thermodynamic and statistical mechanical functions. Moreover, they capture important structural features that are otherwise missed. In particular, a type of mutual information called the excess entropy---an information theoretic measure of memory---serves to detect ordered, low entropy density patterns. It is superior in many respects to other functions used to probe the structure of a configuration distribution, such as magnetization and structure factors. $\epsilon$-Machines---the main objects of computational mechanics---are seen to be the most direct approach to revealing the (group and semigroup) symmetries possessed by the spatial patterns and to estimating the minimum amount of memory required to reproduce the configuration ensemble, a quantity known as the statistical complexity. Finally, we argue that the information theoretic and computational mechanical analyses of spatial patterns capture the intrinsic computational capabilities in spin systems---how they store, transmit, and manipulate configurational information to produce spatial structure.