D. P. Feldman, Computational Mechanics of Classical Spin Systems Ph.D. Dissertation, Physics Department, University of California, Davis, (September, 1998).
How does nature self-organize and how can scientists discover such organization? Is there an objective notion of pattern, or is the discovery of patterns a purely subjective process? And what mathematical vocabulary is appropriate for describing and quantifying pattern, structure, and organization? This dissertation compares and contrasts the way in which statistical mechanics, information theory, and computational mechanics address these questions.
After an in-depth review of the statistical mechanical, information theoretic, and computational mechanical approaches to structure and pattern, I present exact analytic results for the excess entropy and epsilon-machines for one-dimensional, finite-range discrete classical spin systems. The excess entropy, a form of mutual information, is an information theoretic measure of apparent spatial memory. The epsilon-machine---the central object of computational mechanics---is defined as the minimal model capable of statistically reproducing a given configuration, where the model is chosen to belong to the least powerful model class(es) in a stochastic generalization of the discrete computation hierarchy.
These results for one-dimensional spin systems demonstrate 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, the excess entropy serves to detect ordered, low entropy density patterns. It is superior in many respects to other functions used to probe the structure of a distribution, such as structure factors and the specific heat. More generally, epsilon-machines 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. It is shown that the information theoretic and computational mechanical analyses of spatial patterns capture the intrinsic computational capabilities embedded in spin systems---how they store, transmit, and manipulate configurational information to produce spatial structure.
Finally, several approaches to generalizing the excess entropy and epsilon-machines to apply to multi-dimensional configurations are put forth. These measures are then calculated for a few simple, two-dimensional patterns.
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