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What Is a Pattern in Statistical Mechanics? Formalizing Structure and Patterns in One-Dimensional Spin Lattice Models
1Physics Department, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA.
Entropy (Basel, Switzerland)
|January 28, 2026
Summary
This study uses computational mechanics to quantify structure and pattern in spin-lattice models. It finds that information-theoretic measures align with statistical mechanics predictions for these models.
Area of Science:
- Statistical Mechanics
- Information Theory
- Computational Mechanics
- Condensed Matter Physics
Background:
- Spin-lattice models are fundamental in statistical mechanics.
- Understanding their structure and patterns is crucial for predicting material properties.
- Existing methods may not fully capture the complexity of these systems.
Purpose of the Study:
- To formalize structure and pattern in one-dimensional spin-lattice models.
- To apply information-theoretic and computational mechanics frameworks.
- To bridge concepts from statistical mechanics and computational mechanics.
Main Methods:
- Novel derivation of the Boltzmann distribution for spin configurations.
- Recasting the Boltzmann distribution as a stochastic process.
- Analysis using computational mechanics: excess entropy (E) and statistical complexity (Cμ).
- Characterization of the structure-generating mechanism via ϵ-machines.
Main Results:
- Quantified structure and pattern in finite-range Ising, solid-on-solid, and three-body models.
- Demonstrated agreement between information-theoretic measures (E, Cμ) and typical configurations from the Boltzmann distribution.
- Validated the compatibility of computational mechanics with statistical mechanics for these models.
Conclusions:
- Computational mechanics provides a robust framework for analyzing spin-lattice models.
- Information-theoretic measures effectively quantify structural properties.
- The study offers a unified approach to understanding complex systems in physics.
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