Discovering Noncritical Organization: Statistical Mechanical, Information Theoretic, 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. Th...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-09-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/24/9/1282 |
| _version_ | 1797488685445283840 |
|---|---|
| author | David P. Feldman James P. Crutchfield |
| author_facet | David P. Feldman James P. Crutchfield |
| author_sort | David P. Feldman |
| collection | DOAJ |
| description | 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 <i>excess entropy</i>—an information theoretic measure of memory—serves to detect ordered, low entropy density patterns. It is superior in several respects to other functions used to probe structure, such as magnetization and structure factors. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-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 <i>statistical complexity</i>. Finally, we argue 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. |
| first_indexed | 2024-03-10T00:05:50Z |
| format | Article |
| id | doaj.art-fc944dc1f4474fca91f59c90df3f076b |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T00:05:50Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-fc944dc1f4474fca91f59c90df3f076b2023-11-23T16:09:02ZengMDPI AGEntropy1099-43002022-09-01249128210.3390/e24091282Discovering Noncritical Organization: Statistical Mechanical, Information Theoretic, and Computational Views of Patterns in One-Dimensional Spin SystemsDavid P. Feldman0James P. Crutchfield1College of the Atlantic, Bar Harbor, ME 04609, USADepartment of Physics, University of California, Davis, CA 95616, USAWe 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 <i>excess entropy</i>—an information theoretic measure of memory—serves to detect ordered, low entropy density patterns. It is superior in several respects to other functions used to probe structure, such as magnetization and structure factors. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>-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 <i>statistical complexity</i>. Finally, we argue 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.https://www.mdpi.com/1099-4300/24/9/1282spin systemsstatistical mechanicsthermodynamicsinformation theory, computational mechanics |
| spellingShingle | David P. Feldman James P. Crutchfield Discovering Noncritical Organization: Statistical Mechanical, Information Theoretic, and Computational Views of Patterns in One-Dimensional Spin Systems Entropy spin systems statistical mechanics thermodynamics information theory, computational mechanics |
| title | Discovering Noncritical Organization: Statistical Mechanical, Information Theoretic, and Computational Views of Patterns in One-Dimensional Spin Systems |
| title_full | Discovering Noncritical Organization: Statistical Mechanical, Information Theoretic, and Computational Views of Patterns in One-Dimensional Spin Systems |
| title_fullStr | Discovering Noncritical Organization: Statistical Mechanical, Information Theoretic, and Computational Views of Patterns in One-Dimensional Spin Systems |
| title_full_unstemmed | Discovering Noncritical Organization: Statistical Mechanical, Information Theoretic, and Computational Views of Patterns in One-Dimensional Spin Systems |
| title_short | Discovering Noncritical Organization: Statistical Mechanical, Information Theoretic, and Computational Views of Patterns in One-Dimensional Spin Systems |
| title_sort | discovering noncritical organization statistical mechanical information theoretic and computational views of patterns in one dimensional spin systems |
| topic | spin systems statistical mechanics thermodynamics information theory, computational mechanics |
| url | https://www.mdpi.com/1099-4300/24/9/1282 |
| work_keys_str_mv | AT davidpfeldman discoveringnoncriticalorganizationstatisticalmechanicalinformationtheoreticandcomputationalviewsofpatternsinonedimensionalspinsystems AT jamespcrutchfield discoveringnoncriticalorganizationstatisticalmechanicalinformationtheoreticandcomputationalviewsofpatternsinonedimensionalspinsystems |