skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Ergodic properties and thermodynamic behavior of elementary reversible cellular automata. I. Basic properties

Abstract

This is the first part of a series devoted to the study of thermodynamic behavior of large dynamical systems with the use of a family of full-discrete and conservative models named elementary reversible cellular automata (ERCAs). In this paper, basic properties such as conservation laws and phase space structure are investigated in preparation for the later studies. ERCAs are a family of one-dimensional reversible cellular automata having two Boolean variables on each site. Reflection and Boolean conjugation symmetries divide them into 88 equivalence classes. For each rule, additive conserved quantities written in a certain form are regarded as a kind of energy, if they exist. By the aid of the discreteness of the variables, every ERCA satisfies the Liouville theorem or the preservation of phase space volume. Thus, if an energy exists in the above sense, statistical mechanics of the model can formally be constructed. If a locally defined quantity is conserved, however, it prevents the realization of statistical mechanics. The existence of such a quantity is examined for each class and a number of rules which have at least one energy but no local conservation laws are selected as hopeful candidates for the realization of thermodynamic behavior. In addition,more » the phase space structure of ERCAs is analyzed by enumerating cycles exactly in the phase space for systems of comparatively small sizes. As a result, it is revealed that a finite ERCA is not ergodic, that is, a large number of orbits coexist on an energy surface. It is argued that this fact does not necessarily mean the failure of thermodynamic behavior on the basis of an analogy with the ergodic nature of infinite systems.« less

Authors:
 [1]
  1. Kyoto Univ. (Japan)
Publication Date:
OSTI Identifier:
6039645
Resource Type:
Journal Article
Journal Name:
Journal of Statistical Physics; (USA)
Additional Journal Information:
Journal Volume: 56:3-4; Journal ID: ISSN 0022-4715
Country of Publication:
United States
Language:
English
Subject:
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; CRYSTAL MODELS; ERGODIC HYPOTHESIS; COMPUTERIZED SIMULATION; CONSERVATION LAWS; CRYSTAL LATTICES; HAMILTONIANS; MATRICES; ONE-DIMENSIONAL CALCULATIONS; PHASE SPACE; STATISTICAL MECHANICS; SURFACES; SYMMETRY; THERMODYNAMIC PROPERTIES; THERMODYNAMICS; CRYSTAL STRUCTURE; HYPOTHESIS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; MATHEMATICAL SPACE; MECHANICS; PHYSICAL PROPERTIES; QUANTUM OPERATORS; SIMULATION; SPACE; 656002* - Condensed Matter Physics- General Techniques in Condensed Matter- (1987-)

Citation Formats

Takesue, Shinji. Ergodic properties and thermodynamic behavior of elementary reversible cellular automata. I. Basic properties. United States: N. p., 1989. Web. doi:10.1007/BF01044442.
Takesue, Shinji. Ergodic properties and thermodynamic behavior of elementary reversible cellular automata. I. Basic properties. United States. https://doi.org/10.1007/BF01044442
Takesue, Shinji. 1989. "Ergodic properties and thermodynamic behavior of elementary reversible cellular automata. I. Basic properties". United States. https://doi.org/10.1007/BF01044442.
@article{osti_6039645,
title = {Ergodic properties and thermodynamic behavior of elementary reversible cellular automata. I. Basic properties},
author = {Takesue, Shinji},
abstractNote = {This is the first part of a series devoted to the study of thermodynamic behavior of large dynamical systems with the use of a family of full-discrete and conservative models named elementary reversible cellular automata (ERCAs). In this paper, basic properties such as conservation laws and phase space structure are investigated in preparation for the later studies. ERCAs are a family of one-dimensional reversible cellular automata having two Boolean variables on each site. Reflection and Boolean conjugation symmetries divide them into 88 equivalence classes. For each rule, additive conserved quantities written in a certain form are regarded as a kind of energy, if they exist. By the aid of the discreteness of the variables, every ERCA satisfies the Liouville theorem or the preservation of phase space volume. Thus, if an energy exists in the above sense, statistical mechanics of the model can formally be constructed. If a locally defined quantity is conserved, however, it prevents the realization of statistical mechanics. The existence of such a quantity is examined for each class and a number of rules which have at least one energy but no local conservation laws are selected as hopeful candidates for the realization of thermodynamic behavior. In addition, the phase space structure of ERCAs is analyzed by enumerating cycles exactly in the phase space for systems of comparatively small sizes. As a result, it is revealed that a finite ERCA is not ergodic, that is, a large number of orbits coexist on an energy surface. It is argued that this fact does not necessarily mean the failure of thermodynamic behavior on the basis of an analogy with the ergodic nature of infinite systems.},
doi = {10.1007/BF01044442},
url = {https://www.osti.gov/biblio/6039645}, journal = {Journal of Statistical Physics; (USA)},
issn = {0022-4715},
number = ,
volume = 56:3-4,
place = {United States},
year = {Tue Aug 01 00:00:00 EDT 1989},
month = {Tue Aug 01 00:00:00 EDT 1989}
}