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Pecularities of Bethe-like approximations and long-range-interaction Ising models.

James L Monroe1

  • 1Department of Physics, Penn State University--Beaver Campus, 100 University Dr., Monaca, Pennsylvania 15061, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 4, 2003
PubMed
Summary

The study challenges common assumptions about cluster approximations for spin systems. It reveals that for certain 1D Ising models, Bethe cluster approximations can fail to provide upper bounds or improve with larger clusters.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Computational Physics

Background:

  • Mean-field and Bethe approximations are standard tools for studying phase diagrams and critical temperatures in lattice spin systems.
  • These methods are often generalized to cluster approximations, with the expectation of improved accuracy.
  • Typically, these approximations are assumed to provide upper bounds on critical temperatures and improve with larger cluster sizes.

Purpose of the Study:

  • To investigate the validity of commonly held characteristics of cluster mean-field and Bethe approximations.
  • To examine the behavior of these approximations for one-dimensional Ising models with slowly decaying interactions.

Main Methods:

  • Analysis of one-dimensional Ising models.
  • Application and generalization of mean-field and Bethe approximation techniques to cluster versions.

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  • Theoretical investigation of approximation properties.

    • Main Results:

    • Demonstrated that for specific one-dimensional Ising models, Bethe cluster approximations violate expected characteristics.
    • Showed violations of the upper bound property for critical temperature.
    • Observed that increasing cluster size did not necessarily lead to better approximations in these models.

    Conclusions:

    • The findings challenge the universal applicability of standard assumptions regarding cluster approximations in statistical mechanics.
    • Results highlight the importance of model-specific analysis, particularly for systems with long-range interactions.
    • Suggests a need for caution when applying generalized mean-field and Bethe approximations to diverse spin systems.