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  • 1
    Electronic Resource
    Electronic Resource
    Tiling problems and undecidability in the cluster variation method (1988)
    Springer
    Journal of statistical physics 50 (1988), S. 689-714 
    Details
    ISSN: 1572-9613
    Keywords: Classical lattice system ; variational principle ; cluster variation method ; tiling problem ; undecidability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract In cluster approximations for lattice systems the thermodynamic behavior of the infinite system is inferred from that of a relatively small, finite subsystem (cluster), approximations being made for the influence of the surrounding system. In this context we study, for translation-invariant classical lattice systems, the conditions under which a state for a cluster admits an extension to a global translation-invariant state. This extension problem is related to undecidable tiling problems. The implication is that restrictions of global translation-invariant states cannot be characterized purely locally in general. This means that there is an unavoidable element of uncertainty in the application of a cluster approximation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01026496
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    On some variational approximations in two-dimensional classical lattice systems (1985)
    Springer
    Journal of statistical physics 40 (1985), S. 1-27 
    Details
    ISSN: 1572-9613
    Keywords: Classical lattice systems ; variational principle ; translation-invariant equilibrium states ; cluster-variation method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract A new derivation is presented of some variational approximations for classical lattice systems that belong to the class of cluster-variation methods, among them the well-known Bethe-Peierls and Kramers-Wannier approximations. The limiting behavior of a hierarchical sequence of cluster-variation approximations, the so-calledC hierarchy, is discussed. It is shown that this hierarchy provides a monotonically decreasing sequence of upper boundsf n on the free energy per lattice sitef and thatf n → f asn → ∞. Our results are based on extension theorems for states given on subsets of the lattice, which might be of some independent interest, and on an application of transfer matrix concepts to the variational characterization of translation-invariant equilibrium states.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01010524
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Exact variational methods and cluster-variation approximations (1984)
    Springer
    Journal of statistical physics 35 (1984), S. 285-301 
    Details
    ISSN: 1572-9613
    Keywords: Lattice systems ; translation-invariant equilibrium states ; variational principle ; cluster-variation method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract It is shown that for classical,d-dimensional lattice models with finite-range interactions the translation-invariant equilibrium states have the property that their mean entropy is completely determined by their restriction to a subset of the lattice that is infinite in (d−1) dimensions and has a width equal to the range of the interaction in the dth dimension. This property is used to show proper convergence toward the exact result for a hierarchy of approximations of the cluster-variation method that uses one-dimensionally increasing basis clusters in a two-dimensional lattice.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01014385
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Two-sided bounds on the free energy from local states in Monte Carlo simulations (1989)
    Springer
    Journal of statistical physics 56 (1989), S. 247-260 
    Details
    ISSN: 1572-9613
    Keywords: Monte Carlo simulations ; free energy calculation ; lattice models
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We show that a precise assessment of free energy estimates in Monte Carlo simulations of lattice models is possible by using cluster variation approximations in conjunction with the local states approximations proposed by Meirovitch. The local states method (LSM) utilizes entropy expressions which recently have been shown to correspond to a converging sequence of upper bounds on the thermodynamic limit entropy density (i.e., entropy per lattice site), whereas the cluster variation method (CVM) supplies formulas that in some cases have been proven to be, and in other cases are believed to be, lower bounds. We have investigated CVM-LSM combinations numerically in Monte Carlo simulations of the two-dimensional Ising model and the two-dimensional five-states ferromagnetic Potts model. Even in the critical region the combination of upper and lower bounds enables an accurate and reliable estimation of the free energy from data of a single run. CVM entropy approximations are therefore useful in Monte Carlo simulation studies and in establishing the reliability of results from local states methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01044436
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    Paper (German National Licenses)
    Fulltext
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