Maximization of Cluster Entropies Via an Irreversible Algorithm: Applications to the Cluster Variation Method

  • Victor L. Vinograd
  • Udo Becker
  • Andrew Putnis


The problem of searching for the free energy minimum in cluster-variation models which take account of nearest-neighbor (nn) interactions can be reduced to two separate tasks:
  1. 1)

    entropy maximization constrained by fixed nn pair correlation functions,

  2. 2)

    free energy minimization with respect to the nn pair correlation functions.


The first task reduces to the maximization of entropy of a basic cluster, with respect to the correlation functions which correspond to all the subclusters larger than the pair. The simplification becomes possible with the use of an irreversible algorithm for the cluster entropy maximization. With the help of the irreversible algorithm the cluster entropy can be found directly as a function of the nn pair correlation functions, avoiding explicit introduction of correlation functions of higher-order.

The significant decrease in the number of the cluster variables permits an easy evaluation of CVM approximations based on large basic clusters. The method is discussed in relation to the two-dimensional (square lattice) Ising model for which a series of approximations (up to a 16-point basic cluster) is developed.


Ising Model Total Entropy Basic Cluster Pair Probability Cluster Probability 
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Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Victor L. Vinograd
    • 1
  • Udo Becker
    • 1
  • Andrew Putnis
    • 1
  1. 1.Institute of MineralogyMünster UniversityMünsterGermany

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