Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T14:48:45.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Linked-cluster expansions for the correlation functions of lattice systems

Published online by Cambridge University Press:  24 October 2008

John W. Essam
John W. Essam
Affiliation:
Department of Mathematics, Westfield College, University of London
Get access Rights & Permissions [Opens in a new window]

Extract

A method of deriving power series expansions for the correlation functions of lattice systems is described. The concept of lattice constants for rooted graphs is introduced and it is shown how the correlation functions can be expanded in terms of the lattice constants for connected graphs only. The weight functions in the expansions depend on correlation functions for finite systems and may, therefore, be determined by computer methods. This work is an extension of a method previously developed for the free energy. Applications to spin systems, percolation problems and the lattice gas are considered and sum rules for the weight functions are given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 67 , Issue 2 , March 1970 , pp. 523 - 534
Copyright
Copyright © Cambridge Philosophical Society 1970

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Sykes, M. F., Essam, J. W., Heap, B. R. and Hiley, B. J.Lattice constant systems and graph theory. J. Mathematical Phys. 7 (1966), 1557–1572.CrossRefGoogle Scholar
(2)Essam, J. W.Determination of weight factors in linked-cluster expansions for lattice systems. J. Mathematical Phys. 8 (1967), 741–749.CrossRefGoogle Scholar
(3)Oguchi, T.Statistics of the three dimensional ferro-magnet. Phys. Rev. 76 (1950), 1001.CrossRefGoogle Scholar
(4)Fisher, M. E. and Burford, R. J.Theory of critical point scattering and correlations I. Ising model. Phys. Rev. 156 (1967), 583–622.CrossRefGoogle Scholar
(5)Domb, C. and Sykes, M. F.The calculation of lattice constants in crystal statistics. Philos. Mag. 2 (1957), 733–750.CrossRefGoogle Scholar
(6)Broadbent, S. R. and Hammersley, J. M.Percolation processes I. Crystals and mazes. Proc. Cambridge Philos. Soc. 53 (1957), 629–641.CrossRefGoogle Scholar
(7)Essam, J. W. To be published.Google Scholar
(8)Gaunt, D. S. and Fisher, M. E.Hard-sphere lattice gases I. Plane-square lattice. J. Chem. Phys. 43 (1965), 2840–2863.CrossRefGoogle Scholar
(9)Domb, C. and Sykes, M. F.Cluster size in random mixtures and percolation processes. Phys. Rev. 122 (1961), 77–78.CrossRefGoogle Scholar
(10)Elliot, R. J., Heap, B. R., Morgan, B. R. and Rushbrooke, G. S.Equivalence of the critical concentrations in the Ising and Heisenberg models of ferromagnetism. Phys. Rev. Lett. 5 (1960), 366–367.CrossRefGoogle Scholar
(11)Morgan, D. J. and Rushbrooke, G. S.On magnetically dilute Heisenberg and Ising ferromagnetics III. Concentration expansions for the Heisenberg model. Molecular Phys. 6 (1963), 477–488.CrossRefGoogle Scholar