Examples



mdbootstrap.com



 
Статья
2022

Refinement of the Correlation Effects of Interacting Particles in the Ising Model


E. V. VotyakovE. V. Votyakov, Yu. K. TovbinYu. K. Tovbin
Российский журнал физической химии А
https://doi.org/10.1134/S003602442203027X
Abstract / Full Text

A numerical technique has been developed on the basis of the cluster variation method (CVM) for calculating the spatial distribution of particles in the Ising lattice model. The lattice is approximated by a basic cluster, which makes it possible to obtain an exact solution to the problem by increasing the cluster size. Universal expressions for the microscopic distribution of particles in clusters of any size were derived by expanding the cluster probabilities in terms of the correlation factors determined on smaller clusters inside the basic cluster. For the sites of the basic cluster, new variables were introduced in order to preserve the structure of the relationships of probabilistic correlators with smaller clusters at any size of the basic cluster. It was shown that the correlation factor symmetrically reflects multisite spin correlations. It was found that expansion in terms of correlation factors allows one to avoid the difficulties in calculations in the CVM caused by the need to use the method of indefinite Lagrange multipliers for cluster probabilities. The difference between the new spin variables and the conventional variables in the CVM was discussed. The efficiency of the technique was demonstrated using a square planar lattice as an example, for which the exact Onsager solution is known. For clusters of 16 sites, the difference between the critical temperature and the exact value was found to be ~2%.

Author information
  • The Cyprus Institute, Energy Environment and Water Research Center, Nicosia, 2121, CyprusE. V. Votyakov
  • Kurnakov Institute of General and Inorganic Chemistry, Russian Academy of Sciences, 119991, Moscow, RussiaYu. K. Tovbin
References
  1. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics (Cambridge Univ., Cambridge, 1939).
  2. J. O. Hirschfelder, Ch. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954).
  3. E. A. Moelwin-Hughes, Physical Chemistry (Pergamon, London, 1961), Part 2.
  4. V. A. Kireev, Course on Physical Chemistry (Khimiya, Moscow, 1975) [in Russian].
  5. A. M. Krivoglaz and A. A. Smirnov, The Theory of Ordering Alloys (Fizmatlit, Moscow, 1958) [in Russian].
  6. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Clarendon, Oxford, 1971).
  7. A. G. Khachaturyan, The Theory of Phase Transformations and the Structure of Solid Solutions (Nauka, Moscow, 1974) [in Russian].
  8. A. Z. Patashinskii and V. L. Pokrovskii, Fluctuation Theory of Phase Transitions (Nauka, Moscow, 1975) [in Russian].
  9. Sh. Ma, Modern Theory of Critical Phenomena (Benjamin, Reading, MA, 1976).
  10. L. Onsager, Phys. Rev. 65, 117 (1944).
  11. C. Domb, Proc. R. Soc. A 196, 36 (1949).
  12. C. Domb, Adv. Phys. 9, 149 (1960).
  13. T. Hill, Statistical Mechanics: Principles and Selected Applications (Dover, New York, 1987).
  14. K. Huang, Statistical Mechanics (Wiley, New York, 1987).
  15. D. Nicolson and N. G. Parsonage, Computer Simulation and the Statistical Mechanics of Adsorption (Academic, New York, 1982).
  16. Monte Carlo Methods in Statistical Physics, Ed. by K. Binder (Springer, Berlin, 1979).
  17. E. A. Guggenheim, Mixtures: The Theory of The Equlibrium Properties of Some Simple Classes of Mixtures Solutions and Alloys (Clarendon, Oxford, 1952).
  18. J. A. Barker, J. Chem. Phys. 20, 1526 (1952).
  19. I. Prigogine, The Molecular Theory of Solution (North Holland, Amsterdam, 1957).
  20. N. A. Smirnova, Molecular Models of Solutions (Khimiya, Moscow, 1987) [in Russian].
  21. Yu. K. Tovbin, Theory of Physical Chemical Processes at a Gas–Solid Surface Processes (Nauka, Moscow, 1990; CRC, Boca Raton, FL, 1991).
  22. Yu. K. Tovbin, Prog. Surf. Sci. 34, 1 (1990).
  23. R. Kikuchi, Phys. Rev. 81, 988 (1951).
  24. R. Kikuchi, J. Chem. Phys. 19, 1230 (1951).
  25. R. Kikuchi and S. G. Brush, J. Chem. Phys. 47, 195 (1967).
  26. J. A. Barker, Proc. R. Soc. London, Ser. A 216, 45 (1953).
  27. J. Hijmans and J. de Bour, Phys. A (Amsterdam, Neth.) 21, 471 (1955).
  28. J. M. Sanchez and D. de Fontaine, Phys. Rev. B 17, 2926 (1978).
  29. F. Sanchez, F. Ducastelle, and D. Gratias, Phys. A (Amsterdam, Neth.) 128, 334 (1984).
  30. Theory and Applications of the Cluster Variation and Path Probability Methods, Ed. by J. L. Moran-Lopez and J. M. Sanchez (Plenum, New York, 1996).