Solution of lattice gas models in the generalized ensemble on the bethe lattice

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 59, Issue 6, 1999, Pages 6348-6355

  La Nave, Emilia ^a,b   Sastry, Srikanth ^c   Sciortino, Francesco ^a   Tartaglia, Piero ^a  

^a UNIVERSITÀ DI ROMA LA SAPIENZA   (Italy)

^b Boston University   (United States)

^c JAWAHARLAL NEHRU CENTRE FOR ADVANCED SCIENTIFIC RESEARCH   (India)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE;

EID: 4243163288     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.59.6348     Document Type: Article

References (29)

1. G. M. Bell and D. A. Lavis, Statistical Mechanics of Lattice Models (Halsted, Chichester, 1989).
2. K. Huang, Statistical Mechanics (Wiley, New York, 1987).
3. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, London, 1982).
4. P. D. Gujrati, Phys. Rev. Lett. 74, 809 (1995).
5. S. Sastry, P. G. Debenedetti, F. Sciortino, and H. E. Stanley, Phys. Rev. E 53, 6144 (1996).
6. T. L. Hill, Statistical Mechanics (Dover, New York, 1987). The generalized ensemble arises as the appropriate ensemble to describe a physical system when the system is in thermal (exchange of heat), mechanical (change of volume), and material (exchange of particles) equilibrium with its surroundings. A simple example of such a system is a crystal immersed in a solution of the same material. More practical and real world examples may be found, such as the typical setup for osmotic stress experiments 27 that have been used extensively to probe forces organizing biomolecular systems.
7. F. Peruggi, F. di Liberto, and G. Monroy, J. Phys. A 16, 811 (1983).
8. For a recent review of the state of the art in the thermodynamics of supercooled and stretched water, see P. G. Debenedetti, Metastable Liquids (Princeton University Press, Princeton 1997).
9. F. Sciortino, in The Physics of Complex Systems, edited by F. Mallamace and M. E. Stanley (IOS Press, Amsterdam, 1997).
10. Note that no orientational states are associated with an unoccupied site. Thus the orientational degeneracy of an unoccupied site is equal to 1.
11. The exact solution of the model was also obtained in one dimension in Ref. 5, which reduces to the correct result for a simple lattice gas in the appropriate limit of fixed volume per site and orientational state.
12. Restricting attention to nearest neighbor interactions, the equivalence of an exact solution for the Bethe lattice and the Bethe-Peierls approximation can easily be seen by noting that the absence of loops in the Cayley tree makes it possible to represent exactly the influence of the “rest of the system” by an effective chemical potential.
13. Note in this context that the (Formula presented) in the (Formula presented)’s above indicates that the (Formula presented) are evaluated for an arbitrary fixed value of the (Formula presented); the degeneracies are accounted for explicitly as described in points (ii), (iv), and (v) of Sec. III A.
14. When the volume per site is held fixed and the number of orientational states (Formula presented), the above EOS equals that of a simple lattice gas (e. g., see Ref. 2), and in this limit, should coincide with the equivalent special case of the equation of state for the interacting multicomponent polymer mixture obtained by Gujrati 24 on the Bethe lattice.
15. D. Stauffer, Introduction to Percolation Theory (Taylor & Francis, London, 1985).
16. H. E. Stanley and J. Teixeira, J. Chem. Phys. 73, 3404 (1980).
17. D. Bertolini, P. Grigolini, and A. Tani, J. Chem. Phys. 91, 1191 (1989).
18. F. Sciortino and S. Fornili, J. Chem. Phys. 90, 2786 (1989).
19. A. Coniglio and J. W. Essam, J. Phys. A 10, 1917 (1977).
20. J. L. Lebowitz and O. Penrose, J. Stat. Phys. 16, 321 (1977).
21. J. W. Essam, Rep. Prog. Phys. 43, 833 (1980).
22. We note that the same substitution has been used to solve the recurrence equation (23).
23. K. K. Murata, J. Phys. Chem. 12, 81 (1978).
24. P. D. Gujrati, J. Chem. Phys. 108, 6952 (1998).
25. R. J. Speedy, J. Phys. Chem. 86, 982 (1982)
26. J. Phys. Chem.R. J. Speedy86, 3002 (1982).
27. P. G. Debenedetti and M. C. D’Antonio, J. Chem. Phys. 84, 3339 (1986)
28. J. Chem. Phys.P. G. DebenedettiM. C. D’Antonio85, 4005 (1986).
29. V. A. Parsegian, R. P. Rand, and D. C. Rau, in Methods in Enzymology, edited by M. L. Johnson and G. K. Ackers (Academic, San Diego, 1995), Vol. 259, Chap. 3, pp 43–94.