Convergence of fine-lattice discretization for near-critical fluids

Journal of Physical Chemistry B

Volumn 109, Issue 14, 2005, Pages 6824-6837

  Moghaddam, Sarvin ^a,b   Kim, Young C ^b   Fisher, Michael E ^b  

^a UNIVERSITY OF MARYLAND   (United States)

^b Princeton University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CONTINUUM MODEL; CRITICAL TEMPERATURE; FINE-LATTICE DISCRETIZATION; NEAR-CRITICAL FLUIDS;

CONTINUUM MECHANICS; CONVERGENCE OF NUMERICAL METHODS; EXTRAPOLATION; HEURISTIC METHODS; MATHEMATICAL MODELS; PARAMETER ESTIMATION; PHASE SEPARATION; THERMODYNAMIC PROPERTIES;

FLUID MECHANICS;

EID: 17444409648     PISSN: 15206106     EISSN: None     Source Type: Journal    
DOI: 10.1021/jp0406417     Document Type: Article

References (43)

1. Fisher, M. E. J. Stat. Phys. 1995, 75, 1.
2. Panagiotopoulos, A. Z.; Kumar, S. K. Phys. Rev. Lett. 1999, 83, 2981.
3. Panagiotopoulos, A. Z. J. Chem. Phys. 2000, 112, 7132. It should be noted that the gain in speed afforded by the fine discretization technique can be sensitive to programming details, machine architecture, compilation parameters, etc.
4. Panagiotopoulos, A. Z. J. Chem. Phys. 2002, 116, 3007.
5. Moghaddam, S.; Panagiotopoulos, A. Z. J. Chem. Phys. 2003, 118, 7556.
6. Luijten, E.; Fisher, M. E.; Panagiotopoulos, A. Z. Phys. Rev. Lett. 2002, 88, 185701.
7. Cheong, M. D. W.; Panagiotopoulos, A. Z. J. Chem. Phys. 2003, 119, 8526.
8. ζ is employed there in place of its reciprocal. Of course, no difference in the rates of convergence results.
9. max or somewhat larger, such features do not arise in the simple molecular models for ζ ≳ 4; thus they will not be a concern for us here.
10. 0 must be an integer. Alternatively, in terms of the oft-used notation, L* ≡ L/a, one must require that L*ζ is integral.
11. c(ζ) for L → ∞. In such extrapolations, allowance for the nonclassical values of the exponents β, γ, ν, etc., is crucial. Some recent theoretical and numerical studies bearing directly on this issue are: Orkoulas, G.; Fisher, M. E.; Panagiotopoulos, A. Z. Phys. Rev. E 2001, 63, 051507. Luijten, E.; Fisher, M. E.; Panagiotopoulos, A. Z. Phys. Rev. Lett. 2002, 88, 185701. Kim, Y. C.; Fisher, M. E.; Luijten, E. Phys. Rev. Lett. 2003, 91, 065701. Kim, Y. C.; Fisher, M. E. Phys. Rev. E 2003, 68, 041506; J. Phys. Chem. B 2004, 108, 6750 and ref 8.
12. c(ζ) for L → ∞. In such extrapolations, allowance for the nonclassical values of the exponents β, γ, ν, etc., is crucial. Some recent theoretical and numerical studies bearing directly on this issue are: Orkoulas, G.; Fisher, M. E.; Panagiotopoulos, A. Z. Phys. Rev. E 2001, 63, 051507. Luijten, E.; Fisher, M. E.; Panagiotopoulos, A. Z. Phys. Rev. Lett. 2002, 88, 185701. Kim, Y. C.; Fisher, M. E.; Luijten, E. Phys. Rev. Lett. 2003, 91, 065701. Kim, Y. C.; Fisher, M. E. Phys. Rev. E 2003, 68, 041506; J. Phys. Chem. B 2004, 108, 6750 and ref 8.
13. c(ζ) for L → ∞. In such extrapolations, allowance for the nonclassical values of the exponents β, γ, ν, etc., is crucial. Some recent theoretical and numerical studies bearing directly on this issue are: Orkoulas, G.; Fisher, M. E.; Panagiotopoulos, A. Z. Phys. Rev. E 2001, 63, 051507. Luijten, E.; Fisher, M. E.; Panagiotopoulos, A. Z. Phys. Rev. Lett. 2002, 88, 185701. Kim, Y. C.; Fisher, M. E.; Luijten, E. Phys. Rev. Lett. 2003, 91, 065701. Kim, Y. C.; Fisher, M. E. Phys. Rev. E 2003, 68, 041506; J. Phys. Chem. B 2004, 108, 6750 and ref 8.
14. c(ζ) for L → ∞. In such extrapolations, allowance for the nonclassical values of the exponents β, γ, ν, etc., is crucial. Some recent theoretical and numerical studies bearing directly on this issue are: Orkoulas, G.; Fisher, M. E.; Panagiotopoulos, A. Z. Phys. Rev. E 2001, 63, 051507. Luijten, E.; Fisher, M. E.; Panagiotopoulos, A. Z. Phys. Rev. Lett. 2002, 88, 185701. Kim, Y. C.; Fisher, M. E.; Luijten, E. Phys. Rev. Lett. 2003, 91, 065701. Kim, Y. C.; Fisher, M. E. Phys. Rev. E 2003, 68, 041506; J. Phys. Chem. B 2004, 108, 6750 and ref 8.
15. c(ζ) for L → ∞. In such extrapolations, allowance for the nonclassical values of the exponents β, γ, ν, etc., is crucial. Some recent theoretical and numerical studies bearing directly on this issue are: Orkoulas, G.; Fisher, M. E.; Panagiotopoulos, A. Z. Phys. Rev. E 2001, 63, 051507. Luijten, E.; Fisher, M. E.; Panagiotopoulos, A. Z. Phys. Rev. Lett. 2002, 88, 185701. Kim, Y. C.; Fisher, M. E.; Luijten, E. Phys. Rev. Lett. 2003, 91, 065701. Kim, Y. C.; Fisher, M. E. Phys. Rev. E 2003, 68, 041506; J. Phys. Chem. B 2004, 108, 6750 and ref 8.
16. Walfisz, A. Gitterpunkte in mehrdimensionalen Kugeln; PWN-Polish Scientific Publishers: Warsaw, 1957.
17. i(ζ) over a range of ζ values, that is essential in formulating precise analytical results.
18. Bleher, P. M.; Dyson, F. J. Acta Arith. 1994, 68, 383; erratum: 1995, 73, 199.
19. Bleher, P. M.; Dyson, F. J.; Lebowitz, J. L. Phys. Rev. Lett. 1993, 71, 3047.
20. Bleher, P. M. Emerging Applications of Number Theory, The IMA Volumes in Mathematics and its Applications, Vol. 109; Springer-Verlag: New York, 1999; pp 1-38.
21. Fisher, M. E.; Widom, B. J. Chem. Phys. 1969, 50, 3756.
22. See, e.g., Fisher, M. E. in unpublished work reported at the Second Eastern Theoretical Physics Conference, Chapel Hill, N. C., 25 October 1963 and at Institut des Hautes Études Scientifiques, Bures-sur-Yvette, Paris, 26 April 1965.
23. Kac, M.; Uhlenbeck, G. E.; Hemmer, P. J. Math. Phys. 1963, 4, 216, 299. Lebowitz, J. L.; Penrose, O. J. Math. Phys. 1963, 4, 248.
24. Kac, M.; Uhlenbeck, G. E.; Hemmer, P. J. Math. Phys. 1963, 4, 216, 299. Lebowitz, J. L.; Penrose, O. J. Math. Phys. 1963, 4, 248.
25. Lennard-Jones, J. E. Proc. R. Soc. London, Ser. A 1924, 106, 441, 463.
26. Buckingham, R. A. Proc. R. Soc. London, Ser. A 1938, 168, 264.
27. max.
28. Errington, J. R.; Panagiotopoulos, A. Z. J. Chem. Phys. 1998, 109, 1093.
29. See, e.g., McQuarrie, D. A. Statistical Mechanics; Harper Collins: New York, 1976, section 15-2, and Friedman, H. L. Ionic Solution Theory based on cluster expansion methods; Interscience Publishers: New York, 1962.
30. See, e.g., McQuarrie, D. A. Statistical Mechanics; Harper Collins: New York, 1976, section 15-2, and Friedman, H. L. Ionic Solution Theory based on cluster expansion methods; Interscience Publishers: New York, 1962.
31. Mayer, J. E.; Mayer, M. G. Statistical Mechanics; John Wiley: New York, 1940, chapter 13, and, e.g., McQuarrie, D. A. of ref 24, chapter 12.
32. See Fisher, M. E. Physica A 1990, 163, 15; in Proceedings of the Gibbs Symposium, Yale University, May 1989; Caldi, D. G., Mostow, G. D., Eds.; Am. Math. Soc.: Rhode Island, 1990; and references therein.
33. See Fisher, M. E. Physica A 1990, 163, 15; in Proceedings of the Gibbs Symposium, Yale University, May 1989; Caldi, D. G., Mostow, G. D., Eds.; Am. Math. Soc.: Rhode Island, 1990; and references therein.
34. See Fisher, M. E. Physica A 1990, 163, 15; in Proceedings of the Gibbs Symposium, Yale University, May 1989; Caldi, D. G., Mostow, G. D., Eds.; Am. Math. Soc.: Rhode Island, 1990; and references therein.
35. Fisher, M. E.; Felderhof, B. U. Ann. Phys. (N.Y.) 1970, 58, 176, 217. Felderhof, B. U.; Fisher, M. E. Ann. Phys. (N.Y.) 1970, 58, 268; Felderhof, B. U. Ann. Phys. (N.Y.) 1970, 58, 281.
36. Fisher, M. E.; Felderhof, B. U. Ann. Phys. (N.Y.) 1970, 58, 176, 217. Felderhof, B. U.; Fisher, M. E. Ann. Phys. (N.Y.) 1970, 58, 268; Felderhof, B. U. Ann. Phys. (N.Y.) 1970, 58, 281.
37. Fisher, M. E.; Felderhof, B. U. Ann. Phys. (N.Y.) 1970, 58, 176, 217. Felderhof, B. U.; Fisher, M. E. Ann. Phys. (N.Y.) 1970, 58, 268; Felderhof, B. U. Ann. Phys. (N.Y.) 1970, 58, 281.
38. See, e.g., Baxter, R. J. Exactly Solved Models in Statistical Mechanics; Academic Press: London, 1982.
39. Fisher, M. E. In Proceedings of the 1974 AIP Conference No. 24, Magnetism and Magnetic Materials; American Institute of Physics: New York, 1975; p. 273. Aharony, A. in Critical Phenomena, Lecture Notes in Physics No. 186, Ed. Hahne, F. J. W.; Springer: Berlin, 1983; p. 209.
40. Fisher, M. E. In Proceedings of the 1974 AIP Conference No. 24, Magnetism and Magnetic Materials; American Institute of Physics: New York, 1975; p. 273. Aharony, A. in Critical Phenomena, Lecture Notes in Physics No. 186, Ed. Hahne, F. J. W.; Springer: Berlin, 1983; p. 209.
41. For critical loci near bicritical points, see Fisher, M. E.; Nelson, D. R. Phys. Rev. Lett. 1974, 32, 1350. Fisher, M. E. Phys. Rev. Lett. 1975, 34, 1634.
42. For critical loci near bicritical points, see Fisher, M. E.; Nelson, D. R. Phys. Rev. Lett. 1974, 32, 1350. Fisher, M. E. Phys. Rev. Lett. 1975, 34, 1634.
43. Preferred by an anonymous referee.