Abstract
LetS n denote the random total magnetization of ann-site Curie-Weiss model, a collection ofn (spin) random variables with an equal interaction of strength 1/n between each pair of spins. The asymptotic behavior for largen of the probability distribution ofS n is analyzed and related to the well-known (mean-field) thermodynamic properties of these models. One particular result is that at a type-k critical point (S n-nm)/n1−1/2k has a limiting distribution with density proportional to exp[-λs 2k/(2k)!], wherem is the mean magnetization per site and A is a positive critical parameter with a universal upper bound. Another result describes the asymptotic behavior relevant to metastability.
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References
R. H. Brout, inStatistical Physics: Phase Transitions and Superfluidity, M. Chretien, E. P. Gross, S. Deser, eds. (Gordon and Breach, New York, 1968).
M. Blume, V. J. Emery, and R. B. Griffiths,Phys. Rev. A 4:1071 (1971).
F. Dunlop and C. M. Newman,Comm. Math. Phys. 44:223 (1975).
G. G. Emch and H. J. F. Knops,J. Math. Phys. 11:3008 (1970).
R. S. Ellis and C. M. Newman, Limit theorems for sums of dependent random variables occurring in statistical mechanics, U. Mass./Indiana U. preprint (1977), to appear inZ. f. Wahrsch. verw. Geb.
R. S. Ellis and C. M. Newman inProceedings of the International Conference on the Mathematical Problems in Theoretical Physics, Rome, 1977.
G. Gallavotti and G. Jona-Lasinio,Comm. Math. Phys. 41:301 (1975).
G. Gallavotti and H. J. F. Knops,Riv. Nuovo Cimento 5:341 (1975).
G. Jona-Lasinio, Probabilistic approach to critical behavior, 1976 Cargése summer school on “New Developments in Quantum Field Theory and Statistical Mechanics.”
G. Jona-Lasinio,Nuovo Cimento 26B:99 (1975).
M. Kac, inStatistical Physics: Phase Transitions and Superfluidity, M. Chretien, E. P. Gross, S. Deser, eds. (Gordon and Breach, New York, 1968).
L. P. Kadanoff, inPhase Transitions and Critical Phenomena, Vol. 5A, C. Domb and M. S. Green, eds. (Academic, New York, 1976).
S. Karlin and W. J. Studden,Tchebycheff Systems: with Applications in Analysis and Statistics (Interscience, New York, 1966).
T. D. Schultz, D. C. Mattis, and E. H. Lieb,Rev. Mod. Phys. 36:856 (1964).
B. Simon and R. B. GriffithsComm. Math. Phys. 33:145 (1973).
H. E. Stanley,Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press, New York, 1971).
A. H. Stroud,Numerical Quadrature and Solution of Ordinary Differential Equations (Springer, 1974).
C. Thompson,Mathematical Statistical Mechanics (Macmillan, New York, 1972).
K. G. Wilson and M. E. Fischer,Phys. Rev. Lett. 28:240 (1972).
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Research supported in part by National Science Foundation Grants MPS 76-06644 (to RSE) and MPS 74-04870 A01 (to CMN).
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Ellis, R.S., Newman, C.M. The statistics of Curie-Weiss models. J Stat Phys 19, 149–161 (1978). https://doi.org/10.1007/BF01012508
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DOI: https://doi.org/10.1007/BF01012508