Abstract
In the paper, the one-dimensional model with nearest-neighbor interactions I n , n ∈ Z, and the spin values ±1 is considered. It is known that, under some conditions on parameters of I n , a phase transition occurs for this model. We define the notion of a phase separation point between two phases. We prove that the expectation value of this point is zero and its mean-square fluctuation is bounded by a constant C(β) which tends to ¼ as β → ∞, where β = 1/T and T is the temperature.
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References
D. B. Abraham, Surface Structures and Phase-Transitions — Exact Results, vol. 10 of Phase Transitions and Critical Phenomena, C. Domb and J. Lebowits, Eds. (Acad. Press, New York and London, 1986).
J. Bricmont, J. Lebowitz, and C. Pfister, “On the local structure of the phase separation line in the two-dimensional Ising system,” J. Stat. Phys. 26(2), 313–332 (1981).
M. Cassandro, P. Ferrari, I. Merola, and E. Presutti, “Geometry of contours and Peierls estimates in d = 1 Ising models with long range interactions,” J.Math. Phys. 46, 053305–053327 (2005).
R. Dobrushin, “Gibbs state describing coexistence of phases for a three-dimensional Ising model,” Theory Probab. Appl. 17(4), 582–600 (1972).
F. Dyson, “Existence of a phase-transition in a one-dimensional Ising ferromagnet,” Commun. Math. Phys. 12(2), 91–107 (1969).
F. Dyson, “An Ising ferromagnetic with discontinuous long-range order,” Commun. Math. Phys. 21(4), 269–283 (1971).
J. Frohlich and T. Spencer, “The phase transition in the one-dimensional Ising model with 1/r 2 interaction energy,” Commun. Math. Phys. 84(1), 87–101 (1982).
G. Gallavotti, “The phase separation line in the two-dimensional Ising model,” Commun. Math. Phys. 27(2), 103–136 (1972).
G. Gallavotti, Statistical Mechanics. A Short Treatise, Text and Monographs in Physics (Springer, 1999).
H. O. Georgii, Gibbs Measures and Phase Transitions, vol. 9 of de Gruyter Stud. in Math. (Walter de Gruyter, Berlin, 1988).
K. Johansson, “Condensation of a one-dimensional lattice gas,” Commun. Math. Phys. 141(1), 41–61 (1991).
K. Johansson, “On separation of phase in one-dimensional gases,” Commun. Math. Phys. 169(3), 521–561 (1995).
R. Minlos, Introduction to Mathematical Statistical Physics, vol. 19 of Univ. Lect. Ser. (AMS, 2000).
U. Rozikov, “An example of one-dimensional phase transition,” Sib. Adv. Math. 16(2), 121–125 (2006).
D. Ruelle, Statistical Mechanics: Rigorous Results (Benjamin, New York, 1969).
W. Sullivan, “Exponential convergence in dynamic Ising models with Distinct phases,” Phys. Lett. A 53(6), 441–442 (1975).
L. Van Hove, “Sur l’integrale de configuration pour les systems de particles a une dimension,” Physica 16, 137–143 (1950).
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Ganikhodjaev, N.N., Rozikov, U.A. On phase separation points for one-dimensional models. Sib. Adv. Math. 19, 75–84 (2009). https://doi.org/10.3103/S1055134409020011
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DOI: https://doi.org/10.3103/S1055134409020011