Abstract
We consider the one-dimensional planar rotator and classical Heisenberg models with a ferromagnetic Kac potential J γ(r)=γJ(yr), J with compact support. Below the Lebowitz-Penrose critical temperature the limit (mean-field) theory exhibits a phase transition with a continuum of equilibrium states, indexed by the magnetization vectors m β s, s any unit vector and m β the Curie–Weiss spontaneous magnetization. We prove a large-deviation principle for the associated Gibbs measures. Then we study the system in the limit γ ↓ 0 below the above critical temperature. We prove that the norm of the empirical spin average in blocks of order γ−1 converges to m β, uniformly in intervals of order γ−p, for any p ≥ 1. We also give a lower bound to the scale on which the change of phase occurs, by showing that the empirical spin average is approximately constant on intervals having length of order γ-1-λwith λ∈(0,1) small enough.
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REFERENCES
G. Alberti, G. Bellettini, M. Cassandro, and E. Presutti, Surface tension in Ising system with Kac potentials, J. Stat. Phys. 82(3–4):743–796 (1996).
O. Benois, T. Bodineau, P. Buttà, and E. Presutti, On the validity of the van der Waals theory of surface tension, Markov Proc. Rel. Fields 3(2):175–198 (1997).
T. Bodineau, Interface in a one-dimensional Ising spin system, Stoch. Proc. Appl. 61:1–23 (1996).
T. Bodineau, Interface for one-dimensional random Kac potentials, Ann. Inst. H. Poincaré 33(5):559–590 (1997).
T. Bodineau and E. Presutti, Phase diagram of Ising systems with additional long range forces, Commun. Math. Phys. 189:287–298 (1997).
A. Bovier, V. Gayrard, and P. Picco, Large deviation principles for the Hopfield model and the Kac-Hopfield model, Prob. Theor. Rel. Fields 101:511–546 (1995).
A. Bovier, V. Gayrard, and P. Picco, Distribution of overlap profiles in the one-dimensional Kac-Hopfield model, Commun. Math. Phys. 186:323–379 (1997).
A. Bovier and M. Zahradnik, The low temperature phase of Kac-Ising models, J. Stat. Phys. 87:311–332 (1997).
J. Bricmont, J. Fontaine, and J. Landau, On the uniqueness of the equilibrium state for plane rotator, Commun. Math. Phys. 56:281–286 (1977).
P. Buttà, J. Merola, and E. Presutti, On the validity of the van der Waals theory in Ising systems with long range interactions, Markov Proc. Rel. Fields 3(1):63–88 (1997).
M. Cassandro, R. Marra, and E. Presutti, Corrections to the critical temperature in 2d Ising systems with Kac potentials, J. Stat. Phys. 78:1131–1138 (1995).
M. Cassandro, E. Orlandi, and E. Presutti, Interfaces and typical Gibbs configurations for one-dimensional Kac potentials, Prob. Theor. Rel. Fields 96:57–96 (1993).
M. Cassandro and E. Presutti, Phase transitions in Ising systems with long but finite range, Markov Proc. Rel. Fields 2:241–262 (1996).
A. De Masi, E. Orlandi, E. Presutti, and L. Triolo, Glauber evolution with Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics, Nonlinearity 7:1–67 (1994).
R. L. Dobrushin and S. Shlosman, Absence of breakdown of continuous symmetry in two dimensional models of statistical physics, Commun. Math. Phys. 42:31–40 (1975).
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II (Wiley, New York/London, 1963).
J. Fröhlich and C. E. Pfister, On the absence of spontaneous symmetry breaking and crystalline ordering in two dimensional systems, Commun. Math. Phys. 81:277–298 (1981).
J. Fröhlich and C. E. Pfister, Spin waves, vortices, and the structure of equilibrium states in the classical XY model, Commun. Math. Phys. 89:303–327 (1983).
J. Fröhlich, B. Simon, and T. Spencer, Infrared bound, phase transitions and continuous symmetry breaking, Commun. Math. Phys. 50:79–85 (1976).
J. Fröhlich and T. Spencer, The Kosterlitz-Thouless transition in two dimensional abelian spin systems and the Coulomb gas, Comm. Math. Phys. 81:527–607 (1981).
H. O. Georgii, Gibbs Measures and Phase Transitions. De Gruyter Studies in Mathematics, Vol. 9 (Walter de Gruyter, Berlin/New York, 1988).
P. C. Hemmer and J. L. Lebowitz, Systems with weak long-range potentials, in “Phase Transition and Critical Phenomena”, Vol. 5b, Domb and Green, eds. (1976).
K. R. Ito, Clustering in low-dimensional SO(n)-invariant statistical models with long range interactions, J. Stat. Phys. 29:747–760 (1982).
M. Kac, G. Uhlenbeck, and P. C. Hemmer, On the van der Waals theory of vapour-liquid equilibrium. I. Discussion of a one-dimensional model, J. Math. Phys. 4:216–228 (1963).
M. Kac, G. Uhlenbeck, and P. C. Hemmer, On the van der Waals theory of vapour-liquid equilibrium. II. Discussion of the distribution functions, J. Math. Phys. 4:229–247 (1963).
M. Kac, G. Uhlenbeck, and P. C. Hemmer, On the van der Waals theory of vapour-liquid equilibrium. III. Discussion of the critical region, J. Math. Phys. 5:60–74 (1964).
H. Kesten and R. H. Schonmann, Behavior in large dimensions of the Potts and Heisenberg models, Rev. Math. Phys. 1:147–182 (1990).
J. M. Kosterlitz and D. V. Thouless, Ordering metastability and phase transitions in twodimensional systems, J. Phys. C 6:1181–1203 (1973).
J. Lebowitz and O. Penrose, Rigorous treatment of the Van der Waals Maxwell theory of the liquid vapour transition, J. Math. Phys. 7:98–113 (1966).
A. Messager, S. Miracle, and C. E. Pfister, Correlation inequalities and uniqueness of equilibrium state for the planar rotator ferromagnetic model, Commun. Math. Phys. 58:19–29 (1978).
C. E. Pfister, On the symmetry of Gibbs states in two dimensional lattice systems, Commun. Math. Phys. 79:181–188 (1981).
D. Ruelle, Superstable interactions in classical statistical mechanics, Commun. Math. Phys. 18:127 (1970).
D. Ruelle, Probability estimates for continuous spin systems, Commun. Math. Phys. 50:189–194 (1976).
S. B. Shlosman, Decrease of correlations in two-dimensional models with continuous symmetry group, Theor. Math. Phys. 37:1118–1120 (1978).
C. J. Thompson and M. Silver, The classical limit of n-vector spin models, Commun. Math. Phys. 33:53–60 (1973).
G. N. Watson, Theory of Bessel Functions (Cambridge U.P., London, 1944).
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Buttà, P., Picco, P. Large-Deviation Principle for One-Dimensional Vector Spin Models with Kac Potentials. Journal of Statistical Physics 92, 101–150 (1998). https://doi.org/10.1023/A:1023095619236
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DOI: https://doi.org/10.1023/A:1023095619236