zbMATH — the first resource for mathematics

High temperature expansions and dynamical systems. (English) Zbl 0859.58037
Summary: We develop a resummed high-temperature expansion for lattice spin systems with long range interactions, in models where the free energy is not, in general, analytic. We establish uniqueness of the Gibbs state and exponential decay of the correlation functions. Then, we apply this expansion to the Perron-Frobenius operator of weakly coupled map lattices.

58Z05 Applications of global analysis to the sciences
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
Full Text: DOI
[1] Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics470, New York: Springer, 1975 · Zbl 0308.28010
[2] Bricmont, J., Kupiainen, A.: Phase transition in the 3d random field Ising model. Commun. Math. Phys.116, 539–572 (1988) · Zbl 1086.82573
[3] Bricmont, J., Kupiainen, A.: Coupled analytic maps. Nonlinearity8, 379–396 (1995) · Zbl 0836.58027
[4] Brydges, D.C.: A short course on cluster expansions. In: Critical Phenomena, Random Systems, Gauge Theories. Les Houches Session XLIII, Osterwalder, K., Stora, R. eds., Elsevier, 1984, pp. 139–183
[5] Bunimovich, L.A., Sinai, Y.G.: Space-time chaos in coupled map lattices. Nonlinearity1, 491–516 (1988) · Zbl 0679.58028
[6] Bunimovich, L.A., Sinai, Y.G.: Statistical mechanics of coupled map lattices. In: Ref [29]Kaneko, K. (ed): Theory and Applications of Coupled Map Lattices. New York: J. Wiley, 1993 · Zbl 0791.60099
[7] Bunimovich, L.A.: Coupled map lattices: One step forward and two steps back. Preprint (1993), to appear in the Proceedings of the ”Gran Finale” on Chaos, Order and Patterns, Como (1993) · Zbl 0890.58029
[8] Campbell, K.M., Rand, D.A.: A natural spatio-temporal measure for axiom A weakly coupled map lattices. Preprint, Univ. of Warwick (1994)
[9] Cassandro, M., Olivieri, E.: Renormalization group and analyticity in one dimension: A proof of Dobrushin’s theorem. Commun. Math. Phys.80, 255 (1981)
[10] Dobrushin, R.L.: Gibbsian random fields for lattice systems with pairwise interactions. Funct. Anal. Appl.2, 292–301 (1968) · Zbl 1183.82023
[11] Dobrushin, R.L.: The description of a random field by means of conditional probabilities and conditions on its regularity. Theory Prob. Appl.13, 197–224 (1968) · Zbl 0184.40403
[12] Dobrushin, R.L., Martirosyan, M.R.: Nonfinite perturbations of the random Gibbs fields. Theor. Math. Phys.74, 10–20 (1988) · Zbl 1268.60071
[13] Dobrushin, R.L., Martirosyan, M.R.: Possibility of high-temperature phase transitions due to the many particle nature of the potential. Theor. Math. Phys.75, 443–448 (1988)
[14] Dobrushin, R.L., Pecherski, E.A.: Uniqueness conditions for finitely dependent random fields. In: Random Fields, Vol. 1, Fritz, J. et al. (eds.) Amsterdam: North-Holland, 1981, pp. 223–261
[15] Dobrushin, R.L., Shlosman, S.B.: Completely analytical Gibbs fields. In: Statistical Physics and Dynamical Systems (Rigorous Results), Boston: Birkauser, 1985, pp. 371–403 · Zbl 0569.46043
[16] Dobrushin, R.L., Shlosman, S.B.: Completely analytical interactions: A constructive description. J. Stat. Phys.46, 983–1014 (1987) · Zbl 0683.60080
[17] von Dreifus, H., Klein, A., Perez, J.F.: Taming Griffiths’ singularities: Infinite differentiability of quenched correlation functions · Zbl 0820.60086
[18] van Enter, A.C.D., Fernandez, R.: A remark on different norms and analyticity for many particle interactions. J. Stat. Phys.56, 965–972 (1989)
[19] van Enter, A.C.D., Fernandez, R., Sokal, A.: Regularity properties and pathologies of position-space renormalization-group transformations. J. Stat. Phys.72, 879–1167 (1993) · Zbl 1101.82314
[20] Fisher, M.: Critical temperatures for anisotropic Ising lattices. II. General upper bounds. Phys. Rev.162, 480–485 (1967)
[21] Gallavotti, G., Miracle-Sole, S.: Correlation functions of a lattice system. Commun. Math. Phys.7, 274–288 (1968)
[22] Griffiths, R.B.: Non-analytic behavior above the critical point in a random Ising ferromagnet. Phys. Rev. Lett.23, 17–19 (1969)
[23] Gross, L.: Decay of correlations in classical lattice models at high temperature. Commun. Math. Phys.68, 9–27 (1979) · Zbl 0442.60097
[24] Gross, L.: Absence of second-order phase transitions in the Dobrushin uniqueness theorem. J. Stat. Phys.25, 57–72 (1981)
[25] Gundlach, V.M., Rand, D.A.: Spatial-temporal chaos: 1. Hyperbolicity, structural stability, spatial-temporal shadowing and symbolic dynamics. Nonlinearity6, 165–200 (1993); Spatialtemporal chaos: 2. Unique Gibbs states for higher-dimensional symbolic systems. Nonlinearity6, 201–214 (1993); Spatial-temporal chaos: 3. Natural spatial-temporal measures for coupled circle map lattices. Nonlinearity6, 215–230 (1993) · Zbl 0776.58012
[26] Israel, R.B.: High-temperature analyticity in classical lattice systems. Commun. Math. Phys.50, 245–257 (1976)
[27] Jiang, M.: Equilibrium states for lattice models of hyperbolic type. To appear in Nonlinearity · Zbl 0836.58032
[28] Jiang, M., Mazel, A.: Uniqueness of Gibbs states and exponential decay of correlations for some lattice models. Preprint · Zbl 1042.82520
[29] Kaneko, K. (ed): Theory and Applications of Coupled Map Lattices. New York: J. Wiley, 1993 · Zbl 0777.00014
[30] Kaneko, K. (ed): Focus Issue on Coupled Map Lattices. Chaos2 (1993)
[31] Keller, G., Künzle, M.: Transfer operators for coupled map lattices. Erg. Th. and Dyn. Syst.12, 297–318 (1992) · Zbl 0737.58032
[32] Malyshev, V.A., Minlos, R.A.: Gibbs Random Fields. Cluster Expansions. Dordrecht: Kluwer, 1991 · Zbl 0731.60099
[33] Martinelli, F., Olivieri, E.: Approach to equilibrium of Glauber dynamics in the one phase region. Commun. Math. Phys.161, 447–486 (1994) · Zbl 0793.60110
[34] Miller, J., Huse, D.A.: Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled-map lattice. Phys. Rev. E48, 2528–2535 (1993)
[35] Olivieri, E.: On a cluster expansion for lattice spin systems: A finite-size condition for the convergence. J. Stat. Phys.50, 1179–1200 (1988) · Zbl 1084.82523
[36] Olivieri, E., Picco, P.: Cluster expansion for d-dimensional lattice systems and finite volume factorization properties. J. Stat. Phys.59, 221–256 (1990) · Zbl 1083.82509
[37] Pesin, Y.G., Sinai, Y.G.: Space-time chaos in chains of weakly coupled hyperbolic maps. In: Advances in Soviet Mathematics, Vol. 3, ed. Sinai, Y.G., Harwood, 1991
[38] Pomeau, Y.: Periodic behaviour of cellular automata. J. Stat. Phys.70, 1379–1382 (1993) · Zbl 0936.37002
[39] Ruelle, D.: Thermodynamic Formalism. Reading, MA: Addison-Wesley, 1978 · Zbl 0401.28016
[40] Simon, B.: A remark on Dobrushin’s uniqueness theorem. Commun. Math. Phys.68, 183–185 (1979) · Zbl 0435.60099
[41] Simon, B.: The Statistical Mechanics of Lattice Gases. Vol. 1, Princeton, NJ: Princeton Univ. Press, 1994
[42] Sinai, Y.G.: Gibbs measures in ergodic theory. Russ. Math. Surv.27, 21–64 (1972) · Zbl 0246.28008
[43] Sinai, Y.G.: Topics in ergodic theory. Princeton, NJ: Princeton University Press, 1994 · Zbl 0805.58005
[44] Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. New York: Springer, 1988 · Zbl 0662.35001
[45] Volevich, D.L.: Kinetics of coupled map lattices. Nonlinearity4, 37–45 (1991) · Zbl 0778.58043
[46] Volevich, D.L.: The Sinai-Bowen-Ruelle measure for a multidimensional lattice of interacting hyperbolic mappings. Russ. Acad. Dokl. Math.47, 117–121 (1993) · Zbl 0823.58025
[47] Volevich, D.L.: Construction of an analogue of Bowen-Ruelle-Sinai measure for a multidimensional lattice of interacting hyperbolic mappings. Russ. Acad. Math. Sbornik79, 347–363 (1994) · Zbl 0821.58027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.