×

zbMATH — the first resource for mathematics

Interphase Hamiltonian and first-order phase transitions: A generalization of the Lee-Yang theorem. (English. Russian original) Zbl 1157.82319
Theor. Math. Phys. 153, No. 1, 1434-1457 (2007); translation from Teor. Mat. Fiz. 153, No. 1, 98-123 (2007).
Summary: We generalize the Pirogov-Sinai theory and prove the results applicable to first-order phase transitions in the case of both bulk and surface phase lattice models. The region of first-order phase transitions is extended with respect to the chemical activities to the entire complex space \(\mathbb C^{\Phi}\), where \(\Phi \) is the set of phases in the model. We prove a generalization of the Lee-Yang theorem: as functions of the activities, the partition functions with a stable boundary condition have no zeros in \(\mathbb C^{\Phi}\).
MSC:
82B26 Phase transitions (general) in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. A. Pirogov and Ya. G. Sinai, Theor. Math. Phys., 25, 1185–1192 (1975); 26, 39–49 (1976). · doi:10.1007/BF01040127
[2] J. Z. Imbrie, Comm. Math. Phys., 82, 261–304, 305–343 (1981). · doi:10.1007/BF02099920
[3] M. Zahradnik, Comm. Math. Phys., 93, 559–581 (1984). · doi:10.1007/BF01212295
[4] R. Kotecký and D. Preiss, Rend. Circ. Mat. Palermo (2), No. 3 (suppl.), 161–164 (1984).
[5] A. G. Basuev, Theor. Math. Phys., 58, 171–182 (1984). · doi:10.1007/BF01017924
[6] J. Bricmont, K. Kuroda, and J. L. Lebowitz, Comm. Math. Phys., 101, 501–538 (1985). · Zbl 0573.60098 · doi:10.1007/BF01210743
[7] J. Fröhlich, A. Bovier, and U. Glaus, ”Mathematical aspects of the physics of disordered systems,” in: Critical Phenomena, Random Systems, and Gauge Theories (Les Houches, France, 1984, K. Osterwalder and R. Stora, eds.), North-Holland, Amsterdam (1986), pp. 725–893. · Zbl 0669.60098
[8] J. Slawny, ”Low-temperature properties of classical lattice systems: Phase transitions and phase diagrams,” in: Phase Transitions and Critical Phenomena (C. Domb and J. L. Lebowitz, eds.), Vol. 11, Acad. Press, London (1987), pp. 127–205.
[9] A. G. Basuev, Theor. Math. Phys., 64, 716–734 (1985); 72, 861–871 (1987). · doi:10.1007/BF01017040
[10] R. L. Dobrushin and M. Zagradnik, ”Phase diagramm for continuous spin models: An extension Pirogov-Sinai theory,” in: Mathematical Problems of Statistical Mechanics and Dynamics: A Collection of Surveys (R. L. Dobrushin, ed.), Reidel, Dordrecht (1986), pp. 1–123.
[11] F. Koukiou, D. Petritis, and M. Zahradnik, Comm. Math. Phys., 118, 365–383 (1988). · Zbl 0668.58013 · doi:10.1007/BF01466722
[12] Yong Moon Park, Comm. Math. Phys., 114, 219–241 (1988). · Zbl 0663.60076 · doi:10.1007/BF01225036
[13] J. Bricmont and J. Slawny, J. Statist. Phys., 54, 89–161 (1989). · doi:10.1007/BF01023475
[14] J. Fröhlich, L. Rey-Bellet, and D. Ueltschi, Comm. Math. Phys., 224, 33–63 (2001). · Zbl 0987.82005 · doi:10.1007/s002200100530
[15] M. Zahradnik, J. Statist. Phys., 47, 725–755 (1987). · doi:10.1007/BF01206155
[16] A. C. D. van Enter, R. Fernández, and A. D. Sokal, J. Statist. Phys., 72, 879–1167 (1993). · Zbl 1101.82314 · doi:10.1007/BF01048183
[17] R. H. Schonmann and N. Yoshida, Comm. Math. Phys., 189, 299–309 (1997). · Zbl 0888.60089 · doi:10.1007/s002200050203
[18] M. Biskup et al., Comm. Math. Phys., 251, 79–131 (2004). · Zbl 1088.82010 · doi:10.1007/s00220-004-1169-5
[19] S. N. Isakov, Comm. Math. Phys., 95, 427–443 (1984). · doi:10.1007/BF01210832
[20] S. N. Isakov, Theor. Math. Phys., 71, 638–648 (1987). · doi:10.1007/BF01017098
[21] S. Friedli and C. E. Pfister, Comm. Math. Phys., 245, 69–103 (2004). · Zbl 1075.82009 · doi:10.1007/s00220-003-1003-5
[22] M. E. Fisher, Arch. Rational Mech. Anal., 17, 377–410 (1964). · doi:10.1007/BF00250473
[23] D. Ruelle, Statistical Mechanics: Rigorous Results, World Scientific, Singapore (1999). · Zbl 1016.82500
[24] R. B. Griffiths, ”Rigorous results and theorems,” in: Phase Transition and Critical Phenomena (C. Domb and M. S. Green, eds.), Vol. 1, Exact Results, Acad. Press, London (1972), pp. 7–109.
[25] C. N. Yang and T. D. Lee, Phys. Rev. (2), 87, 404–409, 410–419 (1952). · Zbl 0048.43305 · doi:10.1103/PhysRev.87.404
[26] A. G. Basuev, Theor. Math. Phys., 58, 80–91 (1984). · doi:10.1007/BF01031038
[27] A. G. Basuev, Theor. Math. Phys., 39, 343–351 (1979). · doi:10.1007/BF01018947
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.