Rigorous probabilistic analysis of equilibrium crystal shapes

Journal of Mathematical Physics

Volumn 41, Issue 3, 2000, Pages 1033-1098

  Bodineau, T ^a   Ioffe, D ^b   Velenik, Y ^c  

^a UNIVERSITÉ PARIS DESCARTES   (France)

^b TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^c TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034345434     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.533180     Document Type: Article

References (89)

1. G. Wulff, "Zur frage der geschwindigkeit des wachsturms under auflösung der kristallflächen," Z. Kristallogr. 34, 449-530 (1901).
2. C. Rottman and M. Wortis, "Statistical mechanics of equilibrium crystal shapes: Interfacial phase diagrams and phase transitions," Phys. Rep. 103, 59-79 (1984).
3. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory (Cambridge University Press, Cambridge, 1993).
4. J. Taylor, "Crystalline variational problems," Bull. Am. Math. Soc. 84, 568-588 (1978).
5. I. Fonseca, "The Wulff theorem revisited," Proc. R. Soc. London, Ser. A 432, 125-145 (1991).
6. I. Fonseca and S. Mueller, "A uniqueness proof of the Wulff Theorem," Proc. R. Soc. Edinburgh, Sect. A: Math. 119, 125-136 (1991).
7. L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions (CRC Press, London, 1992).
8. L. Ambrosio and A. Braides, "Functionals defined on partitions in sets of finite perimeter II: Semicontinuity, relaxation and homogenization," J. Math. Pures Appl. 69, 307-333 (1990).
9. R. L. Dobrushin, R. Kotecký, and S. Shlosman, "Wulff construction: a global shape from local interaction," AMS translations series, Providence RI, 1992, Vol. 104.
10. R. R. Hall, "A quantitative isoperimetric inequality in n-dimensional space," J. Reine Angew. Math. 428, 161-176 (1992).
11. T. Bodineau, D. Ioffe, and Y. Velenik, in preparation.
12. W. L. Winterbottom, "Equilibrium shape of a small particle n contact with a foreign substrate," Acta Metall. 15, 303-310 (1967).
13. R. Kotecký and C.-E. Pfister, "Equilibrium shapes of crystals attached to walls," J. Stat. Phys. 76, 419-445 (1994).
14. R. B. Griffiths, "Rigorous results and theorems," in Phase Transitions and Critical Phenomena (Academic, London, 1972), Vol. 1.
15. D. G. Kelly and S. Sherman, "General Griffiths inequalities on correlations in Ising ferromagnets," J. Math. Phys. 9, 466-484 (1968).
16. R. B. Griffiths, C. A. Hurst, and S. Sherman, "Concavity of magnetization of an Ising ferromagnet in a positive external field," J. Math. Phys. 11, 790-795 (1970).
17. C. M. Fortuin, P. W. Kasteleyn, and J. Ginibre, "Correlation inequalities on some partially ordered sets," Commun. Math. Phys. 22, 89-103 (1971).
18. C.-E. Pfister and Y. Velenik, "Large deviations and continuum limit in the 2D Ising model," Prob. Theor. Relat. Fields 109, 435-506 (1997).
19. C.-E. Pfister and Y. Velenik, "Interface, surface tension and reentrant pinning transition in the 2D Ising model," Commun. Math. Phys. 204, 269-312 (1999).
20. M. Kac, G. Uhlenbeck, and P. Hemmer, "On the van der Waals theory of the vapor-liquid equilibrium. I. Discussion of a one-dimensional model," J. Math. Phys. 4, 216-228 (1963).
21. J. Lebowitz and O. Penrose, "Rigorous treatment of the van der Waals-Maxwell theory of the liquid-vapor transition," J. Math. Phys. 7, 98-113 (1966).
22. M. Cassandro and E. Presutti, "Phase transitions in Ising systems with long but finite range," Mark. Proc. Relat. Fields 2, 241-262 (1996).
23. A. Bovier and M. Zahradnik, "The low-temperature phase of Kac-Ising models," J. Stat. Phys. 87, 311-332 (1997).
24. T. Bodineau and E. Presutti, "Phase diagram of Ising systems with additional long range forces," Commun. Math. Phys. 189, 287-298 (1997).
25. D. B. Abraham, "Surface structures and phase transitions exact results," Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, London, 1987), Vol. 10, pp. 1-74.
26. C. E. Pfister, "Large deviations and phase separation in the two dimensional Ising model," Helv. Phys. Acta 64, 953-1054 (1991).
27. A. Messager, S. Miracle-Solé, and J. Ruiz, "Convexity properties of the surface tension and equilibrium crystals," J. Stat. Phys. 67, 449-470 (1992).
28. J. Lebowitz and C. E. Pfister, "Surface tension and phase coexistence," Phys. Rev. Lett. 46, 1031-1033 (1981).
29. D. Ioffe, "Large deviations for the 2D Ising model: a lower bound without cluster expansions," J. Stat. Phys. 74, 411-432 (1994).
30. c for the Ising model in two dimensions," Prob. Theor. Relat. Fields 102, 313-330 (1995).
31. R. H. Schonmann and S. Shlosman, "Wulff droplets and the metastable relaxation of kinetic Ising models," Commun. Math. Phys. 194, 389-462 (1998).
32. D. Ioffe and R. Schonmann, "Dobrushin-Kotecký-Shlosman theory up to the critical temperature," Commun. Math. Phys. 199, 117-167 (1998).
33. A. Pisztora, "Surface order large deviations of Ising, Potts and percolation models," Prob. Theor. Relat. Fields 104, 427-466 (1996).
34. T. Bodineau, "The Wulff construction in three and more dimensions," Commun. Math. Phys. 207, 197-229 (1999).
35. R. Cerf and A. Pisztora, "On the Wulff crystal in the Ising model," preprint (1999).
36. R. A. Minlos and Ya. G. Sinai, "The phenomenon of phase separation at low temperatures in some lattice models of a gas I," Math. USSR-Sb. 2, 335-395 (1967).
37. R. A. Minlos and Ya. G. Sinai, "The phenomenon of phase separation at low temperatures in some lattice models of a gas II," Trans. Moscow Math. Soc. 19, 121-196 (1968).
38. K. Kuroda, "The probabilistic treatment of phase separations in lattice models composed of more than two types of particles," Publ. Res. Inst. Math. Sci. 18, 275-305 (1982).
39. R. H. Schonmann, "Second order large deviation estimates for ferromagnetic systems in the phase coexistence region," Commun. Math. Phys. 112, 409-422 (1987).
40. H. Föllmer and M. Ort, "Large deviations and surface entropy for Markov fields," Astérisque 157-158, 173-190 (1988).
41. J. Lebowitz and R. H. Schonmann, "Pseudo free energies and large deviations for non-Gibsian FKG measures," Prob. Theor. Relat. Fields 77, 49-64 (1988).
42. J. T. Chayes, L. Chayes, and R. H. Schonmann, "Exponential decay of connectivities in the two-dimensional Ising model," J. Stat. Phys. 49, 433-445 (1987).
43. K. Alexander, J. T. Chayes, and L. Chayes, "The Wulff construction and asymptotics of the finite cluster distribution for the two-dimensional Bernoulli percolation," Commun. Math. Phys. 131, 1-50 (1990).
44. R. H. Schonmann and S. Shlosman, "Complete analyticity for the 2d Ising model completed," Commun. Math. Phys. 170, 453-482 (1996).
45. R. H. Schonmann and S. Shlosman, "Constrained variational problem with applications to the Ising model," J. Stat. Phys. 83, 867-905 (1996).
46. C.-E. Pfister and Y. Velenik, "Mathematical theory of the wetting phenomenon in the 2D Ising model," Helv. Phys. Acta 69, 949-973 (1996).
47. R. L. Dobrushin and O. Hryniv, "Fluctuations of the phase boundary in the 2D Ising ferromagnet," Commun. Math. Phys. 189, 395-445 (1997).
48. O. Hryniv, in preparation (1999).
49. 2," preprint (1999).
50. S. Shlosman (private communication, 1999).
51. F. Martinelli, "Lectures on Glauber dynamics for discrete spin models," Ecole de probabilités de St Flour 1997, LNM No. 1717, Springer (1999).
52. G. Alberti, G. Bellettini, M. Cassandro, and E. Presutti, "Surface tension in Ising system with Kac potentials," J. Stat. Phys. 82, 743-796 (1996).
53. G. Bellettini, M. Cassandro, and E. Presutti, "Constrained minima of non local free energy functionals," J. Stat. Phys. 84, 1337-1349 (1996).
54. O. Benois, T. Bodineau, P. Butta, and E. Presutti, "On the validity of van der Waals theory of surface tension," Mark. Proc. Relat. Fields 3, 175-198 (1997).
55. O. Benois, T. Bodineau, and E. Presutti, "Large deviations in the van der Waals limit," Stoch. Proc. Appl. 75, 89-104 (1998).
56. A. De Masi, E. Orlandi, E. Presutti, and L. Triolo, "Stability of the interface in a model of phase separation," Proc. R. Soc. Edinburgh 124A, 1013-1022 (1994).
57. A. De Masi, E. Orlandi, E. Presutti, and L. Triolo, "Uniqueness and global stability of the instanton in non local evolution equations," Rend. Mat. 14, 693-723 (1994).
58. R. Cerf, "Large deviations for three dimensional supercritical percolation," preprint (1998).
59. P. Buttà, I. Merola, and E Presutti, "On the validity of the van der Waals theory in Ising systems with long range interactions," Mark. Proc. Relat. Fields 3, 63-88 (1997).
60. G. Ben Arous and J-D. Deuschel, "The construction of the d+1-dimensional Gaussian droplet," Commun. Math. Phys. 179, 467-488 (1996).
61. E. Bolthausen and D. Ioffe, "Harmonic crystal on the wall: a microscopic approach," Commun. Math. Phys. 187, 523-566 (1997).
62. J-D. Deuschel, G. Giacomin, and D. Ioffe, "Large deviation and concentration properties for a class of ∇φ interface models," to appear in Prob. Th. Relat. Fields.
63. F. Dunlop and J. Magnen, "A Wulff shape from constructive field theory," in Mathematical Results in Statistical Mechanics, edited by S. Miracle-Sole, J. Ruiz, and V. Zagrebnov (World Scientific, Singapore, 1999), pp. 31-52.
64. J. de Coninck, F. Dunlop, and V. Rivasseau, "On the microscopic validity of the Wulff construction and of the generalized Young equations," Commun. Math. Phys. 121, 401-419 (1989).
65. T. Funaki and H. Spohn, "Motion by mean curvature from the Ginsburg-Landau ∇φ interface models," Commun. Math. Phys. 185, 1-36 (1997).
66. J. Lebowitz, A. Mazel, and E. Presutti, "Liquid-vapor phase transition for systems with finite range interactions," J. Stat. Phys. 94, 955-1025 (1999).
67. J.-D. Deuschel and A. Pisztora, "Surface order large deviations for high-density percolation," Prob. Theory. Relat. Fields 104, 467-482 (1996).
68. A. Pisztora, "Lectures in IHP," unpublished, Paris, June, 1998.
69. T. Liggett, R. Schonmann, and A. Stacey, "Domination by product measures," Ann. Prob. 25, 71-95 (1997).
70. 2 Ising and Potts model," J. Stat. Phys. 50, 1-40 (1988).
71. G. Grimmett, "The stochastic random cluster process and the uniqueness of random cluster measures," Ann. Prob. 23, 1461-1510 (1995).
72. J. Lebowitz, "Coexistence of phases in Ising ferromagnets," J. Stat. Phys. 16, 463-476 (1977).
73. C. E. Pfister, "On the ergodic decomposition of Gibbs random fields for ferromagnetic Abelian lattice models," Ann. (N.Y.) Acad. Sci. 491, 170-180 (1987).
74. G. Alberti and G. Bellettini. "Asymptotic behavior of a non load anisotropic model for phase transition," J. Math. Ann. 310, 527-560 (1998).
75. R. L. Dobrushin and S. Shlosman, "Large and moderate deviations in the Ising model," Adv. Sov. Math. 20, 91-220 (1994).
76. K. S. Alexander, "Power-law corrections to exponential decay of connectivities and correlations in lattice models," preprint (1998).
77. d," Mark. Proc. Relat. Fields 4, 323-350 (1998).
78. d," preprint (1999).
79. Y. Velenik, "Phase separation as a large deviations problem: a microscopic derivation of surface thermodynamics for some 2D spin systems," Thèse 1712 EPF-L, 1997; available electronically from the author.
80. 2 uniquely minimizes perimeter," Pac. J. Math. 159, 47-59 (1993).
81. F. Morgan and J. Sullivan, "Open problems in soap bubble geometry," Int. J. Math 7, 833-842 (1996).
82. J. Fröhlich and C.-E. Pfister, "Semi-infinite Ising model I. Thermodynamic functions and phase diagram in absence of magnetic field," Commun. Math. Phys. 109, 493-523 (1987).
83. J. Fröhlich and C.-E. Pfister, "Semi-infinite Ising model II. The wetting and layering transitions," Commun. Math. Phys. 112, 51-74 (1987).
84. D. B. Abraham, "Solvable model with a roughening transition for a planar Ising ferromagnet," Phys. Rev. Lett. 44, 1165-1168 (1980).
85. B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model (Harvard University Press, Cambridge, MA, 1973).
86. R. L. Dobrushin, "A statistical behavior of shapes of boundaries of phases," in Phase Transitions: Mathematics, Physics, Biology..., edited by R. Kotecký (World Scientific, Singapore, 1993), pp. 60-70.
87. J. T. Chalker, "The pinning of an interface by a planar defect," J. Phys. A 15, L481-L485 (1982).
88. K. Binder and D. P. Landau, "Wetting and layering in the nearest neighbour simple-cubic Ising lattice: A Monte Carlo investigation," Phys. Rev. B 37, 1745-1765 (1985).
89. N. Akutsu and Y. Akutsu, "Relationship between the anisotropic surface tension, the scaled interface width and the equilibrium shape in two dimensions," J. Phys. A 19, 2813-1820 (1986).