Microstructures and Phase Transitions

Errico Presutti

Bollettino dell'Unione Matematica Italiana (2012)

  • Volume: 5, Issue: 3, page 655-688
  • ISSN: 0392-4041

Abstract

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This is a short survey on some recent developments in the theory of phase transitions and microstructures in a mathematically rigorous context. The issue is discussed at the microscopic, mesoscopic and macroscopic levels recalling the most used mathematical techniques, mainly from probability theory and variational calculus.

How to cite

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Presutti, Errico. "Microstructures and Phase Transitions." Bollettino dell'Unione Matematica Italiana 5.3 (2012): 655-688. <http://eudml.org/doc/290896>.

@article{Presutti2012,
abstract = {This is a short survey on some recent developments in the theory of phase transitions and microstructures in a mathematically rigorous context. The issue is discussed at the microscopic, mesoscopic and macroscopic levels recalling the most used mathematical techniques, mainly from probability theory and variational calculus.},
author = {Presutti, Errico},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {655-688},
publisher = {Unione Matematica Italiana},
title = {Microstructures and Phase Transitions},
url = {http://eudml.org/doc/290896},
volume = {5},
year = {2012},
}

TY - JOUR
AU - Presutti, Errico
TI - Microstructures and Phase Transitions
JO - Bollettino dell'Unione Matematica Italiana
DA - 2012/10//
PB - Unione Matematica Italiana
VL - 5
IS - 3
SP - 655
EP - 688
AB - This is a short survey on some recent developments in the theory of phase transitions and microstructures in a mathematically rigorous context. The issue is discussed at the microscopic, mesoscopic and macroscopic levels recalling the most used mathematical techniques, mainly from probability theory and variational calculus.
LA - eng
UR - http://eudml.org/doc/290896
ER -

References

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  1. ALBERTI, G. - BELLETTINI, G., A non local anisotropic model for phase transition. Part I: the optimal profile problem. Math. Ann., 310 (1998), 527-560. MR1612250DOI10.1007/s002080050159
  2. ALBERTI, G. - BELLETTINI, G., A non local anisotropic model for phase transition: asymptotic behaviour of rescaled energies. European J. Appl. Math., 9 (1998), 261- 284. Zbl0932.49018MR1634336DOI10.1017/S0956792598003453
  3. ALBERTI, G. - BELLETTINI, G. - CASSANDRO, M. - PRESUTTI, E., Surface tension in Ising systems with Kac potentials. J. Stat. Phys., 82 (1996), 743-796. Zbl1042.82539MR1372427DOI10.1007/BF02179792
  4. AMBROSIO, L. - FUSCO, N. - PALLARA, D., Functions of bounded variations and free discontinuity problems. Oxford Science Publications. (2000). Zbl0957.49001MR1857292
  5. BELLETTINI, G. - CASSANDRO, M. - PRESUTTI, E., Constrained minima of non local free energy functionals. J. Stat. Phys., 84 (1996), 1337-1349. Zbl1081.82502MR1412081DOI10.1007/BF02174133
  6. BODINEAU, T., The Wulff construction in three and more dimensions. Comm. Math. Phys., 207 (1999), 197-229. Zbl1015.82005MR1724851DOI10.1007/s002200050724
  7. BRAIDES, A., Gamma-Convergence for Beginners. Series: Oxford Lecture Series in Mathematics and Its Applications, number 22 (2002). Zbl1198.49001MR1968440DOI10.1093/acprof:oso/9780198507840.001.0001
  8. CERF, R. - PISZTORA, A., On the Wulff crystal in the Ising model. Ann. Probab., 28 (2000), 947-1017. Zbl1034.82006MR1797302DOI10.1214/aop/1019160324
  9. DAL MASO, G., An introduction to Gamma convergence. Birkhäuser (1993). Zbl0816.49001MR1201152DOI10.1007/978-1-4612-0327-8
  10. DE MASI, A. - MEROLA, I. - PRESUTTI, E. - VIGNAUD, Y., Coexistence of ordered and disordered phases in Potts models in the continuum. J. Stat. Phys., 134 (2009), 243. Zbl1167.82009MR2485716DOI10.1007/s10955-008-9677-x
  11. DE MASI, A. - ORLANDI, E. - PRESUTTI, E. - TRIOLO, L., Uniqueness of the instanton profile and global stability in non local evolution equations. Rendiconti di Matematica, 14 (1993), 693-723. Zbl0821.45003MR1312824
  12. DE SIMONE, A. - KOHN, R. V. - OTTO, F. - MÜLLER, S., Recent analytical developments in micromagnetics in The Science of Hysteresis II: Physical Modeling, Micromagnetics, and Magnetization Dynamics. G. Bertotti and I. Mayergoyz eds., pp. 269-381, Elsevier (2001). 
  13. DOBRUSHIN, R. L. - KOTECKÝ, R. - SHLOSMAN, S., The Wulff construction: a global shape for local interactions. Amer. Math. Soc.Providence (1992). Zbl0917.60103MR1181197
  14. EVANS, L. C. - GARIEPY, R., Measure theory and fine properties of functions. Studies in Advanced Math.CRC Press, Boca Raton (1992). MR1158660
  15. FERNANDEZ, R. - PROCACCI, A., Cluster expansion for abstract polymer models. New bounds from an old approach, Comm. Math. Phys., 274, n. 1 (2007), 123-140. Zbl1206.82148MR2318850DOI10.1007/s00220-007-0279-2
  16. FERNANDEZ, R. - PROCACCI, A. - SCOPPOLA, B., The Analyticity Region of the Hard Sphere Gas. Improved Bounds. J. Stat. Phys., 128 (2007), 1139-1143. Zbl1206.82099MR2348787DOI10.1007/s10955-007-9352-7
  17. GATES, D. J. - PENROSE, O., The van der Waals limit for classical systems. I. A variational principle. Commun. Math. Phys., 15 (1969), 255-276. Zbl0184.54705MR260283
  18. GATES, D. J. - PENROSE, O., The van der Waals limit for classical systems. II. Existence and continuity of the canonical pressure. Commun. Math. Phys., 16 (1970), 231-237. MR1552568DOI10.1007/BF01646789
  19. GATES, D. J. - PENROSE, O., The van der Waals limit for classical systems. III. Deviation from the van der Waals-Maxwell theory. Commun. Math. Phys., 17 (1970), 194-209. MR1552570DOI10.1007/BF01647090
  20. GIULIANI, A. - LEBOWITZ, J. L. - LIEB, E. H., Periodic Minimizers in 1D Local Mean Field Theory. Commun.Math.Phys., 286 (2009), 163-177. Zbl1173.82008MR2470928DOI10.1007/s00220-008-0589-z
  21. GIULIANI, A. - LEBOWITZ, J. L. - LIEB, E. H., Ising models with long-range antiferromagnetic and short-range ferromagnetic interactions. Phys. Rev. B, 74 (2006), 064-420. 
  22. GIULIANI, A. - LEBOWITZ, J. L. - LIEB, E. H., Striped phases in two dimensional dipole systems. Phys. Rev. B, 76 (2007), 184-426. 
  23. GIULIANI, A. - LEBOWITZ, J. L. - LIEB, E. H., Modulated phases of a 1D sharp interface model in a magnetic field. Phys. Rev. B, 80 (2009), 134-420. 
  24. GIULIANI, A. - LEBOWITZ, J. L. - LIEB, E. H., Checkerboards, stripes, and corner energies in spin models with competing interaction. Phys. Rev. B, 84 (2011), 064205-1-10. 
  25. GURTIN, M. E., Thermodynamics of evolving phase boundaries in the plane. Oxford Science Publications (1993). Zbl0787.73004MR1402243
  26. HANSEN, J. P. - MCDONALD, I. R., Theory of Simple Liquids. Academic, London (1976). 
  27. HEITMANN, R. C. - RADIN, C., The ground state for sticky disks. J. Statist. Phys., 22 (1980), 281-287. MR570369DOI10.1007/BF01014644
  28. HOOVER, W. G. - REE, F. H., J. Chem. Phys., 49 (1968), 3609. 
  29. KAC, M. - UHLENBECK, G. - HEMMER, P. C., On the van der Waals theory of vapor-liquid equilibrium. I. Discussion of a one dimensional model. J. Math. Phys., 4 (1963), 216-228. Zbl0938.82517MR148416DOI10.1063/1.1703946
  30. KAC, M. - UHLENBECK, G. - HEMMER, P. C., On the van der Waals theory of vapor-liquid equilibrium. II. Discussion of the distribution functions. J. Math. Phys., 4 (1963), 229-247. Zbl0938.82518MR148417DOI10.1063/1.1703947
  31. KAC, M. - UHLENBECK, G. - HEMMER, P. C., On the van der Waals theory of vapor-liquid equilibrium. III. Discussion of the critical region. J. Math. Phys., 5 (1964), 60-74. Zbl0938.82519MR157692DOI10.1063/1.1704065
  32. M. KIESSLING, K.-H. - PERCUS, J. K., Hard-sphere fluids with chemical self-potentials. J. Math. Phys., 51 , no. 015206, 42, 82B21 (2010). Zbl1309.82030MR2605839DOI10.1063/1.3279598
  33. KOTECKÝ, R. - PREISS, D., Cluster expansion for abstract polymer models, Comm. Math. Phys., 103 (1986), 491-498. Zbl0593.05006MR832923
  34. LEBOWITZ, J. L. - MAZEL, A. - PRESUTTI, E., Liquid-vapor phase transitions for systems with finite-range interactions. J. Statist. Phys., 94 (1999), 955-1025. Zbl1012.82004MR1694123DOI10.1023/A:1004591218510
  35. LEBOWITZ, J. L. - PENROSE, O., Rigorous treatment of the Van der Waals-Maxwell theory of the liquid vapour transition. J. Math. Phys., 7 (1966), 98-113. Zbl0938.82520MR187835DOI10.1063/1.1704821
  36. LEBOWITZ, J. L. - PRESUTTI, E., Statistical mechanics of unbounded spins. Comm. Math. Phys., 50 (1976), 195-218. MR446251
  37. MAYER, J. E., Theory of Real Gases, Handbuch der Physik, 12 (Springer-Verlag, Berlin, 1958). MR135534
  38. MAYER, J. E. - MAYER, M. G., Statistical Mechanics, New York, John Wiley and Sons (1940). MR674819
  39. MINLOS, R. A. - SINAI, YA. G., Phenomenon of Phase Separation at Low Temperatures in Certain Lattice Models of a Gas. Soviet Physics Doklady, 12 (1968), 688. 
  40. MODICA, L. - MORTOLA, S., Un esempio di Gamma convergenza. Boll.Un. Mat.Ital. B (5), 14 (1977), 285-299. MR445362
  41. PENROSE, O. - LEBOWITZ, J. L., Rigorous treatment of metastable states in the van der Waals-Maxwell theory. J. Stat. Phys., 3 (1971), 211-236. Zbl0938.82521MR293957DOI10.1007/BF01019851
  42. POGHOSYAN, S. - UELTSCHI, D., Abstract cluster expansion with applications to statistical mechanical systems. J. of Math. Phys., 50 (2009), 053-509. Zbl1187.82009MR2531305DOI10.1063/1.3124770
  43. PRESUTTI, E., Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics. Heidelberg: Springer, Theoretical and Mathematical Physics (2008). Zbl1156.82001MR2460018
  44. PULVIRENTI, E. - TSAGKAROGIANNIS, D., Cluster expansion in the canonical ensemble. Submitted to Commun. Math. Phys. Zbl1260.82057MR2993917DOI10.1007/s00220-012-1576-y
  45. RADIN, C., The ground state for soft disks. J. Statist. Phys., 26 (1981), 365-373. MR643714DOI10.1007/BF01013177
  46. RUELLE, D., Statistical Mechanics: rigorous results, World Scientific, Imperial College Press, 1969. Zbl0177.57301MR1747792DOI10.1142/4090
  47. SINAI, YA. G., Theory of phase transitions: rigorous results. Akademiai Kiadó. Budapest (1982). MR691854
  48. SPEEDY, R. J., J. Phys., Condens. Matter, 10 (1998), 4387. 
  49. THEIL, F., A proof of crystallization in two dimensions. Comm. Math. Phys., 262 (2006), 209-236. Zbl1113.82016MR2200888DOI10.1007/s00220-005-1458-7
  50. AU YEUNG, Y. - FRIESECKE, G. - SCHMIDT, B., Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape. arXiv:0909.0927v1 [math.AP] (2009). Zbl1379.74002MR2898772DOI10.1007/s00526-011-0427-6
  51. ZAHRADNIK, M., A short course on the Pirogov-Sinai theory. Rend. Mat. Appl., 18 (1998), 411-486. Zbl0927.60087MR1686840

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