Phase transitions in small systems: Microcanonical vs. canonical ensembles

Physica A: Statistical Mechanics and its Applications

Volumn 370, Issue 2, 2006, Pages 390-406

  Dunkel, Jörn ^a   Hilbert, Stefan ^b  

^a UNIVERSITY OF AUGSBURG   (Germany)

^b HUMBOLDT UNIVERSITY   (Germany)

Author keywords

Lennard Jones chains; Microscopic phase transitions; Small systems

Indexed keywords

ELEMENTARY PARTICLES; MATHEMATICAL MODELS; ONE DIMENSIONAL; OSCILLISTORS; THERMODYNAMICS; THERMOSTATS;

CANONICAL ENSEMBLE; LENNARD JONES CHAINS; MICROSCOPIC PHASE TRANSITIONS; SMALL SYSTEMS;

PHASE TRANSITIONS;

EID: 33748161033     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physa.2006.05.018     Document Type: Article

References (73)

1. Kadanoff L.P., Götze W., Hamblen D., Hecht R., Lewis E.A.S., Palciauskas V.V., Rayl M., Swift J., Aspnes D., and Kane J. Static phenomena near critical points: theory and experiment. Rev. Mod. Phys. 39 2 (1967) 395
2. Lebowitz J.L. Statistical mechanics, two central issues. Rev. Mod. Phys. 71 2 (1999) S351
3. Ehrenfest P., and Ehrenfest T. The conceptual foundation of the statistical approach in mechanics (1959), Cornell University Press, Ithaca, NY (reprint of german 1912 edition)
4. Mayer J.E. The statistical mechanics of condensing systems. I. J. Chem. Phys. 5 (1937) 67
5. Mayer J.E., and Ackermann P.G. The statistical mechanics of condensing systems. II. J. Chem. Phys. 5 (1937) 74
6. Mayer J.E., and Harrison S.F. The statistical mechanics of condensing systems. III. J. Chem. Phys. 5 (1937) 87
7. Streeter S.F., and Mayer J.E. The statistical mechanics of condensing systems. VI. J. Chem. Phys. 7 (1939) 1025
8. Yang C.N., and Lee T.D. Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys. Rev. 87 3 (1952) 404
9. Lee T.D., and Yang C.N. Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Phys. Rev. 87 3 (1952) 410
10. Fisher M.E. Theory of critical point singularities. In: Green M.S. (Ed). Critical Phenomena: Proceedings of the 51st Enrico Fermi Summer School. Varenna, Italy (1971), Academic Press, New York 1
11. Grossmann S., and Rosenhauer W. Temperature dependence near phase transitions in classical and quantum mechanical canonical statistics. Z. Phys. 207 (1967) 138
12. Grossmann S., and Rosenhauer W. Phase transitions and the distribution of temperature zeros of the partition function. Z. Phys. 218 (1969) 437
13. Grossmann S., and Lehmann V. Phase transitions and the distribution of temperature zeros of the partition function. Z. Phys. 218 (1969) 449
14. Casetti L., Cohen E.G.D., and Pettini M. Exact results on topology and phase transitions. Phys. Rev. E 65 (2002) 036112
15. Teixeira A.C.R., and Stariolo D.A. Topological hypothesis: the simplest case. Phys. Rev. E 70 (2004) 016113
16. Kastner M. Unattainability of a purely topological criterion for the existence of a phase transition for nonconfining potentials. Phys. Rev. Lett. 93 (2004) 150601
17. Franzosi R., and Pettini M. Theorem on the origin of phase transitions. Phys. Rev. Lett. 92 (2004) 060601
18. R. Franzosi, M. Pettini, L. Spinelli, Topology and phase transitions I. Theorem on a necessary relation, math-ph/0505057, 2005.
19. R. Franzosi, M. Pettini, Topology and phase transitions II. Entropy and topology, math-ph/0505058, 2005.
20. Angelani L., Casetti L., Pettini M., Ruocco G., and Zamponi F. Topology and phase transitions: from an exactly solvable model to a relation between topology and thermodynamics. Phys. Rev. E 71 (2005) 036152
21. T.L. Hill, Thermodynamics of Small Systems, vol. I/ II, Dover, New York, 1994, reprint.
22. Lebowitz J.L., and Percus J.K. Thermodynamic properties of small systems. Phys. Rev. 124 6 (1961) 1673
23. Wales D.J., and Berry R.S. Coexistence in finite systems. Phys. Rev. Lett. 73 21 (1994) 2875
24. Wales D.J., and Doye J.P.K. Coexistence and phase separation in clusters: from the small to the not-so-small regime. J. Chem. Phys. 103 (1995) 3061
25. D.H.E. Gross, Micro-canonical thermodynamics, World Scientific Lecture Notes in Physics, vol. 66, World Scientific, Singapore, 2001.
26. Klindworth M., Melzer A., Piel A., and Schweigert V.A. Laser-excited intershell rotation of finite coloumb clusters in a dusty plasma. Phys. Rev. B 61 12 (2000) 8404
27. Melzer A., Klindworth M., and Piel A. Normal modes of 2d finite clusters in complex plasmas. Phys. Rev. Lett. 87 (2001) 115002
28. Anderson M.H., Ensher J.R., Matthews M.R., Wiedman C.E., and Cornell E.A. Observation of Bose-Einstein condensation in a dilute atomic vapor. Science 269 (1995) 198
29. Davis K.B., Mewes M.-O., Andrews M.R., van Druten N.J., Durfee D.S., Kurn D.M., and Ketterle W. Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75 (1995) 3969
30. Schmidt M., Hippler T., Donges J., Kronmüller W., von Issendorff B., Haberland H., and Labastie P. Caloric curve across the liquid-to-gas change for sodium clusters. Phys. Rev. Lett. 87 (2001) 203402
31. Bedanov V.M., and Peeters F.M. Ordering and phase transitions of charged particles in a classical finite two-dimensional system. Phys. Rev. B 49 4 (1994) 2667
32. Vaulina O.S., Vladimirov S.V., Petrov O.F., and Fortov V.E. Criteria of phase transitions in a complex plasma. Phys. Rev. Lett. 88 24 (2002) 245002
33. Dalfovo F., Giorgini S., Pitaevskii L.P., and Stringari S. Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71 3 (1999) 463
34. Borrmann P., Mülken O., and Harting J. Classification of phase transitions in small systems. Phys. Rev. Lett. 84 16 (2000) 3511
35. Mülken O., Stamerjohanns H., and Borrmann P. Origins of phase transitions in small systems. Phys. Rev. E 64 (2001) 047105
36. Alves N.A., Ferrite J.P.N., and Hansmann U.H.E. Numerical comparison of two approaches for the study of phase transitions in small systems. Phys. Rev. E 65 (2002) 036110
37. Gross D.H.E. Micro-canonical thermodynamics and statistical fragmentation of dissipative systems-the topological structure of the n-body phase space. Phys. Rep. 279 (1997) 119
38. Janke W., and Kenna R. The strength of first and second order phase transitions from partition function zeroes. J. Stat. Phys. 102 (2001) 1211
39. Ruelle D. Statistical Mechanics (1999), World Scientific, Singapore
40. Pleimling M., and Behringer H. Microcanonical analysis of small systems. Phase Transitions 78 (2005) 787
41. van Hove L. Sur l'integrale de configuration pour les systemes de particules a une dimension. Physica (Amsterdam) 16 (1950) 137
42. Cuesta J.A., and Sánchez A. General non-existence theorem for phase transitions in one-dimensional systems with short range interactions, and physical examples of such transitions. J. Stat. Phys. 115 3/4 (2004) 869
43. Costeniuc M., Ellis R.S., Touchette H., and Turkington B. The generalized canonical ensemble and its universal equivalence with the microcanonical ensemble. J. Stat. Phys. 119 5/6 (2005) 1283
44. C.J. Thomson, Phase transitions and critical phenomena, One-dimensional models-Short Range Forces, vol. 1, Academic Press, 1972, p. 177 (chapter 5).
45. Becker R. Theory of Heat (1967), Springer, New York
46. Huang K. Statistical Mechanics (1963), Wiley, New York
47. A. Münster, Statistical Thermodynamics, vol. 1, Springer, Berlin, Heidelberg, New York, 1969.
48. Pearson E.M., Halicioglu T., and Tiller W.A. Laplace-transform technique for deriving thermodynamic equations from the classical microcanonical ensemble. Phys. Rev. A 32 5 (1985) 3030
49. Cagin T., and Ray J.R. Fundamental treatment of molecular-dynamics ensembles. Phys. Rev. A 37 (1988) 247
50. Hertz P. Über die mechanischen Grundlagen der Thermodynamik. Ann. Phys. (Leipzig) 33 (1910) 225
51. Hertz P. Über die mechanischen Grundlagen der Thermodynamik. Ann. Phys. (Leipzig) 33 (1910) 537
52. V.L. Berdichevsky, Thermodynamics of Chaos and Order, vol. 90, Addison Wesley Longman, Essex, England, 1997.
53. Berdichevsky V.L., and Alberti M.v. Statistical mechanics of Hénon-Heiles oscillators. Phys. Rev. A 44 2 (1991) 858
54. Adib A.B. Adiabatic invariance with first integrals of motion. Phys. Rev. E 66 (2002) 047101
55. Ehrenfest P. Phase changes classified according to singularities of the thermodynamic potential. Proc. Amsterdam Acad. 36 (1933) 153
56. Lavoie J.L., Osler T.J., and Trembley R. Fractional derivatives and special functions. SIAM Rev. 18 2 (1976) 240
57. M. Kastner, O. Schnetz, On the mean-field spherical model, J. Stat. Phys. 122(6) (2006) 1195.
58. Umirzakov I.H. Exact equilibrium statistical mechanics of two particles interacting via Lennard-Jones and Morse potentials. Phys. Rev. E 61 (2000) 7188
59. T. Hahn, Cuba-a library for multidimensional numerical integration, hep-ph/0404043, 2004.
60. W. Nolting, Statistische Physik, Grundkurs Theoretische Physik, vol. 6, Vieweg, Braunschweig, Wiesbaden, 1998.
61. L.D. Landau, E.M. Lifshitz, Statistical Physics, third ed., Course of Theoretical Physics, vol. 5, Butterworth-Heinemann, Oxford, 2003.
62. Chandrasekhar S. Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15 1 (1943)
63. Hänggi P., Talkner P., and Borkovec M. Reaction rate theory: fifty years after Kramers. Rev. Mod. Phys. 62 2 (1990) 251
64. van Kampen N.G. Stochastic Processes in Physics and Chemistry. second ed. (2003), North-Holland, Amsterdam
65. Janke W., Johnston D.A., and Kenna R. Phase transition strength through densities of general distributions of zeroes. Nucl. Phys. B 682 (2004) 618
66. Mathematica 5.0.1.0, Wolfram Research Inc., 〈http://www.wolfram.com〉, 1988-2003.
67. S. Hilbert, J. Dunkel, Singular microscopic phase transitions in the microcanonical ensemble: an exactly solvable 1 d-model for evaporation, cond-mat/0511500, 2005.
68. Ramshaw J.D. Entropy ambiguity in a system in equilibrium with a finite heat bath. Phys. Lett. A 198 (1995) 122
69. Chomaz P., and Gulminelli F. Generalized definitions of phase transitions. Physica A 305 (2002) 330
70. Adib A.B., Moreira A.A., Andrade Jr. J.S., and Almeida M.P. Tsallis thermostatistics for finite systems: a Hamiltonian approach. Physica A 322 (2003) 276
71. Tsallis C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52 (1988) 479
72. Curado E.M.F., and Tsallis C. Generalized statistical mechanics: connection with thermodynamics. J. Phys. A 52 (1991) 479 (corrigenda: 24 (1991) 3187 and 25 (1993) 1019)
73. Gross D.H.E. Non-extensive systems follow Boltzmann's principle not Tsallis statistics-phase transitions, second law of thermodynamics. Physica A 305 (2002) 99