The vlasov continuum limit for the classical microcanonical ensemble

Reviews in Mathematical Physics

Volumn 21, Issue 9, 2009, Pages 1145-1195

  Kiessling, Michael K H ^a  

^a RUTGERS UNIVERSITY   (United States)

Author keywords

Classical statistical mechanics; Entropy; Microcanonical ensemble; N point functions; Unstable interactions; Vlasov continuum limit

Indexed keywords

EID: 70449127197     PISSN: 0129055X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0129055X09003852     Document Type: Article

References (46)

1. E. L. Altschuler, T. J. Williams, E. R. Ratner, R. Tipton, R. Stong, F. Dowla and F. Wooten, Possible global minimum lattice congurations for Thomsons problem of charges on the sphere, Phys. Rev. Lett. 78 (1997) 2681-2685.
2. L. Boltzmann, Vorlesungen über Gastheorie (J. A. Barth, Leipzig, 1896);
3. Lectures on Gas Theory (Univ. California Press, Berkeley, 1964), translated into English by S. G. Brush.
4. E. Caglioti, L. P. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys. 143 (1992) 501-525.
5. E. Caglioti, L. P. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description. II, Comm. Math. Phys. 174 (1995) 229-260.
6. J. L. Campbell and K. O'Neil, Statistics of two-dimensional point vortices and high energy vortex states, J. Stat. Phys. 65 (1991) 495-529.
7. S. Chanillo and M. K.-H. Kiessling, Surfaces with prescribed Gauss curvature, Duke Math. J. 105 (2000) 309-353.
8. P. H. Chavanis, Phase transitions in self-gravitating systems. Self-gravitating fermions and hard spheres models, Phys. Rev. E 65 (2002) 056123.
9. M. Costeniuc, R. S. Ellis, H. Touchette and B. Turkington, The generalized canonical ensemble and its universal equivalence with the microcanonical ensemble, J. Stat. Phys. 119 (2005) 1283-1329.
10. E. B. Dynkin, Klassy ekvivalentnyh slu?ca?inyh veli?cin, Uspeki Mat. Nauk. 6 (1953) 125-134.
11. P. Eichelsbacher and U. Schmock, Large Deviations of U-empirical measures in strong topologies and applications, Ann. Inst. Henri Poincaŕe 38 (2002) 779-797.
12. R. S. Ellis, Entropy, Large Deviations, and Statistical Mechanics (Springer-Verlag, New York, 1985).
13. R. S. Ellis, K. Haven and B. Turkington, Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles, J. Stat. Phys. 101 (2000) 999-1064.
14. G. Eyink and H. Spohn, Negative temperature states and large-scale, long-lived vortices in two-dimensional turbulence, J. Stat. Phys. 70 (1993) 833-886.
15. B. De Finetti, La pŕevision: Ses lois logiques, ses sources subjectives, Ann. Inst. Henri Poincaŕe 7 (1937) 1-68.
16. J. W. Gibbs, Elementary Principles in Statistical Mechanics (Yale Univ. Press, New Haven, 1902); reprint (Dover, New York, 1960).
17. R. B. Griffiths, Microcanonical ensemble in quantum statistical mechanics, J. Math. Phys. 6 (1965) 1447-1461.
18. E. Hewitt and J. L. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc. 80 (1955) 470-501.
19. W. Hoeffding, A class of statistics with asymptotically normal distributions, Ann. Statist. 19 (1948) 293-325; Reprinted (except for Sec. 9.e-9.h), in Breakthroughs in Statistics, Vol.I, eds. S. Kotz and N. L. Johnson (Springer-Verlag, New York, 1992).
20. M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Commun. Pure Appl. Math. 47 (1993) 27-56.
21. M. K.-H. Kiessling, Statistical mechanics approach to some problems in conformal geometry, Physica A 297 (2000) 353-368.
22. M. K.-H. Kiessling, Statistical equilibrium dynamics, in Dynamics and Thermodynamics of Systems with Long-Range Interactions: Theory and Experiments, eds. A. Campa, A. Giansanti, G. Morigi and F. Sylos Labini, AIP Conf. Proc., Vol.970 (American Inst. Phys., 2008), pp. 91-108.
23. M. K.-H. Kiessling, On Ruelle's construction of the thermodynamic limit for the classical microcanonical entropy, J. Stat. Phys. 134 (2009) 19-25.
24. M. K.-H. Kiessling, A note on classical ground state energies, J. Stat. Phys. 136 (2009) 275-284.
25. M. K.-H. Kiessling and C. Lancellotti, in preparation (2009).
26. M. K.-H. Kiessling and J. L. Lebowitz, The microcanonical point vortex ensemble: Beyond equivalence, Lett. Math. Phys. 42 (1997) 43-56.
27. M. K.-H. Kiessling and H. Spohn, A note on the eigenvalue density of random matrices, Comm. Math. Phys. 199 (1999) 683-695.
28. S. Kusuoka and Y. Tamura, Gibbs measures for mean field potentials, J. Fac. Sci. Univ. Tokyo, Sec. IA, Math. 31 (1984) 223-245.
29. O. E. Lanford III, Entropy and equilibrium states in classical statistical physics, in Statistical Mechanics and Mathematical Problems, Conf. Proc. Battelle Seattle Recontres (1971), ed. A. Lenard, Lecture Notes in Physics, Vol.20 (Springer, 1973), pp. 1-113.
30. A. Lenard (ed.), Statistical Mechanics and Mathematical Problems, Conf. Proc. Battelle Seattle Recontres (1971), Lecture Notes in Physics, Vol.20 (Springer, 1973).
31. A. Martin-Löf, Statistical Mechanics and the Foundations of Thermodynamics, eds. J. Ehlers et al., Lecture Notes in Physics, Vol.101 (Springer, 1979).
32. J. Messer, and H. Spohn, Statistical mechanics of the isothermal Lane-Emden equation, J. Stat. Phys. 29 (1982) 561-578.
33. D. Lynden-Bell and R. M. Lynden-Bell, Exact general solutions to extraordinary N-body problems, Proc. R. Soc. Lond. Ser. A 445 (1999) 475-489.
34. D. Lynden-Bell and R. M. Lynden-Bell, Relaxation to a perpetually pulsating equilibrium, J. Stat. Phys. 117 (2004) 199-209.
35. K. O'Neil and A. R. Redner, On the limiting distribution of pair-summable potential functions in many-particle systems, J. Stat. Phys. 62 (1991) 399-410.
36. L. Onsager, Statistical hydrodynamics, Nuovo. Cimerto. Sup. 6 (1949) 279-287.
37. O. Penrose, Foundations of Statistical Mechanics: A Deductive Treatment (Pergamon Press, Oxford, 1970); reprinted (Dover, 2005).
38. A. Ṕerez-Garrido,M. J. W. Dodgson, M. A.Moore,M. Ortuño and A. D́?az-Śanchez, Comment on "Possible global minimum lattice configurations for Thomson's problem of charges on a sphere", Phys. Rev. Lett 79 (1997) 1417.
39. M. Reed and B. Simon, Methods of Modern Mathematical Physics I (Academic Press, New York, 1980).
40. W. D. Robinson and D. Ruelle, Mean entropy of states in classical statistical mechanics, Comm. Math. Phys. 5 (1967) 288-300.
41. D. Ruelle, Statistical Mechanics: Rigorous Results (Benjamin, New York, 1969); reprinted, Advanced Book Classics series (Addison-Wesley, Reading, 1989).
42. H. Spohn, Large Scale Dynamics of Interacting Particles, Texts and Monographs in Physics (Springer, 1991).
43. B. Stahl, M. K. H. Kiessling and K. Schindler, Phase transitions in gravitating systems and the formation of condensed objects, Planet. Space Sci. 43 (1995) 271-282.
44. J. J. Thomson, On the structure of the atom: An investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure, Philos. Mag. 7 (1904) 237-265.
45. H. Touchette, Review on large deviation theory and statistical mechanics, Phys. Rep. 478 (2009) 1-69.
46. S. R. S. Varadhan, Large Deviations and Applications, CBMS-NSF Regional Conf. Ser. Appl. Math., Vol.46 (SIAM, 1984).