Statistical mechanics approach to some problems in conformal geometry

Physica A: Statistical Mechanics and its Applications

Volumn 279, Issue 1, 2000, Pages 353-368

  Kiessling, Michael K H ^a  

^a RUTGERS UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ELECTROMAGNETIC FIELD THEORY; GRAVITATION; MAGNETOPLASMA; MAXWELL EQUATIONS; NONLINEAR EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PROBABILITY DISTRIBUTIONS; QUANTUM THEORY;

CONFORMAL DIFFERENTIAL GEOMETRY; LIOUVILLE EQUATION; MICROCANONICAL ENSEMBLE; NIRENBERG'S PROBLEM; ONSAGER'S POINT VORTEX DISTRIBUTIONS; PANEITZ EQUATION; POISSON-BOLTZMANN EQUATION; QUANTUM GRAVITY;

STATISTICAL MECHANICS;

EID: 0033737272     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0378-4371(99)00515-4     Document Type: Article

References (67)

1. L. Onsager, Statistical hydrodynamics, Nuovo Cimento. Suppl. 6 (1949) 279-287.
2. Kirchhoff, Vorlesungen über mathematische, Physik. Mechanik Teubner, Leipzig, 1877.
3. C.C. Lin, On the Motion of Vortices in two Dimensions, The University of Toronto Press, Toronto, Canada, 1943.
4. C. Marchioro, M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Springer, Berlin, 1994.
5. N.F. Ramsey, Thermodynamics and statistical mechanics at negative absolute temperatures, Phys. Rev. 103 (1956) 20-28.
6. D.C. Montgomery, Two-dimensinal vortex motion and "negative temperatures", Phys. Lett. A 39 (1972) 7-8.
7. G. Joyce, D.C. Montgomery, Negative temperature states for the two-dimensional guiding center plasma, J. Plasma Phys. 10 (1973) 107-121.
8. S. Kida, Statistics of the system of line vortices, J. Phys. Soc. Japan 39 (1975) 1395-1404.
9. T.S. Lundgren, Y.B. Pointin, Statistical mechanics of two-dimensional vortices, J. Stat. Phys. 17 (1977) 323-355.
10. D.C. Montgomery, G. Joyce, Statistical mechanics of "negative temperature" states, Phys. Fluids 17 (1974) 1139-1145.
11. Y.B. Pointin, T.S. Lundgren, Statistical mechanics of two-dimensional vortices in a bounded container, Phys. Fluids 19 (1976) 1459-1470.
12. J. Fröhlich, D. Ruelle, Statistical mechanics of vortices in an inviscid two-dimensional fluid, Commun. Math. Phys. 87 (1982) 1-36.
13. J. Messer, H. Spohn, Statistical mechanics of the isothermal Lane-Emden equation, J. Stat. Phys. 29 (1982) 561-578.
14. E. Caglioti, P.L. Lions, C. Marchioro, M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Commun. Math. Phys. 143 (1992) 501-525.
15. M.K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Commun. Pure Appl. Math. 47 (1993) 27-56.
16. G. Eyink, H. Spohn, Negative temperature states and large-scale, long-lived vortices in two-dimensional turbulence, J. Stat. Phys. 70 (1993) 833-886.
17. E. Caglioti, P.L. Lions, C. Marchioro, M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description, Part II, Commun. Math. Phys. 174 (1995) 229-260.
18. M.K.-H. Kiessling, J.L. Lebowitz, The microcanonical point vortex ensemble: beyond equivalence, Lett. Math. Phys. 42 (1997) 43-56.
19. M.K.-H. Kiessling, Negative temperature bounds for 2D vorticity compounds, Lett. Math. Phys. 34 (1995) 49-57.
20. L.J. Campbell, K. O'Neil, Statistics of two-dimensional point vortices and high energy vortex states, J. Stat. Phys. 65 (1991) 495-529.
21. K. O'Neil, R.A. Redner, On the limiting distribution of pair-summable potential functions in many-particle systems, J. Stat. Phys. 62 (1991) 399-110.
22. J. Moser, On a nonlinear problem in differential geometry, in: M. Peixoto (Ed.), Dynamical Systems, Academic Press, New York, 1973.
23. C. Bandle, Isoperimetric Inequalities and Applications, Pitman, Boston, 1980.
24. D.J. Struik, ectures on Classical Differential Geometry, 2nd Edition, Dover, New York, 1988.
25. 2 = 0, J. Math. Pures Appl. 18 (1853) 71-72.
26. R. Osserman, On the inequality Δu ≥ f(u), Pac. J. Math. 7 (1957) 1641-1647.
27. u, J. Math. 4 (1898) 137-230.
28. E.A. Carlen, M. Loss, Competing symmetries, the logarithmic HLS inequality, and Onofri's inequality on S", Geom. Funct. Anal. 2 (1992) 90-104.
29. S. Chanillo, M.K.-H. Kiessling, Conformally invariant systems of nonlinear PDEs of Liouville type, Geom. Funct. Anal. 5 (1995) 924-947.
30. W. Chen, C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991) 615-622.
31. u = 0 in a punctured disc, Pac. J. Math. 163 (1994) 269-276.
32. 2, Invent. Math. 83 (1986) 519-544.
33. S. Chanillo, M.K.-H. Kiessling, Rotational symmetry of solutions to some nonlinear problems in statistical mechanics and geometry, Commun. Math. Phys. 160 (1994) 217-238.
34. S. Chanillo, M.K.-H. Kiessling, Surfaces with radially symmetric prescribed Gauss curvature, Duke Math. J. (2000), in press.
35. 2, Duke Math. J. 71 (1993) 427-139.
36. 2, Duke Math. J. 62 (1991) 721-737.
37. K.-S. Cheng, C.-S. Lin, On the asymptotic behavior of solutions of the conformal Gaussian curvature equations on K2, Math. Ann. 308 (1997) 119-139.
38. 2 with prescribed Gaussian curvature and positive total curvature, Indiana Univ. Math. J. 34 (1985) 97-104.
39. 2u = 0 and conformal metrics with prescribed Gaussian curvatures, Invent. Math. 66 (1982) 343-352.
40. 2: symmetry and a uniqueness result, Proc. Roy. Soc. Edinburgh (2000), in press.
41. 2 with prescribed curvatures, Indiana Univ. Math. J. 22 (1972) 1-4.
42. 2, Acta Math. 159 (1987) 215-259.
43. 2, J. Diff. Geom. 27 (1988) 256-296.
44. 2, Duke Math. J. 61 (1990) 679-703.
45. J. Kazdan, F. Warner, Curvature functions for compact 2-manifolds, Ann. Math. 99 (1974) 14-42.
46. u on a compact Riemann surface, Univ. Leipzig Preprint, February 1998.
47. T. Aubin, Meilleures constantes dans le theoreme d'inclusion de Sobolev et un theoreme de Fredholm non lineaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal. 32 (1979) 148-174.
48. J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971) 1077-1092.
49. E. Onofri, On the positivity of the effective action in a theory of random surfaces, Commun. Math. Phys. 86 (1982) 321-326.
50. M.K.-H. Kiessling, H. Spohn, A note on the eigenvalue density of random matrices, Commun. Math. Phys. 199 (1999) 683-695.
51. S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, Preprint, MIT, 1983.
52. T. Branson, Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987) 199-293.
53. T. Branson, An anomaly associated with 4-dimensional quantum gravity, Commun. Math. Phys. 178 (1996) 301-309.
54. T. Branson, S.Y.A. Chang, P. Yang, Estimates and extremal problems for the log-determinant on 4-manifolds, Commun. Math. Phys. 149 (1992) 241-262.
55. S.Y.A. Chang, On a fourth order PDE in conformal geometry, Preprint, 1999.
56. S.Y.A. Chang, J. Qing, The zeta functional determinants on four manifolds with boundary. I -the formula, J. Funct. Anal. 147 (1997) 327-362.
57. S.Y.A. Chang, P. Yang, Extremal metrics for the zeta function determinants on 4-manifolds, Ann. Math. 142 (1995) 171-212.
58. 4, Preprint, 1999.
59. W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. Math. 138 (1993) 213-242.
60. S.Y.A. Chang, P. Yang, On uniqueness of solutions of a n-th order differential equation in conformal geometry, Preprint, 1999.
61. N.S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473-183.
62. E.B. Dynkin, Klassy ekvivalentnyh slucainyhvelic in, Usp. Mat. Nauk 6 (1953) 125-134.
63. B. de Finetti, Funzione caratteristica di un fenomeno aleatorio, Atti R. Accad. Naz. Lincei, Ser. 6, Mem. Cl. Sci. Fis., Mat. Nat. 4 (1931) 251-300.
64. P. Diaconis, D. Freedman, A dozen de Finetti-style results in search of a theory, Ann. Inst. Henri Poincare 23 (1987) 397-123.
65. E. Hewitt, L.J. Savage, Symmetric measures on Cartesian products, Trans. Am. Math. Soc. 80 (1955) 470-501.
66. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
67. E.B. Saff, V. Totik, Logarithmic Potentials with External Fields, Grundl. Math. Wiss., Vol. 316, Springer, New York, 1997.