×

zbMATH — the first resource for mathematics

Statistical mechanics of Euler equations in two dimensions. (English) Zbl 1050.82553
Summary: We formulate the statistical mechanics of a two-dimensional inviscid incompressible fluid in a manner which, for the first time, respects all conservation laws. For a special case, we demonstrate that a mean-field theory is exact. A consequence of our arguments is that, in an inviscid fluid evolving from initial conditions to statistical equilibrium, only the energy and certain one-body integrals appear to be conserved. Our methods may be applied to a variety of Hamiltonian systems possessing an infinite number of conservation laws.

MSC:
82D15 Statistical mechanical studies of liquids
76A02 Foundations of fluid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] G. Kirchoff, in: Lectures on Mathematical Physics, Mechanics (1877)
[2] D. Forster, Phys. Rev. A 16 pp 732– (1977) · doi:10.1103/PhysRevA.16.732
[3] P. G. Saffman, Annu. Rev. Fluid. Mech. 11 pp 95– (1979) · doi:10.1146/annurev.fl.11.010179.000523
[4] A. Leonard, J. Comput. Phys. 37 pp 289– (1980) · Zbl 0438.76009 · doi:10.1016/0021-9991(80)90040-6
[5] L. Onsager, Nuovo Cimento Suppl. 6 pp 279– (1949) · doi:10.1007/BF02780991
[6] R. H. Kraichnan, Rep. Prog. Phys. 43 pp 547– (1980) · doi:10.1088/0034-4885/43/5/001
[7] J. Fröhlich, Commun. Math. Phys. 87 pp 1– (1982) · Zbl 0505.76037 · doi:10.1007/BF01211054
[8] R. H. Kraichnan, Phys. Fluids 10 pp 1417– (1967) · doi:10.1063/1.1762301
[9] R. Salmon, Annu. Rev. Fluid Mech. 20 pp 225– (1988) · doi:10.1146/annurev.fl.20.010188.001301
[10] D. Holm, Phys. Rep. C 123 pp 1– (1985) · Zbl 0717.76051 · doi:10.1016/0370-1573(85)90028-6
[11] V. I. Arnold, in: Mathematical Methods of Classical Mechanics (1978) · Zbl 0386.70001 · doi:10.1007/978-1-4757-1693-1
[12] P. J. Olver, in: Applications of Lie Groups to Differential Equations (1986) · Zbl 0588.22001 · doi:10.1007/978-1-4684-0274-2
[13] V. E. Zak%harov, Sov. Phys. JETP 33 pp 927– (1971)
[14] T. D. Lee, Q. Appl. Math. 10 pp 69– (1952) · Zbl 0047.19601 · doi:10.1090/qam/51081
[15] D. Lynden-Bell, Mon. Not. Roy. Astron. Soc. 136 pp 101– (1967)
[16] S. Tremaine, Mon. Not. Roy. Astron. Soc. 219 pp 285– (1986)
[17] J. Binney, in: Galactic Dynamics (1987)
[18] W. Saslaw, in: Gravitational Physics of Stellar and Galactic Systems (1985) · doi:10.1017/CBO9780511564239
[19] P. S. Marcus, Nature (London) 331 pp 693– (1988) · doi:10.1038/331693a0
[20] P. S. Marcus, J. Fluid Mech. 215 pp 393– (1990) · Zbl 0698.76030 · doi:10.1017/S0022112090002695
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.