메뉴 건너뛰기




Volumn 31, Issue 1-3, 1996, Pages 299-322

Boltzmann's ergodic hypothesis, A conjecture for centuries?

Author keywords

Billiards; Boltzmann's ergodic hypothesis; Hard ball systems; Sinai's conjecture

Indexed keywords


EID: 1542574947     PISSN: 00816906     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article
Times cited : (44)

References (76)
  • 1
    • 0001725488 scopus 로고
    • Time evolution for infinitely many hard spheres
    • [A(1976)] MR 55 #7252
    • [A(1976)] ALEXANDER, R., Time evolution for infinitely many hard spheres, Comm. Math. Phys. 49 (1976), 217-232. MR 55 #7252
    • (1976) Comm. Math. Phys. , vol.49 , pp. 217-232
    • Alexander, R.1
  • 2
    • 0010860380 scopus 로고
    • Certain smooth ergodic systems
    • [A-S(1967)] in Russian. MR 37 #370
    • [A-S(1967)] ANOSOV, D. V. and SINAI, YA. G., Certain smooth ergodic systems, Us- pehi Mat. Nauk 22 (1967), no. 5 (137), 107-172 (in Russian). MR 37 #370
    • (1967) Us- Pehi Mat. Nauk , vol.22 , Issue.5-137 , pp. 107-172
    • Anosov, D.V.1    Sinai, Y.A.G.2
  • 3
    • 0002764803 scopus 로고
    • Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian
    • [A(1963)] (in Russian). MR 29 #328
    • [A(1963)] ARNOLD, V. I., Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi. Mat. Nauk 18 (1963), no. 5 (113), 13-40 (in Russian). MR 29 #328
    • (1963) Uspehi. Mat. Nauk , vol.18 , Issue.5-113 , pp. 13-40
    • Arnold, V.I.1
  • 4
    • 0001650497 scopus 로고
    • Proof of the ergodic theorem
    • [Bi(1931)] Zbl 3, 256
    • [Bi(1931)] BIRKHOFF, G. D., Proof of the ergodic theorem, Proc. Nat. Acad. Sci. U.S.A. 17 (1931), 656-660. Zbl 3, 256
    • (1931) Proc. Nat. Acad. Sci. U.S.A. , vol.17 , pp. 656-660
    • Birkhoff, G.D.1
  • 5
    • 0009187229 scopus 로고
    • Recent contributions to ergodic theory
    • [Bi-K(1932)] Zbl, 4, 311
    • [Bi-K(1932)] BIRKHOFF, G. D. and KOOPMAN, B. O., Recent contributions to ergodic theory, Proc. Nat. Acad. Sci. U.S.A. 18 (1932), 279-282. Zbl, 4, 311
    • (1932) Proc. Nat. Acad. Sci. U.S.A. , vol.18 , pp. 279-282
    • Birkhoff, G.D.1    Koopman, B.O.2
  • 6
    • 0000241369 scopus 로고
    • Ergodic properties of a semi-infinite one-dimensional system of statistical mechanics
    • [B-P-P-S-S(1985)] MR 87f:82002
    • [B-P-P-S-S(1985)] BOLDRIGHINI, C., PELLEGRINOTTI, A., PRESUTTI, E., SINAI, YA. G. and SOLOVEICHIK , M. R., Ergodic properties of a semi-infinite one-dimensional system of statistical mechanics, Comm. Math. Phys. 101 (1985), 363-382. MR 87f:82002
    • (1985) Comm. Math. Phys. , vol.101 , pp. 363-382
    • Boldrighini, C.1    Pellegrinotti, A.2    Presutti, E.3    Sinai, Y.A.G.4    Soloveichik, M.R.5
  • 7
    • 0012446231 scopus 로고
    • Einige allgemeine Sätze über das Wärmegleichgewicht
    • [B(1871)]
    • [B(1871)] BOLTZMANN, L., Einige allgemeine Sätze über das Wärmegleichgewicht, Wien. Ber. 63 (1871), 679-711. Jahrbuch Fortschritte Math. 3, 536
    • (1871) Wien. Ber. , vol.63 , pp. 679-711
    • Boltzmann, L.1
  • 8
    • 1542738012 scopus 로고    scopus 로고
    • [B(1871)] BOLTZMANN, L., Einige allgemeine Sätze über das Wärmegleichgewicht, Wien. Ber. 63 (1871), 679-711. Jahrbuch Fortschritte Math. 3, 536
    • Jahrbuch Fortschritte Math. , vol.3 , pp. 536
  • 9
    • 0000229982 scopus 로고
    • Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen
    • [B(1872)]
    • [B(1872)] BOLTZMANN, L., Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, Wien. Ber. 66 (1872), 275-370. Jahrbuch Fortschritte Math. 4, 566-567
    • (1872) Wien. Ber. , vol.66 , pp. 275-370
    • Boltzmann, L.1
  • 10
    • 1542527958 scopus 로고    scopus 로고
    • [B(1872)] BOLTZMANN, L., Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, Wien. Ber. 66 (1872), 275-370. Jahrbuch Fortschritte Math. 4, 566-567
    • Jahrbuch Fortschritte Math. , vol.4 , pp. 566-567
  • 11
    • 84942397575 scopus 로고    scopus 로고
    • Über die Eigenschaften monocyklischer und anderer damit verwandter Systeme
    • [B(1884)]
    • [B(1884)] BOLTZMANN, L., Über die Eigenschaften monocyklischer und anderer damit verwandter Systeme, Kronecker J. = J. Reine Angew. Math. 98 (1884), 68-94. Jahrbuch Fortschritte Math. 16, 863
    • (1884) Kronecker J. = J. Reine Angew. Math. , vol.98 , pp. 68-94
    • Boltzmann, L.1
  • 12
    • 84942397575 scopus 로고    scopus 로고
    • [B(1884)] BOLTZMANN, L., Über die Eigenschaften monocyklischer und anderer damit verwandter Systeme, Kronecker J. = J. Reine Angew. Math. 98 (1884), 68-94. Jahrbuch Fortschritte Math. 16, 863
    • Jahrbuch Fortschritte Math. , vol.16 , pp. 863
  • 13
    • 0039423777 scopus 로고
    • Ergodic systems of n balls in a billiard table
    • [B-L-P-S(1992)] MR 93e:58107
    • [B-L-P-S(1992)] BUNIMOVICH, L. A., LIVERANI, C., PELLEGRINOTTI, A. and SUKHOV, Yu. M., Ergodic systems of n balls in a billiard table, Comm. Math. Phys. 146 (1992), 357-396. MR 93e:58107
    • (1992) Comm. Math. Phys. , vol.146 , pp. 357-396
    • Bunimovich, L.A.1    Liverani, C.2    Pellegrinotti, A.3    Sukhov, Yu.M.4
  • 14
    • 0000182012 scopus 로고
    • The fundamental theorem of the theory of scattering billiards
    • [B-S(1973)] (in Russian). MR 51 #3395
    • [B-S(1973)] BUNIMOVICH, L. A. and SINAI, YA. G., The fundamental theorem of the theory of scattering billiards, Mat. Sb. (N. S.) 90(132) (1973), 415-431 (in Russian). MR 51 #3395
    • (1973) Mat. Sb. (N. S.) , vol.90 , Issue.132 , pp. 415-431
    • Bunimovich, L.A.1    Sinai, Y.A.G.2
  • 15
    • 0141820481 scopus 로고
    • [CH(1991)] (in Russian). Translation in Functional Anal. Appl. 25 (1991), 204-219. MR 93c:58110
    • [CH(1991)] CHERNOV, N. I., A new proof of Sinai's formula for entropy of hyperbolic billiards. Its application to Lorentz gas and stadium, Joint Inst. Nuclear Res., Dubna, 1990. MR 91k:58103; A new proof of the Sinai's formula for computing the entropy of hyperbolic billiards. Application to Lorentz gas and to the Bunimovich stadium, Funktional Anal. i Prilozhen. 25 (1991), no. 3, 50-69 (in Russian). Translation in Functional Anal. Appl. 25 (1991), 204-219. MR 93c:58110
    • (1991) Funktional Anal. i Prilozhen. , vol.25 , Issue.3 , pp. 50-69
    • Chernov, N.I.1
  • 17
    • 0040950948 scopus 로고
    • On Poisson's law for distribution of particles in space
    • [D(1956)] (in Russian). MR 18-341
    • [D(1956)] DOBRUSHIN, R. L., On Poisson's law for distribution of particles in space, Ukrain. Mat. Ž. 8 (1956), 127-134 (in Russian). MR 18-341
    • (1956) Ukrain. Mat. Ž , vol.8 , pp. 127-134
    • Dobrushin, R.L.1
  • 18
    • 33846039615 scopus 로고
    • Potentials on the two-torus for which the Hamiltonian flow is ergodic
    • [D-L(1991)] MR 92a:58082
    • [D-L(1991)] DONNAY, V. and LIVERANI C., Potentials on the two-torus for which the Hamiltonian flow is ergodic, Comm. Math. Phys. 135 (1991), 267-302. MR 92a:58082
    • (1991) Comm. Math. Phys. , vol.135 , pp. 267-302
    • Donnay, V.1    Liverani, C.2
  • 19
    • 34249961241 scopus 로고    scopus 로고
    • Ergodic properties of the multidimensional Rayleigh gas with a semipermeable barrier
    • [E-T(199?)] to appear
    • [E-T(199?)] ERDÖS, L. and TUYEN, D. Q., Ergodic properties of the multidimensional Rayleigh gas with a semipermeable barrier, J. Stat. Phys. (to appear).
    • J. Stat. Phys.
    • Erdös, L.1    Tuyen, D.Q.2
  • 22
    • 21344497680 scopus 로고
    • Stationary states of random Hamiltonian systems
    • [F-F-L(1994)]
    • [F-F-L(1994)] FRITZ, J., FUNAKI, T. and LEBOWITZ, J. L., Stationary states of random Hamiltonian systems, Probab. Theory Related Fields 99 (1994), 211-236.
    • (1994) Probab. Theory Related Fields , vol.99 , pp. 211-236
    • Fritz, J.1    Funaki, T.2    Lebowitz, J.L.3
  • 25
    • 0010187590 scopus 로고
    • Ordered and chaotic motions in Hamiltonian systems and the problem of energy partition
    • [G(1985)] ed. by J. R. Buchler
    • [G(1985)] GALGANI, L., Ordered and chaotic motions in Hamiltonian systems and the problem of energy partition, Chaos in astrophysics, ed. by J. R. Buchler, 1985, 245-257.
    • (1985) Chaos in Astrophysics , pp. 245-257
    • Galgani, L.1
  • 26
    • 0001276861 scopus 로고
    • On the problem of energy equipartition for large systems of the Fermi-Pasta-Ulam type: Analytical and numerical estimates
    • [G-G-M-V(1993)] MR 93h:58135
    • [G-G-M-V(1993)] GALGANI, L., GIORGILLI, A., MARTINOLI, A. and VANZINI, S., On the problem of energy equipartition for large systems of the Fermi-Pasta-Ulam type: analytical and numerical estimates, Phys. D 59 (1992), 334-348. MR 93h:58135
    • (1992) Phys. D , vol.59 , pp. 334-348
    • Galgani, L.1    Giorgilli, A.2    Martinoli, A.3    Vanzini, S.4
  • 27
    • 21844522223 scopus 로고
    • Ergodicity, ensembles, irreversibility in Boltzmann and beyond
    • [G(1995)]
    • [G(1995)] GALLAVOTTI, G., Ergodicity, ensembles, irreversibility in Boltzmann and beyond, J. Stat. Phys. 78 (1995), 1571-1589.
    • (1995) J. Stat. Phys. , vol.78 , pp. 1571-1589
    • Gallavotti, G.1
  • 28
    • 0001838706 scopus 로고
    • Ergodic properties of a system in contact with a heat bath: A one-dimensional model
    • [G-L-R(1982)] MR 84k:82053
    • [G-L-R(1982)] GOLDSTEIN, S., LEBOWITZ, J. L. and RAVISHANKAR, K., Ergodic properties of a system in contact with a heat bath: a one-dimensional model, Comm. Math. Phys. 85 (1982), 419-427. MR 84k:82053
    • (1982) Comm. Math. Phys. , vol.85 , pp. 419-427
    • Goldstein, S.1    Lebowitz, J.L.2    Ravishankar, K.3
  • 29
    • 0001585656 scopus 로고
    • The dynamics of geodesic flows
    • [HE(1939)] Zbl 20, 403-404
    • [HE(1939)] HEDLUND, G. A., The dynamics of geodesic flows, Bull. Amer. Math. Soc. 45 (1939), 241-260. Zbl 20, 403-404
    • (1939) Bull. Amer. Math. Soc. , vol.45 , pp. 241-260
    • Hedlund, G.A.1
  • 30
    • 0038903941 scopus 로고
    • Numerical exploration of Hamiltonian systems
    • [H(1983)] Les Houches, 1981, North-Holland, Amsterdam-New York, MR 84k:58074
    • [H(1983)] HÉNON, M., Numerical exploration of Hamiltonian systems, Chaotic behavior of deterministic systems (Les Houches, 1981), North-Holland, Amsterdam-New York, 1983, 53-170. MR 84k:58074
    • (1983) Chaotic Behavior of Deterministic Systems , pp. 53-170
    • Hénon, M.1
  • 32
    • 0001946992 scopus 로고
    • Mathematische Probleme
    • [H(1900)]
    • [H(1900)] HILBERT, D., Mathematische Probleme, Gött. Nachr. 1900, 253-297. Jahrbuch Fortschritte Math. 31, 68-69
    • (1900) Gött. Nachr. , pp. 253-297
    • Hilbert, D.1
  • 33
    • 1542527945 scopus 로고    scopus 로고
    • [H(1900)] HILBERT, D., Mathematische Probleme, Gött. Nachr. 1900, 253-297. Jahrbuch Fortschritte Math. 31, 68-69
    • Jahrbuch Fortschritte Math. , vol.31 , pp. 68-69
  • 34
    • 0000683114 scopus 로고
    • Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung
    • [Ho(1939)] Zbl 24, 80-81
    • [Ho(1939)] HOPF, E., Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. sächs. Akad. Leipzig 91 (1939), 261-304. Zbl 24, 80-81
    • (1939) Ber. Verh. Sächs. Akad. Leipzig , vol.91 , pp. 261-304
    • Hopf, E.1
  • 35
    • 34247542154 scopus 로고
    • Ergodic and topological properties of Coulombic periodic potentials
    • [K(1987)] MR 88f:58119
    • [K(1987)] KNAUF, A., Ergodic and topological properties of Coulombic periodic potentials, Comm. Math. Phys. 110 (1987), 89-112. MR 88f:58119
    • (1987) Comm. Math. Phys. , vol.110 , pp. 89-112
    • Knauf, A.1
  • 36
    • 0001246972 scopus 로고
    • On conservation of conditionally periodic motions for a small change in Hamilton's function
    • [K(1954)] (in Russian). MR 16-924
    • [K(1954)] KOLMOGOROV, A. N., On conservation of conditionally periodic motions for a small change in Hamilton's function. Dokl. Akad. Nauk SSSR (N. S.) 98 (1954), 527-530 (in Russian). MR 16-924
    • (1954) Dokl. Akad. Nauk SSSR (N. S.) , vol.98 , pp. 527-530
    • Kolmogorov, A.N.1
  • 37
    • 0001046226 scopus 로고
    • Hamiltonian systems and transformations in Hilbert space
    • [K(1931)] Zbl 2, 57
    • [K(1931)] KOOPMAN, B. O., Hamiltonian systems and transformations in Hilbert space, Proc. Nat. Acad. Sci. U.S.A. 17 (1931), 315-318. Zbl 2, 57
    • (1931) Proc. Nat. Acad. Sci. U.S.A. , vol.17 , pp. 315-318
    • Koopman, B.O.1
  • 38
    • 0000593586 scopus 로고
    • Ergodic properties of semi-dispersing billiards I. Two cylindric scatterers in the 3D torus
    • [K-S-Sz(1989)] MR 90f:58138
    • [K-S-Sz(1989)] KRÁMLI, A., SIMÁNYI, N. and SZÁSZ, D., Ergodic properties of semi-dispersing billiards I. Two cylindric scatterers in the 3D torus, Nonlinearity 2 (1989), 311-326. MR 90f:58138
    • (1989) Nonlinearity , vol.2 , pp. 311-326
    • Krámli, A.1    Simányi, N.2    Szász, D.3
  • 39
    • 0000059523 scopus 로고
    • A "transversal" fundamental theorem for semi-dispersing billiards
    • [K-S-Sz(1990)] MR 91b:58125; see also MR 92e:58119
    • [K-S-Sz(1990)] KRÁMLI, A., SIMÁNYI, N. and SZÁSZ, D., A "transversal" fundamental theorem for semi-dispersing billiards, Comm. Math. Phys. 129 (1990), 535-560. MR 91b:58125; see also MR 92e:58119
    • (1990) Comm. Math. Phys. , vol.129 , pp. 535-560
    • Krámli, A.1    Simányi, N.2    Szász, D.3
  • 40
    • 0002884213 scopus 로고
    • The K-property of three billiard balls
    • [K-S-Sz(1991)] MR 92b:58130
    • [K-S-Sz(1991)] KRÁMLI, A., SIMÁNYI, N. and SZÁSZ, D., The K-property of three billiard balls, Ann. of Math. (2) 133 (1991), 37-72. MR 92b:58130
    • (1991) Ann. of Math. (2) , vol.133 , pp. 37-72
    • Krámli, A.1    Simányi, N.2    Szász, D.3
  • 41
    • 0001027451 scopus 로고
    • The K-property of four billiard balls
    • [K-S-Sz(1992)] MR 93a:58107
    • [K-S-Sz(1992)] KRÁMLI, A., SIMÁNYI, N. and SZÁSZ, D., The K-property of four billiard balls, Comm. Math. Phys. 144 (1992), 107-148. MR 93a:58107
    • (1992) Comm. Math. Phys. , vol.144 , pp. 107-148
    • Krámli, A.1    Simányi, N.2    Szász, D.3
  • 42
    • 26744476393 scopus 로고
    • On the convergence to equilibrium of the Lorentz gas
    • [K-Sz(1983)] Budapest
    • [K-Sz(1983)] KRÁMLI, A. and SZÁSZ, D., On the convergence to equilibrium of the Lorentz gas, Functions, series, operators, Vol. I-II (Budapest, 1980), Colloq. Math. Soc. J. Bolyai, Vol. 35, North-Holland, Amsterdam-New York, 1983, 757-766. MR 85i:82011
    • (1980) Functions, Series, Operators , vol.1-2
    • Krámli, A.1    Szász, D.2
  • 43
    • 1542737999 scopus 로고
    • North-Holland, Amsterdam-New York, MR 85i:82011
    • [K-Sz(1983)] KRÁMLI, A. and SZÁSZ, D., On the convergence to equilibrium of the Lorentz gas, Functions, series, operators, Vol. I-II (Budapest, 1980), Colloq. Math. Soc. J. Bolyai, Vol. 35, North-Holland, Amsterdam-New York, 1983, 757-766. MR 85i:82011
    • (1983) Colloq. Math. Soc. J. Bolyai , vol.35 , pp. 757-766
  • 44
    • 0002090271 scopus 로고
    • The problem of recurrence for Lorentz processes
    • [K-Sz(1985)] MR 86m:60063
    • [K-Sz(1985)] KRÁMLI, A. and SZÁSZ, D., The problem of recurrence for Lorentz processes, Comm. Math. Phys. 98 (1985), 539-552. MR 86m:60063
    • (1985) Comm. Math. Phys. , vol.98 , pp. 539-552
    • Krámli, A.1    Szász, D.2
  • 45
    • 0009293971 scopus 로고
    • The processes of relaxation of statistical systems and the criterion of mechanical instability
    • [K(1942)] Thesis, 1942. Princeton University Press, Princeton, NJ
    • [K(1942)] KRYLOV, N. S., The processes of relaxation of statistical systems and the criterion of mechanical instability, Thesis, 1942. In: Development of Krylov's ideas, Princeton University Press, Princeton, NJ, 1977, 193-238.
    • (1977) Development of Krylov's Ideas , pp. 193-238
    • Krylov, N.S.1
  • 46
    • 84972539502 scopus 로고
    • Perturbed billiard systems. I. The ergodicity of the motion of a particle in a compound central field
    • [K(1976)] MR 55#6486
    • [K(1976)] KUBO, I., Perturbed billiard systems. I. The ergodicity of the motion of a particle in a compound central field, Nagoya Math. J. 61 (1976), 1-57. MR 55#6486
    • (1976) Nagoya Math. J. , vol.61 , pp. 1-57
    • Kubo, I.1
  • 47
    • 84972500417 scopus 로고
    • Perturbed billiard systems. II. Bernoulli properties
    • [K-M(1981)] MR 83b:58043
    • [K-M(1981)] KUBO, I. and MURATA, H., Perturbed billiard systems. II. Bernoulli properties, Nagoya Math. J. 81 (1981), 1-25. MR 83b:58043
    • (1981) Nagoya Math. J. , vol.81 , pp. 1-25
    • Kubo, I.1    Murata, H.2
  • 48
    • 0001518823 scopus 로고
    • Time evolution of large classical systems
    • [L(1975)] Recontres, Battelle Res. Inst., Seattle, Wash., 1974, Lecture Notes in Phys., Vol. 38, Springer, Berlin, MR 57 #18653
    • [L(1975)] LANFORD, O. E., Time evolution of large classical systems, Dynamical systems, theory and applications (Recontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Phys., Vol. 38, Springer, Berlin, 1975, 1-111. MR 57 #18653
    • (1975) Dynamical Systems, Theory and Applications , pp. 1-111
    • Lanford, O.E.1
  • 49
    • 0001043657 scopus 로고    scopus 로고
    • Ergodicity in Hamiltonian systems
    • [L-W(1994)] to appear
    • [L-W(1994)] LIVERANI, C. and WOJTKOWSKI, M., Ergodicity in Hamiltonian systems, Dynamics Reported (to appear).
    • Dynamics Reported
    • Liverani, C.1    Wojtkowski, M.2
  • 50
    • 0009108966 scopus 로고
    • Von Neumann and the early days of ergodic theory
    • [M(1990)] Hempstead, NY, 1988, Proc. Sympos. Pure Math., 50, American Mathematical Society, Providence, RI, MR 92a:01056.
    • [M(1990)] MACKEY, G. W., Von Neumann and the early days of ergodic theory, The legacy of John von Neumann (Hempstead, NY, 1988), Proc. Sympos. Pure Math., 50, American Mathematical Society, Providence, RI, 1990, 25-38. MR 92a:01056. See also: The scope and history of commutative and noncommutative harmonic analysis, History of Mathematics, 5, American Mathematical Society, Providence, RI; London Mathematical Society, London, 1992, 311-324. MR 93g:22006
    • (1990) The Legacy of John Von Neumann , pp. 25-38
    • Mackey, G.W.1
  • 51
    • 0042273002 scopus 로고
    • History of Mathematics, 5, American Mathematical Society, Providence, RI; London Mathematical Society, London, MR 93g:22006
    • [M(1990)] MACKEY, G. W., Von Neumann and the early days of ergodic theory, The legacy of John von Neumann (Hempstead, NY, 1988), Proc. Sympos. Pure Math., 50, American Mathematical Society, Providence, RI, 1990, 25-38. MR 92a:01056. See also: The scope and history of commutative and noncommutative harmonic analysis, History of Mathematics, 5, American Mathematical Society, Providence, RI; London Mathematical Society, London, 1992, 311-324. MR 93g:22006
    • (1992) The Scope and History of Commutative and Noncommutative Harmonic Analysis , pp. 311-324
  • 52
    • 0006452128 scopus 로고
    • Generic Hamiltonian dynamical systems are neither integrable nor ergodic
    • M-M(1974)] American Math. Society, Providence, RI, MR 56 #9588
    • (M-M(1974)] MARKUS, L. and MEYER, K. R., Generic Hamiltonian dynamical systems are neither integrable nor ergodic, Memoirs Amer. Math. Soc., No. 144, American Math. Society, Providence, RI, 1974. MR 56 #9588
    • (1974) Memoirs Amer. Math. Soc., No. 144 , vol.144
    • Markus, L.1    Meyer, K.R.2
  • 53
    • 0001851184 scopus 로고
    • On invariant curves of area-preserving mappings of an annu- Lus
    • [M(1962)] MR 26 #5255
    • [M(1962)] MOSER, J., On invariant curves of area-preserving mappings of an annu- lus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962, 1-20. MR 26 #5255
    • (1962) Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II , pp. 1-20
    • Moser, J.1
  • 54
    • 0003095621 scopus 로고    scopus 로고
    • Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik
    • [N(1929)]
    • [N(1929)] NEUMANN, J. v., Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik, Z. f. Physik 57 (1929), 30-70. Jahrbuch Fortschritte Math. 55, 523
    • (1929) Z. F. Physik , vol.57 , pp. 30-70
    • Neumann, J.V.1
  • 55
    • 0003095621 scopus 로고    scopus 로고
    • [N(1929)] NEUMANN, J. v., Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik, Z. f. Physik 57 (1929), 30-70. Jahrbuch Fortschritte Math. 55, 523
    • Jahrbuch Fortschritte Math. , vol.55 , pp. 523
  • 56
    • 0002286325 scopus 로고
    • Proof of the quasi-ergodic hypothesis
    • [N(1932)] Zbl 4, 310-311
    • [N(1932)] NEUMANN, J. v., Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. U.S.A. 18 (1932), 70-82. Zbl 4, 310-311
    • (1932) Proc. Nat. Acad. Sci. U.S.A. , vol.18 , pp. 70-82
    • Neumann, J.V.1
  • 57
    • 21344483632 scopus 로고
    • Hydrodynamic limit for a Hamiltonian system with weak noise
    • [O-V-Y(1993)]
    • [O-V-Y(1993)] OLLA, S., VARADHAN, S. R. S. and YAU, H. T., Hydrodynamic limit for a Hamiltonian system with weak noise, Comm. Math. Phys. 155 (1993), 523-560.
    • (1993) Comm. Math. Phys. , vol.155 , pp. 523-560
    • Olla, S.1    Varadhan, S.R.S.2    Yau, H.T.3
  • 58
    • 0000616278 scopus 로고
    • The K-property of N billiard balls. I
    • [S(1992)] MR 93m:58066a
    • [S(1992)] SIMÁNYI, N., The K-property of N billiard balls. I, Invent. Math. 108 (1992), 521-548. MR 93m:58066a; The K-property of N billiard balls, II. Computation of neutral linear spaces, Invent. Math. 110 (1992), 151-172. MR 93m:58066b
    • (1992) Invent. Math. , vol.108 , pp. 521-548
    • Simányi, N.1
  • 59
    • 0000944602 scopus 로고
    • The K-property of N billiard balls, II. Computation of neutral linear spaces
    • MR 93m:58066b
    • [S(1992)] SIMÁNYI, N., The K-property of N billiard balls. I, Invent. Math. 108 (1992), 521-548. MR 93m:58066a; The K-property of N billiard balls, II. Computation of neutral linear spaces, Invent. Math. 110 (1992), 151-172. MR 93m:58066b
    • (1992) Invent. Math. , vol.110 , pp. 151-172
  • 60
    • 26744440835 scopus 로고    scopus 로고
    • The K-property of 4D billiards with nonorthogonal cylindric scatterers
    • [S-Sz(1994-A)] to appear
    • [S-Sz(1994-A)] SIMÁNYI, N. and SZÁSZ, D., The K-property of 4D billiards with nonorthogonal cylindric scatterers, J. Statist. Phys. (to appear).
    • J. Statist. Phys.
    • Simányi, N.1    Szász, D.2
  • 61
    • 26744445439 scopus 로고    scopus 로고
    • [S-Sz(1994-B)] in preparation
    • [S-Sz(1994-B)] SIMÁNYI, N. and SZÁSZ, D. (in preparation).
    • Simányi, N.1    Szász, D.2
  • 62
    • 0039110667 scopus 로고
    • Fifteen problems in mathematical physics
    • [S(1984)] Birkhäuser, Basel-Boston, Mass., MR 86g:81003
    • [S(1984)] SIMON, B., Fifteen problems in mathematical physics, Perspectives in mathematics, Birkhäuser, Basel-Boston, Mass., 1984, 423-454. MR 86g:81003
    • (1984) Perspectives in Mathematics , pp. 423-454
    • Simon, B.1
  • 63
    • 0001738670 scopus 로고
    • On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics
    • [S(1963)] (in Russian).
    • [S(1963)] SINAI, YA. G., On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics, Dokl. Akad. Nauk SSSR 153 (1963), 1261-1264 (in Russian). Translated as Soviet Math. Dokl. 4 (1963), 1818-1822. RŽ Mat 1964 #5 B497. MR 35 #5576
    • (1963) Dokl. Akad. Nauk SSSR , vol.153 , pp. 1261-1264
    • Sinai, Y.A.G.1
  • 64
    • 0001200214 scopus 로고
    • RŽ Mat 1964 #5 B497. MR 35 #5576
    • [S(1963)] SINAI, YA. G., On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics, Dokl. Akad. Nauk SSSR 153 (1963), 1261-1264 (in Russian). Translated as Soviet Math. Dokl. 4 (1963), 1818-1822. RŽ Mat 1964 #5 B497. MR 35 #5576
    • (1963) Soviet Math. Dokl. , vol.4 , pp. 1818-1822
  • 65
    • 0001379329 scopus 로고
    • Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards
    • [S(1970)] in Russian
    • [S(1970)] SINAI, YA. G., Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Uspehi Mat. Nauk 25 (1970), no. 2 (152), 141-192 (in Russian).
    • (1970) Uspehi Mat. Nauk , vol.25 , Issue.2-152 , pp. 141-192
    • Sinai, Y.A.G.1
  • 67
    • 0040071746 scopus 로고
    • Ergodic properties of a Lorentz gas
    • [S(1979)] in Russian. MR 81b:28018
    • [S(1979)] SINAI, YA. G., Ergodic properties of a Lorentz gas, Funktsional. Anal. i Prilozhen. 13 (1979), no. 3, 46-59 (in Russian). MR 81b:28018
    • (1979) Funktsional. Anal. i Prilozhen. , vol.13 , Issue.3 , pp. 46-59
    • Sinai, Y.A.G.1
  • 68
    • 0009281855 scopus 로고
    • Ergodic properties of some systems of two-dimensional disks and three-dimensional balls
    • [S-CH(1987)] MR 89c:58097
    • [S-CH(1987)] SINAI, YA. G. and CHERNOV, N. I., Ergodic properties of some systems of two-dimensional disks and three-dimensional balls, Uspekhi Mat. Nauk 42 (1987), 153-174. MR 89c:58097
    • (1987) Uspekhi Mat. Nauk , vol.42 , pp. 153-174
    • Sinai, Y.A.G.1    Chernov, N.I.2
  • 69
    • 0002298909 scopus 로고
    • Ergodicity of classical hard balls
    • [Sz(1993)]
    • [Sz(1993)] SZÁSZ, D., Ergodicity of classical hard balls, Physica A 194 (1993), 86- 92.
    • (1993) Physica A , vol.194 , pp. 86-92
    • Szász, D.1
  • 70
    • 21344475213 scopus 로고
    • The K-property of "orthogonal" cylindric billiards
    • [Sz(1994)]
    • [Sz(1994)] SZÁSZ, D., The K-property of "orthogonal" cylindric billiards, Comm. Math. Phys. 160 (1994), 581-597.
    • (1994) Comm. Math. Phys. , vol.160 , pp. 581-597
    • Szász, D.1
  • 71
    • 0040690714 scopus 로고
    • Ergodic properties of an ideal gas with an infinite number of degrees of freedom
    • [V-S(1971)] (in Russian). MR 44 #6289
    • [V-S(1971)] VOLKOVISSKIǏ, K. L. and SINAI, YA. G., Ergodic properties of an ideal gas with an infinite number of degrees of freedom, Funkcional. Anal. i Priložen. 5 (1971), no. 3, 19-21 (in Russian). MR 44 #6289
    • (1971) Funkcional. Anal. i Priložen. , vol.5 , Issue.3 , pp. 19-21
    • Volkovisskiǐ, K.L.1    Sinai, Y.A.G.2
  • 72
    • 0000889723 scopus 로고
    • A system of one-dimensional balls with gravity
    • [W(1990-A)] MR 91h:58066
    • [W(1990-A)] WOJTKOWSKI, M. P., A system of one-dimensional balls with gravity, Comm. Math. Phys. 126 (1990), 507-533. MR 91h:58066
    • (1990) Comm. Math. Phys. , vol.126 , pp. 507-533
    • Wojtkowski, M.P.1
  • 73
    • 0000308852 scopus 로고
    • The system of one-dimensional balls in an external field. II
    • [W(1990-B)] , MR 91k:58101
    • [W(1990-B)], WOJTKOWSKI, M. P., The system of one-dimensional balls in an external field. II, Comm. Math. Phys. 127 (1990), 425-432. MR 91k:58101
    • (1990) Comm. Math. Phys. , vol.127 , pp. 425-432
    • Wojtkowski, M.P.1
  • 74
    • 0001676415 scopus 로고
    • Linearly stable orbits in 3-dimensional billiards
    • [W(1990-C)] MR 91b:58128
    • [W(1990-C)] WOJTKOWSKI, M. P., Linearly stable orbits in 3-dimensional billiards, Comm. Math. Phys. 129 (1990), 319-327. MR 91b:58128
    • (1990) Comm. Math. Phys. , vol.129 , pp. 319-327
    • Wojtkowski, M.P.1
  • 75
    • 0010092610 scopus 로고    scopus 로고
    • Travaux de Herman sur les tores invariants
    • [Y(1992)]
    • [Y(1992)] Yoccoz, J-CH., Travaux de Herman sur les tores invariants, Séminaire Bourbaki, Vol. 1991/92, 44; Astérisque No. 206 (1992), Exp. No. 754, 311-344. MR 94g:58195
    • Séminaire Bourbaki , vol.1991 , Issue.92 , pp. 44
    • Yoccoz, J.-C.H.1
  • 76
    • 0010092610 scopus 로고    scopus 로고
    • Exp. No. 754, MR 94g:58195
    • [Y(1992)] Yoccoz, J-CH., Travaux de Herman sur les tores invariants, Séminaire Bourbaki, Vol. 1991/92, 44; Astérisque No. 206 (1992), Exp. No. 754, 311-344. MR 94g:58195
    • (1992) Astérisque , vol.206 , pp. 311-344


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.