Aspects of Ergodic, Qualitative and Statistical Theory of Motion

1 General Qualitative Properties.- 2 Ergodicity and Ergodic Points.- 3 Entropy and Complexity.- 4 Markovian Pavements.- 5 Gibbs Distributions.- 6 General Properties of Gibbs and SRB Distributions.- 7 Analyticity, Singularity and Phase Transitions.- 8 Special Ergodic Theory Problems in Nonchaotic Dynamics.- 9 Some Special Topics in KAM Theory.- 10 Special Problems in Chaotic Dynamics.- A Nonequilibrium Thermodynamics? Twenty-Seven Comments.- Name Index.- Citations Index.

[1]  C. Shen A Functional Calculus Approach to the Ursell‐Mayer Functions , 1972 .

[2]  Joel L. Lebowitz,et al.  Boltzmann's Entropy and Time's Arrow , 1993 .

[3]  Absence of phase transitions in one-dimensional systems with hard cores , 1968 .

[4]  Absence of Second-Order Phase Dobrushin Uniqueness Region , 1981 .

[5]  P. Pearce,et al.  Mathematical properties of position-space renormalization-group transformations , 1979 .

[6]  D. Saari,et al.  Stable and Random Motions in Dynamical Systems , 1975 .

[7]  Pierre Collet,et al.  Universal properties of maps on an interval , 1980 .

[8]  Charles B. Morrey,et al.  On the derivation of the equations of hydrodynamics from statistical mechanics , 1955 .

[9]  R. L. Dobrushin,et al.  Gibbsian random fields for lattice systems with pairwise interactions , 1968 .

[10]  David Ruelle,et al.  Extension of the Lee-Yang Circle Theorem , 1971 .

[11]  David P Ruelle Extending the definition of entropy to nonequilibrium steady states , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[12]  Methods in the theory of quasi periodic motions , 1997 .

[13]  D. Lindley,et al.  Boltzmann’s Atom: The Great Debate That Launched a Revolution in Physics , 2001 .

[14]  R. Bowen,et al.  MARKOV PARTITIONS FOR AXIOM A DIFFEOMORPHISMS. , 1970 .

[15]  Giovanni Gallavotti,et al.  A local fluctuation theorem , 1998, chao-dyn/9808005.

[16]  G. Gentile,et al.  Majorant series convergence for twistless KAM tori , 1995, Ergodic Theory and Dynamical Systems.

[17]  David Ruelle,et al.  Some remarks on the ground state of infinite systems in statistical mechanics , 1969 .

[18]  S. Miracle-Sole,et al.  Analyticity properties of a lattice gas , 1967 .

[19]  Taro Asano,et al.  Theorems on the Partition Functions of the Heisenberg Ferromagnets , 1970 .

[20]  James A. Yorke,et al.  Expanding maps on sets which are almost invariant. Decay and chaos , 1979 .

[21]  Michał Misiurewicz,et al.  Absolutely continuous measures for certain maps of an interval , 1981 .

[22]  A. Rényi Representations for real numbers and their ergodic properties , 1957 .

[23]  Michael E. Fisher,et al.  Zeros of the Partition Function for the Heisenberg, Ferroelectric, and General Ising Models , 1971 .

[24]  Large Deviations in Rarefied Quantum Gases , 2001, cond-mat/0107295.

[25]  David Ruelle,et al.  Observables at infinity and states with short range correlations in statistical mechanics , 1969 .

[26]  L. Van Hove,et al.  WITHDRAWN: SUR L'INTÉGRALE DE CONFIGURATION POUR LES SYSTÉMES DE PARTICULES Á UNE DIMENSION , 1966 .

[27]  J. Bricmont,et al.  High temperature expansions and dynamical systems , 1995 .

[28]  P. L. Dobruschin The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity , 1968 .

[29]  E. Olivieri On a cluster expansion for lattice spin systems: A finite-size condition for the convergence , 1988 .

[30]  D. Ornstein Ergodic theory, randomness, and dynamical systems , 1974 .

[31]  David Ruelle,et al.  Statistical mechanics of a one-dimensional lattice gas , 1968 .

[32]  Giovanni Gallavotti Renormalization group in Statistical Mechanics and Mechanics: gauge symmetries and vanishing beta functions , 2001 .

[33]  M. Feigenbaum The universal metric properties of nonlinear transformations , 1979 .

[34]  D. Ruelle States of Classical Statistical Mechanics , 1967 .

[35]  K. Jacobs Die Übertragung diskreter Informationen durch periodische und fastperiodische Kanäle , 1959 .

[36]  A. Wehrl General properties of entropy , 1978 .

[37]  L. Onsager Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[38]  D. Iagolnitzer,et al.  Decrease properties of truncated correlation functions and analyticity properties for classical lattices and continuous systems , 1973 .

[39]  R. Dobrushin,et al.  Completely analytical interactions: Constructive description , 1987 .

[40]  L. Russo,et al.  A family of codes between some Markov and Bernoulli schemes , 1975 .

[41]  R. Israel High-temperature analyticity in classical lattice systems , 1976 .

[42]  Scaling near resonances and complex rotation numbers for the standard map , 1994 .

[43]  R. MacKay,et al.  Approximate renormalization with codimension-one fixed point for the break-up of some three-frequency tori , 2000 .

[44]  Meir Smorodinsky,et al.  Bernoulli schemes of the same entropy are finitarily isomorphic , 1979 .

[45]  S. M. Ulam,et al.  On Combination of Stochastic and Deterministic Processes , 1947 .

[46]  R. Minlos,et al.  THE PHENOMENON OF “PHASE SEPARATION” AT LOW TEMPERATURES IN SOME LATTICE MODELS OF A GAS. I , 1967 .

[47]  齐景福,et al.  Schwartz综合征一例 , 1989 .

[48]  Lyapunov spectra and nonequilibrium ensembles equivalence in 2D fluid mechanics , 2002, nlin/0209039.

[49]  G. Gallavotti Instabilities and phase transitions in the ising model. A review , 1972 .

[50]  David Ruelle,et al.  On the use of “small external fields” in the problem of symmetry breakdown in statistical mechanics☆ , 1972 .

[51]  Guido Gentile,et al.  Scaling properties for the radius of convergence of a Lindstedt series: the standard map , 1999 .

[52]  Pierre Collet,et al.  On the abundance of aperiodic behaviour for maps on the interval , 1980 .

[53]  Guido Gentile,et al.  Lindstedt series, ultraviolet divergences and Moser's theorem , 1995, chao-dyn/9511009.

[54]  M. Feigenbaum Quantitative universality for a class of nonlinear transformations , 1978 .

[55]  D. Ruelle Correlation functions of classical gases , 1963 .

[56]  T. D. Lee,et al.  Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model , 1952 .

[57]  E. Olivieri,et al.  Renormalization group and analyticity in one dimension: A proof of Dobrushin's theorem , 1981 .

[58]  A. Mazel,et al.  Uniqueness and exponential decay of correlations for some two-dimensional spin lattice systems , 1996 .

[59]  Approximate symmetries and their spontaneous breakdown , 1975 .

[60]  SRB States and Nonequilibrium Statistical Mechanics Close to Equilibrium , 1996, chao-dyn/9612002.

[61]  Elliptic equations and Gaussian processes , 1980 .

[62]  Existence of invariant measures for piecewise continuous transformations , 1981 .

[63]  J. Groeneveld,et al.  Two theorems on classical many-particle systems , 1962 .

[64]  Barry Simon,et al.  The statistical mechanics of lattice gases , 1993 .

[65]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[66]  Chetana Prakash High-temperature differentiability of lattice Gibbs states by Dobrushin uniqueness techniques , 1983 .

[67]  E. Olivieri,et al.  Some probabilistic techniques in field theory , 1978 .

[68]  K. Wilson MODEL OF COUPLING-CONSTANT RENORMALIZATION. , 1970 .

[69]  Analyticity and clustering properties of unbounded spin systems , 1978 .

[70]  G. Birkhoff Proof of a Recurrence Theorem for Strongly Transitive Systems. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[71]  H. Poincaré,et al.  Sur les courbes définies par les équations différentielles(III) , 1885 .

[72]  P. Mazur,et al.  Non-equilibrium thermodynamics, , 1963 .

[73]  Michael Keane,et al.  A class of finitary codes , 1977 .

[74]  Stephen G. Brush,et al.  Gadflies and Geniuses in the History of Gas Theory , 1999, Synthese.

[75]  Oliver Penrose,et al.  Convergence of Fugacity Expansions for Fluids and Lattice Gases , 1963 .

[76]  J. Lebowitz,et al.  On the (Boltzmann) entropy of non-equilibrium systems , 2003, cond-mat/0304251.

[77]  L. Chierchia,et al.  Compensations in small divisor problems , 1996 .

[78]  Jürgen Moser,et al.  Convergent series expansions for quasi-periodic motions , 1967 .

[79]  B. L. Waerden Die lange Reichweite der regelmäßigen Atomanordnung in Mischkristallen , 1941 .

[80]  P. Résibois H-theorem for the (modified) nonlinear Enskog equation , 1978 .

[81]  G. Gallavotti The phase separation line in the two-dimensional Ising model , 1972 .

[82]  A. Davie,et al.  The critical function for the semistandard map , 1994 .

[83]  Elisa Ercolessi,et al.  The Renormalization Group and Critical Phenomena , 2001 .

[84]  H. Kunz,et al.  General properties of polymer systems , 1971 .

[85]  Peter V. E. McClintock,et al.  Ludwig Boltzmann : the man who trusted atoms. , 2006 .

[86]  R. Bowen Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms , 1975 .

[87]  David Ruelle,et al.  A MEASURE ASSOCIATED WITH AXIOM-A ATTRACTORS. , 1976 .

[88]  V. Arnold SMALL DENOMINATORS AND PROBLEMS OF STABILITY OF MOTION IN CLASSICAL AND CELESTIAL MECHANICS , 1963 .

[89]  Robert B. Griffiths,et al.  Strict convexity (“continuity”) of the pressure in lattice systems , 1971 .

[90]  G. Gallavotti Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods , 1985 .

[91]  S. Miracle-Sole,et al.  Correlation functions of a lattice system , 1968 .

[92]  G. Gentile,et al.  Tree expansion and multiscale analysis for KAM tori , 1995 .

[93]  Jürgen Moser Is the solar system stable? , 1978 .

[94]  Garrett S. Sylvester Weakly coupled Gibbs measures , 1979 .

[95]  Anthony Manning,et al.  Axiom A Diffeomorphisms have Rational Zeta Functions , 1971 .

[96]  J. Yorke,et al.  On the existence of invariant measures for piecewise monotonic transformations , 1973 .

[97]  D. Ruelle,et al.  Applications conservant une mesure absolument continue par rapport àdx sur [0, 1] , 1977 .

[98]  Giovanni Gallavotti,et al.  TWISTLESS KAM TORI, QUASI FLAT HOMOCLINIC INTERSECTIONS, AND OTHER CANCELLATIONS IN THE PERTURBATION SERIES OF CERTAIN COMPLETELY INTEGRABLE HAMILTONIAN SYSTEMS: A REVIEW , 1993, chao-dyn/9304012.

[99]  D. Ruelle Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics , 1998, chao-dyn/9812032.

[100]  G. Pianigiani First return map and invariant measures , 1980 .

[101]  K. Parthasarathy,et al.  Probability measures on metric spaces , 1967 .

[102]  David Ruelle,et al.  Some remarks on the location of zeroes of the partition function for lattice systems , 1973 .

[103]  L. H. Eliasson,et al.  Absolutely convergent series expansions for quasi periodic motions. , 1996 .

[104]  G. Gallavotti,et al.  One dimensional lattice gases with rapidly decreasing interaction , 1970 .

[105]  Jackson,et al.  Constrained Euler system for Navier-Stokes turbulence. , 1993, Physical review letters.

[106]  Freeman J. Dyson,et al.  Existence of a phase-transition in a one-dimensional Ising ferromagnet , 1969 .

[107]  R. Holley Remarks on the FKG inequalities , 1974 .

[108]  L. Breiman The Individual Ergodic Theorem of Information Theory , 1957 .

[109]  R. Becker Electromagnetic fields and interactions , 1982 .

[110]  J. Bricmont,et al.  Infinite-dimensional SRB measures , 1995, chao-dyn/9509022.

[111]  Derek W. Robinson,et al.  Statistical mechanics of quantum spin systems. III , 1968 .

[112]  D. Ruelle Turbulence, strange attractors, and chaos , 1995 .

[113]  Guido Gentile,et al.  Bryuno Function and the Standard Map , 1998, chao-dyn/9810035.

[114]  Camillo Cammarota,et al.  Decay of correlations for infinite range interactions in unbounded spin systems , 1982 .

[115]  I. Bendixson Sur les courbes définies par des équations différentielles , 1901 .

[116]  Giovanni Gallavotti,et al.  Twistless KAM tori , 1993, chao-dyn/9306003.