Dynamical Systems of Statistical Mechanics

The motion of a system of N particles in d dimensions is described in Statistical Mechanics by means of a Hamiltonian system of 2Nd differential equations, which generates the group of transformations of the phase space. The object of the investigation is the time evolution of probability measures on the phase space determined by this group of transformations. The principal feature of problems in Statistical Mechanics is the fact that one deals with systems consisting of a large number of particles of the same type (a mole of a gas contains 6–1023 particles). Therefore, only those results in which all estimates are uniform with respect to the number of degrees of freedom are of interest here. This restriction, which is unusual from the point of view of the standard theory of dynamical systems, specifies the mathematical feature of the problems of Statistical Mechanics.

[1]  E. Hille.  An existence theorem , 1924 .

[2]  J. Lebowitz,et al.  Ergodic properties of an infinite one dimensional hard rod system , 1975 .

[3]  V. I. Arnolʹd,et al.  Ergodic problems of classical mechanics , 1968 .

[4]  B. Tirozzi,et al.  Time evolution of infinite classical systems with singular, long range, two body interactions , 1976 .

[5]  Carlo Cercignani,et al.  Kinetic Theories and the Boltzmann Equation , 1984 .

[6]  Ergodic properties of an ideal gas with an infinite number of degrees of freedom , 1971 .

[7]  O. Lanford.  Ergodic Theory and Approach to Equilibrium for Finite and Infinite Systems , 1973 .

[8]  R. Siegmund-Schultze.  On non-equilibrium dynamics of multidimensional infinite particle systems in the translation invariant case , 1985 .

[9]  David Ruelle,et al.  Superstable interactions in classical statistical mechanics , 1970 .

[10]  J. Lebowitz,et al.  Ergodic properties of infinite systems , 1975 .

[11]  R. Dobrushin,et al.  Non-equilibrium dynamics of two-dimensional infinite particle systems with a singular interaction , 1977 .

[12]  Asymptotic behaviour of time evolutions of infinite particle systems , 1980 .

[13]  Hans Zessin,et al.  The method of moments for random measures , 1983 .

[14]  D. Petrina.  Mathematical description of the evolution of infinite systems of classical statistical physics. I. Locally perturbed one-dimensional systems , 1979 .

[15]  O. Pazzis.  Ergodic properties of a semi-infinite hard rods system , 1971 .

[16]  L. Bunimovich,et al.  Markov Partitions for dispersed billiards , 1980 .

[17]  C. Boldrighini,et al.  Convergence to stationary states for infinite harmonic systems , 1983 .

[18]  R. Dobrushin,et al.  On the problem of the mathematical foundation of the Gibbs postulate ie classical statistical mechanics , 1978 .

[19]  Y. Sinai,et al.  Dynamical systems with elastic reflections , 1970 .

[20]  R. Lang.  On the asymptotic behaviour of infinite gradient systems , 1979 .

[21]  J. Doob.  Stochastic processes , 1953 .

[22]  Y. Sinai.  Ergodic properties of a gas of one-dimensional hard rods with an infinite number of degrees of freedom , 1972 .

[23]  Y. Suhov,et al.  Random point processes and DLR equations , 1976 .

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

[25]  L. Landau,et al.  The Theory of Phase Transitions , 1936, Nature.

[26]  Ergodic properties of the lorentz gas , 1979 .

[27]  Generalized solutions of Gibbs type for the Bogolyubov-Strel'tsova diffusion hierarchy , 1984 .

[28]  O. Lanford.  The classical mechanics of one-dimensional systems of infinitely many particles , 1968 .

[29]  Local stability and hydrodynamical limit of Spitzer's one dimensional lattice model , 1982 .

[30]  O. Lanford.  Time evolution of large classical systems , 1975 .

[31]  R. Dobrushin,et al.  One-dimensional hard rod caricature of hydrodynamics , 1983 .

[32]  E. H. Hauge.  What can one learn from Lorentz models , 1974 .

[33]  J. Lebowitz,et al.  Time evolution and ergodic properties of harmonic systems , 1975 .

[34]  R. Illner,et al.  Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum , 1986 .

[35]  Viktor Ivanovich Gerasimenko,et al.  Statistical mechanics of quantum-classical systems. Nonequilibrium systems , 1980 .

[36]  A. Lenard,et al.  Correlation functions and the uniqueness of the state in classical statistical mechanics , 1973 .

[37]  O. Bratteli.  Operator Algebras And Quantum Statistical Mechanics , 1979 .

[38]  S. V. Fomin,et al.  Ergodic Theory , 1982 .

[39]  J. Lebowitz,et al.  Thermodynamic Limit of Time‐Dependent Correlation Functions for One‐Dimensional Systems , 1970 .

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

[41]  N. Bogolyubov,et al.  Mathematical description of the equilibrium state of classical systems on the basis of the canonical ensemble formalism , 1969 .

[42]  R. Siegmund-Schultze,et al.  The Hydrodynamic Limit for Systems of Particles with Independent Evolution , 1982 .

[43]  David Griffeath,et al.  Additive and Cancellative Interacting Particle Systems , 1979 .

[44]  A. Lenard,et al.  States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures , 1975 .

[45]  R. Alexander.  Time evolution for infinitely many hard spheres , 1976 .

[46]  S. Vladut,et al.  Number of points of an algebraic curve , 1983 .

[47]  Leonid A. Bunimovich,et al.  Statistical properties of lorentz gas with periodic configuration of scatterers , 1981 .

[48]  H. Spohn.  Hydrodynamical theory for equilibrium time correlation functions of hard rods , 1982 .

[49]  Gibbsian Description of Point Random Fields , 1977 .

[50]  R. Dobrušin.  CONDITIONS FOR THE ABSENCE OF PHASE TRANSITIONS IN ONE-DIMENSIONAL CLASSICAL SYSTEMS , 1974 .

[51]  H.-G. Schöpf.  V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics. (The Mathematical Physics Monograph Series) IX + 286 S. m. Fig. New York/Amsterdam 1968. W. A. Benjamin, Inc. Preis geb. $ 14.75, brosch. $ 6.95 . , 1970 .

[52]  T. Liggett,et al.  The stochastic evolution of infinite systems of interacting particles , 1977 .

[53]  A. Lenard,et al.  States of classical statistical mechanical systems of infinitely many particles. I , 1975 .

[54]  H. Beijeren.  Transport properties of stochastic Lorentz models , 1982 .

[55]  Y. M. Sukhov.  Steady solutions of the BBGKY hierarchy and first integrals of the motion of a system of classical particles One-dimensional case , 1983 .

[56]  R. Dobrushin.  Gibbsian random fields for particles without hard core , 1970 .

[57]  J. Percus.  Exact Solution of Kinetics of a Model Classical Fluid , 1969 .

[58]  Time evolution of infinite anharmonic systems , 1977 .

[59]  George Papanicolaou,et al.  A limit theorem for turbulent diffusion , 1979 .

[60]  H. Grad.  Principles of the Kinetic Theory of Gases , 1958 .

[61]  O. Lanford.  The classical mechanics of one-dimensional systems of infinitely many particles , 1969 .

[62]  Mario Pulvirenti,et al.  Vortex Methods in Two-Dimensional Fluid Dynamics , 1984 .

[63]  M. Pulvirenti.  On the time evolution of the states of infinitely extended particles systems , 1982 .

[64]  S. Kaniel,et al.  The Boltzmann equation , 1978 .

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

[66]  R. Dobrushin,et al.  Non-equilibrium dynamics of one-dimensional infinite particle systems with a hard-core interaction , 1977 .

[67]  Bernard Howard Lavenda,et al.  Nonequilibrium Statistical Thermodynamics , 1985 .

[68]  S. Miracle-Sole,et al.  Absence of Phase Transitions in Hard‐Core One‐Dimensional Systems with Long‐Range Interactions , 1970 .

[69]  R. Temam.  Navier-Stokes Equations , 1977 .

[70]  Ya. G. Sinai.  Construction of dynamics in one-dimensional systems of statistical mechanics , 1972 .

[71]  F. Zirilli,et al.  Scattering states and bound states in λ℘(φ)2 , 1976 .

[72]  V. Maslov,et al.  Asymptotics of the Kolmogorov-Feller equation for a system of a large number of particles , 1983 .

[73]  B. Gurevich,et al.  Stationary solutions of the bogoliubov hierarchy equations in classical statistical mechanics. 2 , 1977 .

[74]  O. Kallenberg.  On the asymptotic behavior of line processes and systems of non-interacting particles , 1978 .

[75]  H. Spohn.  Kinetic equations from Hamiltonian dynamics: Markovian limits , 1980 .

[76]  V. Malyshev,et al.  Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi-gas , 1983 .

[77]  J. Lebowitz,et al.  Steady state self-diffusion at low density , 1982 .

[78]  W. Braun,et al.  The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles , 1977 .

[79]  Some remarks on nonequilibrium dynamics of infinite particle systems , 1984 .

[80]  Infinite lattice systems of interacting diffusion processes, existence and regularity properties , 1982 .

[81]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[82]  Rick Durrett,et al.  An introduction to infinite particle systems , 1981 .