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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Communications in mathematical physics 50 (1976), S. 113-132 
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We prove the uniqueness of a solution of the Dobrushin-Lanford-Ruelle equation for random point processes when the generating function (interaction potential) has no hard cores, is non-negative and rapidely decreasing.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In this paper we study the time evolution of a regular class of states of an infinite classical system of anharmonic oscillators. The conditional probabilities are investigated and an explicit form for these is given.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract This is the forth and final paper of a series in which we investigate the stationary solutions of the BBGKY equations. Herein we prove a lemma which forms the basic step in the proof of our Main Theorem characterizing the stationary solutions of these equations which was stated in the first of this series.
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  • 4
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: We study the spectrum of the operator generating an infinite-dimensional diffusion process Ξ (t), in space . Here ν is a “natural”Ξ (t)-invariant measure on which is a Gibbs distribution corresponding to a (formal) Hamiltonian H of an anharmonic crystal, with a value of the inverse temperature β 〉 0. For β small enough, we establish the existence of an L-invariant subspace such that has a distinctive character related to a “quasi-particle” picture. In particular, has a Lebesgue spectrum separated from the rest of the spectrum of L and concentrated near a point κ1〉0 giving the smallest non-zero eigenvalue of a limiting problem associated with β= 0. An immediate corollary of our result is an exponentially fast L 2-convergence to equilibrium for the process Ξ(t) for small values of β.
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We continue the analysis of the “conjugate” equation for the generating function of a Gibbs random point field corresponding to a stationary solution of the classical BBGKY hierarchy. This equation was established and partially investigated in the preceding papers under the same title. In the present paper we reduce a general theorem about the form of solutions of the “conjugate” equation to a statement which relates to a special case where the interacting particles constitute a “quasi”—one dimensional configuration.
    Type of Medium: Electronic Resource
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  • 6
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract A general model of a branching random walk inR 1 is considered, with several types of particles, where the branching occurs with probabilities determined by the type of a parent particle. Each new particle starts moving from the place where it was born, independently of other particles. The distribution of the displacement of a particle, before it splits, depends on its type. A necessary and sufficient condition is given for the random variable $$X^0 = \mathop {\sup max}\limits_{ n \geqq 0 1 \leqq k \leqq N_n } X_{n,k} $$ to be finite. Here,X n, k is the position of thek th particle in then th generation,N n is the number of particles in then th generation (regardless of their type). It turns out that the distribution ofX 0 gives a minimal solution to a natural system of stochastic equations which has a linearly ordered continuum of other solutions. The last fact is used for proving the existence of a monotone travelling-wave solution to systems of coupled non-linear parabolic PDE's.
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Queueing systems 6 (1990), S. 391-403 
    ISSN: 1572-9443
    Keywords: Packet-switching networks ; infinite graph ; FCFS discipline ; small perturbation ; receiving time ; stochastic network equations ; stationary solution ; majorizing families ; decay of dependence (correlations)
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract A class of open queueing networks with packet switching is discussed. The configuration graph of the network may be finite or infinite. The external messages are divided into standard pieces (packets) each of which is transmitted as a single message. The messages are addressed, as a rule, to nearest neighbours and thereby the network may be treated as a small perturbation of the collection of isolated servers. The switching rule adopted admits overtaking: packets which appeared later may reach the delivery node earlier. The transmission of a message is completed when its last packet reaches the destination node. The main result of this paper is that the network possesses a unique stationary working regime. Weak dependence properties of this regime are established as well as the continuity with respect to the parameters of the external message flows.
    Type of Medium: Electronic Resource
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  • 8
    ISSN: 1572-9613
    Keywords: Harmonic oscillators ; Gibbs states and conserved quantities ; hydrodynamic limit.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We derive the hydrodynamic (Euler) approximation for the harmonic time evolution of infinite classical oscillator system on one-dimensional lattice ℤ1 It is known that equilibrium (i.e., time-invariant attractive) states for this model are translationally invariant Gaussian ones, with the mean 0, which satisfy some linear relations involving the interaction quadratic form. The natural “parameter” characterizing equilibrium states is the spectral density matrix function (SDMF)F(θ), θ∃[− π, π). Time evolution of a space “profile” of local equilibrium parameters is described by a space-time SDMFF(t;x, θ) t, x∃R 1. The hydrodynamic equation forF(t; x, θ) which we derive in this paper means that the “normal mode” profiles indexed byθ are moving according to linear laws and are mutually independent. The procedure of deriving the hydrodynamic equation is the following: We fix an initial SDMF profileF(x, θ) and a familyP ɛ,ɛ〉0 of mean 0 states which satisfy the two conditions imposed on the covariance of spins at various lattice points: (a) the covariance at points “close” to the valueɛ −1 x in the stateP ɛ is approximately described by the SDMFF(x, θ); (b) The covariance (on large distances) decreases with distance quickly enough and uniformly inɛ. Given nonzerot∃R 1, we consider the states P ɛ−1τ ɛ ,ɛ〉0, describing the system at the time momentsɛ −1 t during its harmonic time evolution. We check that the covariance at lattice points close toɛ −1 x in the state P ɛ−1τ ɛ is approximately described by a SDMFF(t;x, θ) and establish the connection betweenF(t; x, θ) andF(x,θ).
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  • 9
    ISSN: 1572-9613
    Keywords: Nonequilibrium quantum statistical mechanics ; convergence to a stationary state ; hydrodynamic limit ; one-dimensionalXY model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We prove theorems on convergence to a stationary state in the course of time for the one-dimensionalXY model and its generalizations. The key point is the well-known Jordan-Wigner transformation, which maps theXY dynamics onto a group of Bogoliubov transformations on the CARC *-algebra overZ 1. The role of stationary states for Bogoliubov transformations is played by quasifree states and for theXY model by their inverse images with respect to the Jordan-Wigner transformation. The hydrodynamic limit for the one-dimensionalXY model is also considered. By using the Jordan-Wigner transformation one reduces the problem to that of constructing the hydrodynamic limit for the group of Bogoliubov transformations. As a result, we obtain an independent motion of “normal modes,” which is described by a hyperbolic linear differential equation of second order. For theXX model this equation reduces to a first-order transfer equation.
    Type of Medium: Electronic Resource
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  • 10
    ISSN: 1572-9613
    Keywords: Random surfaces ; SOS model with symmetric constraints ; dominant ground states
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider some models of classical statistical mechanics which admit an investigation by means of the theory of dominant ground states. Our models are related to the Gibbs ensemble for the multidimensional SOS model with symmetric constraints ∣φ x ∣ ⩽m/2. The main result is that for β⩾β0, where β0 does not depend onm, the structure of thermodynamic phases in the model is determined by dominant ground states: for an evenm a Gibbs state is unique and for an oddm the number of space-periodic pure Gibbs states is two.
    Type of Medium: Electronic Resource
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