Yu. M. Suhov

Stationary solutions of the Bogoliubov hierarchy equations in classical statistical mechanics. I

In the preceding paper under the same title we have formulated a theorem which describes the set of states (i.e., probability measures on phase space of an infinite system of particles inRv) corresponding to stationary solutions of the BBGKY hierarchy. We have proved the following statement: ifG is a Gibbs measure (Gibbs random point field) corresponding to a stationary solution of the BBGKY hierarchy, then its generating function satisfies a differential equation which is “conjugated” to the BBGKY hierarchy. The present paper deals with the investigation of the “conjugated” equation for the generating function in particular cases.

Random surfaces with two-sided constraints: An application of the theory of dominant ground states

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 ∣φx∣ ⩽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.

One-dimensional harmonic lattice caricature of hydrodynamics

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∃R1. 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 familyPɛ,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−1x in the statePɛ is approximately described by the SDMFF(x, θ); (b) The covariance (on large distances) decreases with distance quickly enough and uniformly inɛ. Given nonzerot∃R1, we consider the states Pɛ−1τɛ,ɛ>0, describing the system at the time momentsɛ−1t during its harmonic time evolution. We check that the covariance at lattice points close toɛ−1x in the state Pɛ−1τɛ is approximately described by a SDMFF(t;x, θ) and establish the connection betweenF(t; x, θ) andF(x,θ).

Higher-order Lindley equations

A model of a queueing network is proposed which leads to a stochastic equation generalizing a standard Lindley equation for a single FCFS server. We study the problem of the existence and uniqueness of a stationary solution to this equation and its connection with random processes on a Cayley tree.

On convergence to equilibrium distribution, II. The wave equation in odd dimensions, with mixing

The paper considers the wave equation, with constant or variable coefficients in ℝn, with odd n≥3. We study the asymptotics of the distribution μt of the random solution at time t ∈ ℝ as t → ∞. It is assumed that the initial measure μ0 has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that μ0 satisfies a Rosenblatt- or Ibragimov–Linnik-type space mixing condition. The main result is the convergence of μt to a Gaussian measure μ∞ as t → ∞, which gives a Central Limit Theorem (CLT) for the wave equation. The proof for the case of constant coefficients is based on an analysis of long-time asymptotics of the solution in the Fourier representation and Bernsteins “room-corridor” argument. The case of variable coefficients is treated by using a version of the scattering theory for infinite energy solutions, based on Vainbergs results on local energy decay.

Dobrushin's approach to queueing network theory

R.L. Dobrushin (1929-1995) made substantial contributions to Queueing Network Theory (QNT). A review of results from QNT which arose from his ideas or were connected to him in other ways is given. We also comment on various related open problems.

A Mean-Field Limit for a Class of Queueing Networks

A model of centralized symmetric message-switched networks is considered, where the messages having a common address must be served in the central node in the order which corresponds to their epochs of arrival to the network. The limitN → ∞ is discussed, whereN is the branching number of the network graph. This procedure is inspired by an analogy with statistical mechanics (the mean-field approximation). The corresponding limit theorems are established and the limiting probability distribution for the network response time is obtained. Properties of this distribution are discussed in terms of an associated boundary problem.

Random point processes and DLR equations

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.

The Boundedness of Branching Markov Processes

This chapter continues [5]–[7], and its main concepts originally arose from problems in queueing network theory. In the course of later development, the relations with the theory of branching processes were discovered. We are very glad to note that the subject of this chapter is closely related to the problems which were and are the focus of attention of Professor E. B. Dynkin; see [1] and references therein. One author (F.I.K) was a student of Evgenii Borisovich; the others had opportunities to attend his lecture courses and seminars which always had an enormous impact on listeners and participants (not to mention the experience of being examined by him, remembered by everyone for life). We hope that these results will develop further in the direction of Professor Dynkin’s interest.

A Criterion of Boundedness of Discrete Branching Random Walk

A general model of a branching random walk in Z is considered, where the branching and displacements occur with probabilities determined by the position of a parent. A necessary and sufficient condition is given for the random variable