A quantum Mermin--Wagner theorem for quantum rotators on two--dimensional graphs

This is the first of a series of papers considering symmetry properties of quantum systems over 2D graphs or manifolds, with continuous spins, in the spirit of the Mermin--Wagner theorem. In the model considered here (quantum rotators) the phase space of a single spin is a $d-$dimensional torus, and spins (or particles) are attached to sites of a graph satisfying a special bi-dimensionality property. The kinetic energy part of the Hamiltonian is minus a half of the Laplace operator. We assume that the interaction potential is C$^2$-smooth and invariant under the action of a connected Lie group {\ttG}. A part of our approach is to give a definition (and a construction) of a class of infinite-volume Gibbs states for the systems under consideration (the class \fG). This class contains the so-called limit Gibbs states, with or without boundary conditions. We use ideas and techniques originated from various past papers, in combination with the Feynman--Kac representation, to prove that any state lying in the class \fG (defined in the text) is {\ttG}-invariant. An example is given where the interaction potential is singular and there exists a Gibbs state which is not {\ttG}-invariant. In the next paper under the same title we establish a similar result for a bosonic model where particles can jump from a vertex of the graph to one of its neighbors (a generalized Hubbard model).Comment: 27 page

On Convergence to Equilibrium Distribution, I. The Klein - Gordon Equation with Mixing

Consider the Klein-Gordon equation (KGE) in $\R^n$, $n\ge 2$, with constant or variable coefficients. We study the distribution $\mu_t$ of the random solution at time $t\in\R$. We assume that the initial probability measure $\mu_0$ has zero mean, a translation-invariant covariance, and a finite mean energy density. We also asume that $\mu_0$ satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The main result is the convergence of $\mu_t$ to a Gaussian probability measure as $t\to\infty$ which gives a Central Limit Theorem for the KGE. 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 Bernstein's `room-corridor' argument. The case of variable coefficients is treated by using an `averaged' version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.Comment: 30 page

Fast Jackson-Type Networks with Dynamic Routing

We propose a new class of models of queueing networks with load-balanced dynamic routing. The paper extends earlier works, including [FC], [FMcD], [VDK], where systems with no feedback were considered. The main results are: (a) a sufficient condition for positive recurrence of the arising Markov process and (b) a limiting mean-field picture where the process becomes deterministic and is described by a system of non-linear ODEs

One-Dimensional Hard-Rod Caricature of Hydrodynamics: Navier-Stokes Correction

One-dimensional system of hard-rod particles of length a is studied in the hydrodynamical limit. The Navier-Stokes correction to Euler's equation is found for an initial locally-equilibrium family of states of constant density ρ ϵ [0,a^(-1)). The correction is given, at t~0, by the non-linear second-order differential operator (Bf)(q,v) = (a^2/2)(∂/∂q)[∫dw|v-w|f(q,w)(∂/∂q)f(q,v) - f(q,v)∫dw|v-w|(∂/∂q)f(q,w)](1-ρa)^(-1) where f(q,v) is the (hydrodynamical) density at a point q ϵ R^1 of the species of particles with velocity v ϵ R^1

Towards Time - Dynamics for Bosonic Systems in Quantum Statistical Mechanics

Consider a one-dimensional lattice boson system with the Hamiltonian in a finite box Λ, H_Λ = K_Λ + U_Λ. Here K_Λ is the kinetic energy and U_Λ is the potential energy corresponding to a finite-range pair interaction. For a class of states T of the infinite system, we prove the existence of the limit T_t(A) = lim_(Λ→Z) T(e^(itH_Λ)*Ae^(-itH_Λ)) for any t ϵ R^4 and any local observable A. Thereby a family {T_t, t ϵ R^4} of locally normal states is determined which describes the time-evolution of the initial state T