33 research outputs found
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 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-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 , , with constant
or variable coefficients. We study the distribution of the random
solution at time . We assume that the initial probability measure
has zero mean, a translation-invariant covariance, and a finite mean
energy density. We also asume that satisfies a Rosenblatt- or
Ibragimov-Linnik-type mixing condition. The main result is the convergence of
to a Gaussian probability measure as 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 Ï Ï” [0,a^(-1)). The correction is given, at t~0, by the non-linear second-order differential operator (Bf)(q,v) = (a^2/2)(â/âq)[â«dw|v-w|f(q,w)(â/âq)f(q,v) - f(q,v)â«dw|v-w|(â/âq)f(q,w)](1-Ïa)^(-1) where f(q,v) is the (hydrodynamical) density at a point q Ï” R^1 of the species of particles with velocity v Ï” 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 Î, H_Î = K_Î + U_Î. Here K_Î is the kinetic energy and U_Î 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_(ÎâZ) T(e^(itH_Î)*Ae^(-itH_Î)) for any t Ï” R^4 and any local observable A. Thereby a family {T_t, t Ï” R^4} of locally normal states is determined which describes the time-evolution of the initial state T