Lawrence E. Thomas

Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics

Abstract: We continue the study of a model for heat conduction [6] consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite-dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators.

EXACT GROUND STATE ENERGY OF THE STRONG-COUPLING POLARON

Abstract: The polaron has been of interest in condensed matter theory and field theory for about half a century, especially the limit of large coupling constant, α. It was not until 1983, however, that a proof of the asymptotic formula for the ground state energy was finally given by using difficult arguments involving the large deviation theory of path integrals. Here we derive the same asymptotic result, , and with explicit error bounds, by simple, rigorous methods applied directly to the Hamiltonian. Our method is easily generalizable to other settings, e.g., the excitonic and magnetic polarons.

Asymptotic Behavior of Thermal Nonequilibrium Steady States for a Driven Chain of Anharmonic Oscillators

Abstract: We consider a model of heat conduction introduced in [6], which consists of a finite nonlinear chain coupled to two heat reservoirs at different temperatures. We study the low temperature asymptotic behavior of the invariant measure. We show that, in this limit, the invariant measure is characterized by a variational principle. The main technical ingredients are some control theoretic arguments to extend the Freidlin–Wentzell theory of large deviations to a class of degenerate diffusions.

Fluctuations of the Entropy Production in Anharmonic Chains

Abstract. We prove the Gallavotti-Cohen fluctuation theorem for a model of heat conduction through a chain of anharmonic oscillators coupled to two Hamiltonian reservoirs at different temperatures.

Exponential convergence to equilibrium for a class of random-walk models

We prove exponential convergence to equilibrium (L2 geometric ergodicity) for a random walk with inward drift on a sub-Cayley rooted tree. This randomwalk model generalizes a Monte Carlo algorithm for the self-avoiding walk proposed by Berretti and Sokal. If the number of vertices of levelN in the tree grows asCN~μNNγ−1, we prove that the autocorrelation time τ satisfies 〈N〉2 ≲ τ ≲ 〈N〉1+γ

Expansions and phase transitions for the ground state of quantum Ising lattice systems

Expansions for the ground state of some transverse Ising-like models are developed. These expansions are easily estimated by the solutions to some simple implicit equations. Short range or long range order obtains, depending on the coupling constants of the models.

Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. II. An explicit Plancherel formula

In its ground state representation, the infinite, spin 1/2 Heisenberg chain provides a model for spin wave scattering, which entails many features of the quantum mechanicalN-body problem. Here, we give a complete eigenfunction expansion for the Hamiltonian of the chain in this representation, forall numbers of spin waves. Our results resolve the questions of completeness and orthogonality of the eigenfunctions given by Bethe for finite chains, in the infinite volume limit.

Ground state representation of the infinite one-dimensional Heisenberg ferromagnet

In its ground state representation, the infinite, spin 1/2 Heisenberg chain provides a model for spin wave scattering, which entails many features of the quantum mechanicalN-body problem. Here, we give a complete eigenfunction expansion for the Hamiltonian of the chain in this representation, forall numbers of spin waves. Our results resolve the questions of completeness and orthogonality of the eigenfunctions given by Bethe for finite chains, in the infinite volume limit.

Semiclassical approximation for Schrödinger operators on a two‐sphere at high energy

Let H=−ΔS+V be a Schrodinger operator acting in L2(S), with S the two‐dimensional unit sphere, ΔS the spherical Laplacian, and V a smooth potential. Approximate eigenfunctions and eigenvalues for H are obtained involving expansions in inverse powers of the classical angular momentum variables, provided that these variables are in a region of phase space where the corresponding classical Hamiltonian is nearly integrable. The analysis is carried out in a Bargmann representation, where ΔS becomes a quadratic expression in the sum of two quantum harmonic oscillator Hamiltonians, and V becomes a pseudodifferential operator.

Bipolaron and N-Polaron Binding Energies

The binding of polarons, or its absence, is an old and subtle topic. Here we prove two things rigorously. First, the transition from many-body collapse to the existence of a thermodynamic limit for N polarons occurs precisely at U=2α, where U is the electronic Coulomb repulsion and α is the polaron coupling constant. Second, if U is large enough, there is no multipolaron binding of any kind. Considering the known fact that there is binding for some U>2α, these conclusions are not obvious and their proof has been an open problem for some time.