Correlation Inequalities for Spin Glasses

We prove a correlation type inequality for spin systems with quenched symmetric random interactions. This gives monotonicity of the pressure with respect to the strength of the interaction for a class of spin glass models. Consequences include existence of the thermodynamic limit for the pressure and bounds on the surface pressure. We also describe other conjectured inequalities for such systems. Submitted: January 6, 2007. Accepted: May 10, 2007.     

Time Asymptotics of the Schrödinger Wave Function in Time-Periodic Potentials

Journal of Statistical Physics - We study the transition to the continuum of an initially bound quantum particle in ? d , d=1,2,3, subjected, for t≥0, to a time periodic forcing of...     

Statistical mechanics of the nonlinear Schrödinger equation. II. Mean field approximation

We investigate a mean field approximation to the statistical mechanics of complex fields with dynamics governed by the nonlinear Schrödinger equation. Such fields, whose Hamiltonian is unbounded below, may model plasmas, lasers, and other physical systems. Restricting ourselves to one-dimensional systems with periodic boundary conditions, we find in the mean field approximation a phase transition from a uniform regime to a regime in which the system is dominated by solitons. We compute explicitly, as a function of temperature and density (L ² norm), the transition point at which the uniform configuration becomes unstable to local perturbations; static and dynamic mean field approximations yield the same result.     

A Laudatio for Elliott Lieb on His Receiving the Poincaré Medal at the International Congress on Mathematical Physics in Lisbon,July 30, 2003

Journal of Statistical Physics -     

Spectral and stability aspects of quantum chaos

The relation between the spectrum of a generalized quasienergy operator and the stability of quantum systems driven by quasiperiodic time-dependent forces is discussed.     

Diffusion in a periodic potential with a local perturbation

We consider the diffusion of a particle at X_t in a drift field derived from a smooth potential of the formV+B, whereV is periodic andB is a bump of compact support. With no bump,B=0, the mean squared displacementE(t) E |X _t – X₀|² =D(V)t +C +O(e ^–t ),>0, in any dimension. WhenB0, we establish in one dimension the asymptotic expansion , 0, ast. Our analysis relies on the Nash estimates developed in previous work for the transition density of the process and their consequences for the analytic structure,of the Laplace transform ofE(t).