We investigate the dynamics of a boson gas with three-body interactions in dimensions d = 1, 2. We prove that in the limit where the particle number N tends to infinity, the BBGKY hierarchy of k-particle marginals converges to a limiting (Gross-Pitaevskii (GP)) hierarchy for which we prove existence and uniqueness of solutions. The solutions of the GP hierarchy are shown to be determined by sol...

A spin glass describes a system of spins on a lattice (or a crystal) where the interactions are frustrated as well as disordered. Studying such systems presents many theoretical difficulties, but over the last twenty years considerable advances have been made. In this paper I will discuss some of the experimental results that motivated theorists and some of the concepts that were developed to d...

The sample-to-sample fluctuations Delta FN of the free-energy in the Sherrington-Kirkpatrick model are shown rigorously to be related to bond chaos. Via this connection, the fluctuations become analytically accessible by replica methods. The replica calculation for bond chaos shows that the exponent mu governing the growth of the fluctuations with system size N, Delta FN approximately Nmu, is b...

In [8] the large deviations limit limN→∞(Na) −1 log EZa N for the moments of the partition function ZN in the Sherrington-Kirkpatrick model [5] was computed for all real a ≥ 0. For a ≥ 1 this result extends the classical physicist’s replica method that corresponds to integer a. We give a new proof for a ≥ 1 in the case of the pure p-spin SK model that provides a strong exponential control of th...

We investigate the large deviation behavior of the overlap probability density in the Sherrington–Kirkpatrick model using the coupled replica scheme, and compare with the results of a large scale numerical simulation. In the spin glass phase we show that generically, for any model with continuous RSB, 1/N log PN (q) ≈ −A ((|q| − qEA), and compute the first correction to the expansion of A in po...

V. E. Young, A. J. Patterson, U. Sadat, D. J. Bowden, M. J. Graves, A. N. Priest, T. Y. Tang, J. N. Skepper, P. J. Kirkpatrick, and J. H. Gillard University Department of Radiology, Addenbrookes Hospital, Cambridge, Cambridgeshire, United Kingdom, Department of Medical Physics, Addenbrookes Hospital, Cambridge, Cambridgeshire, United Kingdom, Multi-imaging Centre, University of Cambridge, Cambr...

Brian Kirkpatrick, Wayne S. Fenton, William T. Carpenter, Jr., and Stephen R. Marder Department of Psychiatry and Health Behavior, Medical College of Georgia; Division of Adult Translational Research and Development, National Institute of Mental Health, National Institutes of Health, Department of Health and Human Services; MarylandPsychiatricResearchCenter,DepartmentofPsychiatry, University of...

We study the Kac version of the Sherrington-Kirkpatrick (SK) model of a spin glass, i.e. a spin glass with long but nite range interaction on ZZ d and gaussian mean zero couplings. We prove that for all < 1, the free energy of this model converges to that of the SK model as the range of the interaction tends to innnity. Moreover, we prove that for all temperatures for which the innnite volume G...

This article describes the design, testing, and use of Kirkpatrick-Baez (KB) x-ray microscopes on OMEGA. These microscopes consist of grazingincidence mirror pairs arranged to reflect and focus x rays, thereby providing images of the x-ray-emitting region. KB microscopes are one of several practical devices that can be used to focus and collect x rays. Simpler x-ray-imaging devices are the pinh...

The mean-field theory of a spin glass with a specific form of nearestand next-nearest-neighbor interactions is investigated. Depending on the sign of the interaction matrix chosen we find either the continuous replica symmetry breaking seen in the Sherrington-Kirkpartick model or a one-step solution similar to that found in structural glasses. Our results are confirmed by numerical simulations ...