Stefan Adams (Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Germany)
"On probabilistic approaches to the Gross-Pitaevskii theory for dilute systems of Bosons."
Abstract. In this talk we first review some basic facts about the Gross-Pitaveskii theory for dilute systems of interacting Bosons in inhomogenous magnetic traps. Then we present a probabilistic approach and interpretation for the Gross-Pitaveskii variational formula. Starting point are N interacting Bosons moving in an external field which keeps them in a bounded region of space. Our probabilistic ansatz starts with a transformed path measure for N Brownian particles given by a Feynman-Kac formula. The aim is then to study two limits, one of vanishing temperature (infinite time for the the probabilistic model) and one for diverging number of Bosons (Brownian particles). In particular we introduce two probabilistic models for N interacting Brownian motions moving in a trap under mutually repellent forces. The first pair interaction exhibits a particle repellency, while the second one imposes a path repellency. We analyse both models in the limit of diverging time with fixed number N of Brownian motions. In particular, we prove large deviations principles for the normalised occupation measures. The minimisers of the rate functions are related to a certain associated operator, the Hamilton operator for a system of N interacting trapped particles. More precisely, in the particle-repellency model, the minimiser is its ground state, and in the path-repellency model, the minimisers are its ground product-states. In the case of path-repellency, we also discuss the case of a Dirac-type interaction, which is rigorously defined in terms of Brownian intersection local times. We prove a large-deviation result for a discrete variant of the model. This study is a contribution to the search for a mathematical formulation of the quantum system of N trapped interacting bosons as a model for Bose-Einstein condensation, motivated by the success of the famous 1995 experiments. Recently, Lieb et al. described the large-N behaviour of the ground state in terms of the well-known Gross-Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-N behaviour of the ground product-states is also described by the Gross-Pitaevskii formula, however with the scattering length of the pair potential replaced by its integral.
Yurii Suhov (Cambridge University)
"Anderson localisation for multi-particle systems."
Abstract Anderson localisation is an important phenomenon describing a transition between insulation and conductivity. The problem is to analyse the spectrum of a Schroedinger operator with a random potential in the Euclidean space or on a lattice. We say that the system exhibits (exponential) localisation if with probability one the spectrum is pure point and the corresponding eigen-functions decay exponentially fast. So far in the literature one considered a single-particle model where the potential at different sites is IID or has a controlled decay of correlations. The present talk aims at $N$-particle systems (bosons or fermions) where the potential sums over different sites, and the traditional approach needs serious modifications. The main result is that if the `randomness' is strong enough, the $N$-particle system exhibits localisation. The proof exploits the muli-scale analysis scheme going back to Froehlich, Martinelli, Scoppola and Spencer and refined by von Drefus and Klein. No preliminary knowledge of the related material will be assumed from the audience, apart from basic facts. This is a joint work with V. Chulaevsky (University of Reims, France).
Alan Sokal (New York University)
Chromatic polynomials, Tutte polynomials, Potts models, and all that.
Abstract Chromatic polynomials, Tutte polynomials, Potts models, and all that The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information about the graph indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte polynomial --- and therefore also its one-variable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial --- but is considerably more flexible. I begin by giving an introduction to all these problems, stressing the advantages of working with the multivariate version. I then discuss some questions concerning the complex zeros of the multivariate Tutte polynomial, along with their physical interpretations in statistical mechanics (in connection with the Yang--Lee approach to phase transitions) and electrical circuit theory. Along the way I mention numerous open problems. This survey is intended to be understandable to mathematicians with no prior knowledge of physics.
Colm Connaughton (ENS and CNLS LANL)
Constant Flux Relations in Non-Equilibrium Statistical Mechanics.
Abstract The presence of constant flux of a conserved quantity in the steady state of a turbulent system often allows an exact determination of the scaling exponent of a particular correlation function, namely the one measuring average flux. Such universal scaling laws generalize the well known 4/5 law of Navier-Stokes turbulence and are not the result of any mean field approximations. They must be respected by any effective description of non-equilibrium statistical systems. In this talk we shall introduce the general philosophy which leads to such constant flux relations and discuss the conditions under which the theory is applicable. We shall then study in more detail the application of these ideas to turbulence-like cascades in cluster-cluster aggregation and wave turbulence.