Abstracts of the invited speakers

László Erdös: Derivation of nonlinear evolution equations from the dynamics of interacting quantum particles We study the evolution of $N$ quantum particles interacting via weak pair interactions. In a certain scaling limit, the evolution of the marginal density of the system can be described by an effective nonlinear one-body equation. For bosonic particles, this equation is the non-linear Hartree equation with a self-consistent effective potential, for fermions it is the corresponding semiclassical limiting equation, the Vlasov equation. For very singular potentials, we derive the Gross-Pitaevskii equation that is especially important for describing the dynamics of the Bose-Einstein condensate. Most of this work is joint with H.T. Yau and B. Schlein. Fabio Martinelli: Kinetically constrained spin models Kinetically constrained spin models (KSCM) are interacting particle systems with Glauber-like dynamics, no static interactions beyond hard core and reversible w.r.t. simple i.i.d product measure. The essential feature is that the creation/destruction of a particle on $x$ can occur only if the configuration around $x$ satisfies certain constraints (which completely define each specific model). KCSM have been introduced and studied in physical literature as examples of models showing a slow relaxation and in the attempt to model liquid/glass transition. Because of the constraints KCSM may show ergodicity breakdown (dynamical phase transition) without any equilibrium counterpart and it is this phenomenon that has been the main object of investigation in the physical community. In these lectures, starting from the basics of reversible continuous time Markov chains, we will develop rigorous tools to analyze the density and volume dependence of the relaxation time for a fairly general class of KCSM.

Abstracts of the talks

Nicoletta Cancrini: Kinetically constrained spin models: the east model Kinetically constrained spin models have been introduced in physics literature as models which can have some properties of glass transition. They are lattice particle systems with Glauber like dynamics reversible with respect to the product Bernoulli measure. Creation/annihilation of one particle on a site can happen iff the configuration around the site satisfies a particular constraint which completely defines the model. I will discuss one of the models studied with F. Martinelli, C. Roberto, C. Toninelli: the east model (one dimensional). Each site changes its occupation iff its right neighbor is empty. In particular I will show the results we obtained on the dependence of the relaxation time to equilibrium on the system size and on the particle density. Anna Maria Cherubini: Asymptotic Behaviour for a collision model The most popular models of collisions between particles (in granular fluids or elsewhere) are liable to 'inelastic collapse': clusters of particles are subject to an infinite number of collisions in a finite time. This pathology is common when using a restitution coefficient to model the energy dissipation caused by an impact. We propose a simple model for a bouncing ball that takes explicitely into account the deformability of the body and the energy dissipation due to internal friction, and show that it doesn't incur anelastic collapse. Michele Correggi: Rapidly Rotating Bose Gases in Anharmonic Traps Starting from the full many body Hamiltonian we derive the leading order energy and density asymptotics for the ground state of a dilute, rotating Bose gas in an anharmonic trap in the Thomas Fermi (TF) limit when the Gross-Pitaevskii coupling parameter and/or the rotation velocity tend to infinity. Although the many-body wave function is expected to have a complicated phase, the leading order contribution to the energy can be computed by minimizing a simple functional of the density alone. Domenico Finco: Two scale approximation for a class of Many Body Schrödinger equation We consider a non relativistic quantum system consisting of $K$ heavy and $N$ light particles in dimension three, where each heavy particle interacts with the light ones via a two-body potential $\alpha V$. No interaction is assumed among particles of the same kind. Choosing an initial state in a product form we characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. The proof relies on a perturbative analysis and exploits a generalized version of the standard dispersive estimates for the Schrödinger group. Exploiting the asymptotic formula, it can be outlined an application to the problem of the decoherence effect produced on a heavy particle by the interaction with the light ones. Alessandro Michelangeli:TBA Daniela Morale: Stochastic many particle systems in biology and medicin In biology and medicine it is possible to observe a wide spectrum of formation of patterns and clustering, usually due to self-organization phenomena. Some interesting example may be found in the process of tumour growth and in particular in angiogenesis, where at an individual level, cells interact and perform a branching process during the formation of new vessels, under the stimulus of a chemical field produced by a tumour. In this way formation of aggregating networks are shown as a consequence of collective behaviour. Aggregation patterns are usually explained in terms of forces, external and/or internal, acting upon individuals. Over the past couple of decades, a large amount of literature has been devoted to the mathematical modelling of self-organizing populations, based on the concepts of short-range/long-range social interaction" at the individual level. The main interest has been in catching the main features of the interaction at the lower scale of single individuals that are responsible, at a larger scale, for a more complex behaviour that leads to the formation of aggregating patterns. Here we discuss the modelling of the dynamics of some self-organization population via a system of N stochastic differential equations. We consider two working examples: animal grouping and angiogenesis Laura Morato: Stochastic quantization for a system of N interacting Bose particles We apply Stochastic Quantization to a system of $N$ interacting identical Bosons in an external potential $\Phi$, by means of a general stationary-action principle. The collective motion is described in terms of a Markovian diffusion on $\R^{3N}$, with joint density $\hat\rho$ and entangled current velocity field $\hat V$, in principle of non-gradient form, related one to the other by the continuity equation. Dynamical equations relax to those of canonical quantization, in some analogy with Parisi-Wu stochastic quantization. Thanks to the identity of particles, the one-particle marginal densities $\rho$, in the physical space $\R^{3}$, are all the same and it is possible to give, under mild conditions, a natural definition of the single-particle current velocity, which is related to $\rho$ by the continuity equation in $\R^{3}$. The motion of single particles in the physical space comes to be described in terms of a non-Markovian three-dimensional diffusion with common density $\rho $ and, at least at dynamical equilibrium, common current velocity $v$. The three-dimensional drift is perturbed by zero-mean terms depending on the whole configuration of the $N$-boson interacting system. Finally we discuss in detail under which conditions the one-particle dynamical equations, which in their general form allow rotational perturbations, can be particularized, up to a change of variables, to Gross-Pitaevskii equations. Benjamin Schlein: Rate of convergence towards Hartree dynamics The nonlinear Hartree equation can be used to describe the macroscopic time-evolution of bosonic quantum system in the so called mean-field limit. In this talk I am going to present recent bounds on the error associated with the Hartree approximation, obtained in collaboration with I. Rodnianski. Petra Scudo: Entanglement in the quantum Ising model We study the asymptotic scaling of the entanglement of a block of spins for the ground state of the one-dimensional quantum Ising model with transverse magnetic field. When the field is sufficiently strong, the entanglement grows at most logarithmically in the number of spins. The proof utilises a transformation to a model of classical probability called the continuum random-cluster model, a close relative of bond percolation. The generality of this technique allows to apply our analysis to a large class of disordered interactions. Fabio Zucca: Ecological Equilibrium for Restrained Branching Random Walks We study a generalized branching random walk where particles breed at a rate which depends on the number of neighbouring particles. Under general assumptions on the breeding rates we prove the existence of a phase where the population survives without exploding. We construct a non trivial invariant measure for this case (joint work with D.Bertacchi and G.Posta).

Posters

Lucie Fajfrová: Zero range process on a binary tree Equilibrium behaviour of Zero rage processes is studied in the case when the set of sites, on which the particles move, has a structure different from the usually considered set $\mathbb{Z}^d$. We have chosen the tree structure, since the zero range process corresponds to an infinite system of queues and the arrangement of servers in the tree structure is natural in a number of situations. The main result of this work is a characterisation of invariant measures for some important cases of site-disordered zero range processes on a binary tree. Namely, the case when the single particle law is a simple random walk on a binary tree. Another result is connected with the speed of convergence to equilibrium for the latter zero range process.