**On the fluctuating Boltzmann equation**

Thierry Bodineau *CMAP-Ecole Polytechnique*

Abstract: We consider a hard sphere dynamics at equilibrium and show that in the dilute gas limit, the fluctuations are described by the fluctuating Boltzmann equation. The convergence is valid for long times so that the fluctuating Fourier and Stokes equations can be also derived in a hydrodynamic scaling.

This is joint work with I. Gallagher, L. Saint-Raymond,S. Simonella

** Currents in some quantum non-equilibrium steady states**

Eric Carlen *Rutgers University*

Abstract: This talk presents joint work with Joel Lebowitz on the structure of quantum non-equilibrium steady states, a topic which has recently attracted the attention of a number of researchers. We discuss the subject form its beginnings in the 1970's, assuming no special background, and present some new results and discuss some open problems.

**Continuous box ball system**

Pablo Ferrari *Universidad de Buenos Aires*

Abstract: The box-ball system is a one-dimensional transport cellular automaton that exhibits soliton behavior. The slot decomposition of a ball configuration reveals soliton components travelling ballistically. This linearization yields generalized hydrodynamic theorems, as those discussed in this meeting. Furthermore, it provides methods for constructing invariant measures through the slot diagrams of the excursions. In this talk, we will see that the slot diagram of a continuous soliton-weighted random excursion is a nonhomogeneous Poisson process in the positive quadrant of the plane, and that the slot decomposition of the corresponding translation-invariant measure is a Poisson process in the upper semiplane with intensity dx q(y) dy , where q is an integrable explicit function, proportional to the law of the maximum of the excursion.

Based on works with Inés Armendáriz, Pablo Blanc, Davide Gabrielli, Chi Nguyen, Leo Rolla and Minmin Wang.

**On the Smoluchowski equation for aggregation phenomena: stationary non-equilibrium solutions**

Alessia Nota *University of L'Aquila*

Abstract: Smoluchowski’s coagulation equation, an integro-differential equation of kinetic type, is a classical model for mass aggregation phenomena extensively used in the analysis of problems of polymerization, particle aggregation in aerosols, drop formation in rain and several other situations. The rate at which aggregation takes place is determined by a rate kernel that encodes the details of the particle aggregation process under consideration. The solutions of the equation exhibit rich behavior depending on the rate of coagulation considered, such as gelation (formation of particles with infinite mass in finite time) or self-similarity (preservation of the shape over time).

In this talk I will first discuss some fundamental properties of the Smoluchowski equation. I will then present recent results on the problem of existence or non-existence of stationary solutions, both for single and multi-component systems, under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. The most striking feature of these stationary solutions is that, whenever they exist, the solutions to multi-component systems exhibit an unusual “spontaneous localization” phenomenon: they localize along a line in the composition space as the total size of the particles increase. This localization is a universal property of multicomponent systems and it has also been recently proved to occur in time dependent solutions to mass conserving coagulation equations.

(Based on joint works with M.Ferreira, J.Lukkarinen and J. Velázquez)

**Kinetic limit and dynamical cluster expansion**

Sergio Simonella *Sapienza University of Rome*

Abstract: We consider the hard sphere gas at low density with random initial data far away from equilibrium. The goal is to compute dynamical correlations in fine detail. To this effect, we introduce and study a partition function on a space of cluster paths. On a finite time span, the combinatorial method of statistical mechanics is adapted naturally to the nonequilibrium gas. This method can be used to derive large deviation estimates around the Boltzmann equation and to construct an infinite-dimensional dynamics.