Schedule

Giovanni Gallavotti

Ergodic hypothesis and Boltzmann's equation

1866-1872 the 7 years from the Ergodic hypothesis through Ensembles and Boltzmann's equation (BE) reviewed in selected papers by Boltzmann (preceding BE) and by Maxwell (preceding and following it).

Antonio Prados

Beating the natural timescale for relaxation with optimal control

Many physical systems irreversibly relax towards an equilibrium state or a non-equilibrium steady state--as first predicted by Boltzmann 150 years ago for a dilute granular gas. This irreversible evolution takes place over a certain timescale, which is characteristic of each physical system and measured by the so-called relaxation time. Is it possible to beat this natural timescale for relaxation by smartly engineering time-dependent protocols for the physical quantities controlling the time evolution of the system? We address this question in this talk, bringing to bear tools and results of optimal control theory to analyse two different physical situations: (i) a uniformly heated granular gas, described at the kinetic level by the inelastic Boltzmann equation, and (ii) a Brownian particle confined in a d-dimensional harmonic well, described at the mesoscopic level by the Fokker-Planck equation.

Maria Groppi

A new mixed Boltzmann-BGK model for gas mixtures

Relaxation-time approximations of BGK-type for gas mixtures description constitute the most used simplified kinetic models of the Boltzmann equations, since they retain the most significant mathematical and physical features of the Boltzmann description, but are computationally more manageable. The BGK models for mixtures available in the literature may be divided into two classes, the former assuming the kinetic equation for each species governed by a unique relaxation operator [1,2], and the latter showing a sum of binary relaxation operators, preserving thus the structure of the original Boltzmann system [3].

In order to preserve as much as possible the accuracy of the Boltzmann description, but with a kinetic system manageable from the computational point of view, we propose a mixed Boltzmann-BGK model for an inert gas mixture of monoatomic gases. In this setting, collisions occurring within the same species (intra-species) are modelled by Boltzmann operators, while interactions between different components (inter-species) are described by the BGK operators given in [3], that represent the relaxation model for mixtures with the closest structure to the Boltzmann one. We prove consistency of the model, in particular a Boltzmann H-Theorem holds true, prescribing convergence of solutions to equilibrium Maxwellians with all species sharing a common mean velocity and a common temperature. The structure of the model allows us to formally derive hydrodynamic equations in different collisional regimes, namely with all collisions dominant, or with only collisions within the same species playing the dominant role, the latter leading to multitemperature and multivelocity Euler and Navier Stokes equations. Some results relevant to the particular case of a two-species mixture [4] are presented. This is a joint work with M. Bisi, E. Lucchin, G. Martalò (University of Parma- Italy).

Bruce Boghosian

Exotic Distributions as Weak Solutions to Kinetic Equations

It is well known that certain kinetic equations, including Fokker-Planck equations derived in the grazing-collision limit of Boltzmann equations, can have generalized distributions as weak solutions. These may appear either as initial conditions, time-asymptotic states, or singularities that manifest themselves in finite time. The most well-known of these generalized distributions is the Dirac delta, but in recent years it has become clear that there are others of more exotic form, and they have physical significance ranging from conservation of heat energy in infinite domains, to wealth distribution in the presence of oligarchy, to the energy distribution of runaway electrons. The mathematical tools used to understand such exotic distributions range from Sobolev-Schwarz distribution theory to nonstandard analysis, to measure theory. This talk will begin with an introduction to such exotic generalized distributions at a level accessible to an advanced undergraduate, and gradually ramp up to some of the unsolved problems of the field. (This is joint work with Adrian Devitt-Lee, David Cohen, and Sauro Succi.)

Giuseppe Toscani

H-theorem and entropy dissipation estimates

in kinetic equations

In a founding paper Cercignani [1] stated a conjecture about Boltzmann's entropy production for the full Boltzmann equation which would imply an exponentially fast trend to equilibrium, a conjecture almost exhaustively proven seventeen years later by the author with Villani [2]. A prominent role in the proof is played by the Fokker-Planck kinetic equation, a model which has recently applied to study relaxation to equilibrium in socio-

economic phenomena [3]. In this talk, after a short review of some classical result about entropy production estimates, we show how Boltzmann's strategy can be fruitfully applied

to study the trend to equilibrium in the new field of econophysics [4].

Mario Pulvirenti

Some considerations on the validity problem for the Boltzmann equation.

In this talk I try to discuss what is known on the rigorous derivation of the Boltzmann equation starting from mechanical particle systems (including quantum systems) with emphasis on recent results and perspectives.