Boltzmann equation:

mathematics, modeling and simulations

In memory of Carlo Cercignani

Institut Henri Poincaré, Paris

9-11 February 2011

Titles and abstracts

Kazuo Aoki. Some remarks on collisionless gases

We investigate the behavior of a collisionless (or highly rarefied) gas in the following two problems:
(i) We consider a gas around a group of bodies at rest with arbitrary configurations and temperature distributions. It is known that the flows induced by temperature fields (such as the thermal creep flow), which are peculiar to a rarefied gas, do not take place in the collisionless limit if the gas-surface interaction is described by the Maxwell-type condition [Y. Sone, J. Mec. Theor. Appl. 4, 1 (1985)]. We show numerically that such flows do not vanish if the Cercignani-Lampis boundary condition is used. We also give some general discussions on the thermally induced flows in the collisionless limit. This is a joint work with S. Kosuge, S. Takata, and others.
(ii) A collisionless gas is enclosed in a closed domain bounded by a diffusely reflecting wall with a uniform temperature. The approach of the gas to an equilibrium state at rest, caused by the interaction of gas molecules with the wall, is investigated numerically. It is shown that the approach is slow and proportional to an inverse power of time. Some theoretical considerations are also made. This is a joint work with F. Golse and T. Tsuji.

Claude Bardos. About incompressible Euler limit of solutions of Navier-Stokes and Boltzmann equation in the presence of boundary effects

This is a report on a on going joint work with François Golse. It is well know that in any dimension (even in 2d the limit (when the Reynold number goes to $\infty$ ) is in presence of boundary a challenging open problem... Results are simpler when the fluid satisfies a Navier boundary condition and the problem is completely open when the fluid satisfies for finite Reynold number a Dirichlet boundary condition. The only general (always valid) mathematical result being a classical theorem of Tosio Kato. It has been observed by several researchers from Yoshio Sone to Laure Saint Raymond that convenient scalings of the Boltzmann equation leads to the incompressible Euler equation... Hence we try to adapt to this limit, in presence of boundary with accomodation, what is known or conjectured at the level of the Navier Stokes limit.

Emanuele Caglioti. Some considerations on the long time behavior of solutions of Vlasov-like Equations

José Antonio Carrillo. Global Classical Solutions Close to Equilibrium to the Vlasov-Fokker-Planck-Euler System

We are concerned with the global well-posedness of a two-phase flow system arising in the modelling of fluid-particle interactions. This system consists of the Vlasov-Fokker-Planck equation for the dispersed phase (particles) coupled to the incompressible Euler equations for a dense phase (fluid) through the friction forcing. Global existence of classical solutions to the Cauchy problem in the whole space is established when initial data is a small smooth perturbation of a constant equilibrium state, and moreover an algebraic rate of convergence of solutions toward equilibrium is obtained under additional conditions on initial data. The proof is based on the macro-micro decomposition and Kawashima's hyperbolic-parabolic dissipation argument. This result is generalized to the periodic case, when particles are in the torus, improving the rate of convergence to exponential. This is a work in collaboration with R. Duan and A. Moussa.

Colm Connaughton. Instantaneous Gelation in Smoluchowski's Coagulation Equation Revisited

We study the solutions of the Smoluchowski coagulation equation with a regularisation term which removes clusters from the system when their mass exceeds a specified cut-off size, M. We focus primarily on collision kernels which would exhibit an instantaneous gelation transition in the absence of any regularisation. Numerical simulations demonstrate that for such kernels with monodisperse initial data, the regularised gelation time decreases as M increases, consistent with the expectation that the gelation time is zero in the unregularised system. This decrease appears to be a logarithmically slow function of M, indicating that instantaneously gelling kernels may still be justifiable as physical models despite the fact that they are highly singular in the absence of a cut-off. We also study the case when a source of monomers is introduced in the regularised system. In this case a stationary state is reached. We present a complete analytic description of this regularised stationary state for the model kernel, K(m_1,m_2)=max{m_1,m_2}^v, which gels instantaneously when M tends to infinity if v>1. The stationary cluster size distribution decays as a stretched exponential for small cluster sizes and crosses over to a power law decay with exponent v for large cluster sizes. The total particle density in the stationary state slowly vanishes as (Log M^(v-1))^-1/2 when M gets large. The approach to the stationary state is non-trivial : oscillations about the stationary state emerge from the interplay between the monomer injection and the cut-off, M, which decay very slowly when M is large.

Raffaele Esposito. Stability of coexisting phases in a kinetic binary mixture

We study a kinetic binary mixture of colliding particles, with long range repulsive interaction between different species, on a large interval, modeled by a couple of Vlasov-Boltzmann equations. This system undergoes a segregation phase transition below a critical temperature. We show that, below the critical temperature, the equilibrium corresponding to pure phases and coexisting segregated phases is exponential asymptotic stability while the mixed phase is unstable.

Irene M. Gamba. Analysis for the non-linear Boltzmann equation and its numerical approximations

We introduce a spectral conservative method for the approximation of the elastic and inelastic Boltzmann problem. The method is based on a representation formula for the Fourier transform of the collisional operator and a Lagrangian optimization correction used for conservation of mass, momentum and energy. We presents an analysis on the accuracy and consistency of the method, for both elastic and inelastic collisions, alongside discussion of the L^1 - L^2 theory for the scheme in the elastic case. This analysis allow us to present L^2 and Sobolev error estimates for the approximating sequence. The estimates are based on recent progress of convolution and gain of integrability estimates by the authors and corresponding isomoment inequalities for the discretized collision operator.
This is work in collaboration with Ricardo Alonso and Harsha Tharskabhushanam

Pierre Gaspard. Time-reversal symmetry relation for nonequilibrium flows ruled by the fluctuating Boltzmann equation

Time-reversal symmetry relations known as fluctuation theorems have been proved for fluctuating currents flowing across classical or quantum nonequilibrium systems. Here, a fluctuation theorem is considered for nonequilibrium flows of rarefied gases ruled by the fluctuating Boltzmann equation. This latter is the master equation for the stochastic process of random jumps between states defined in terms of the numbers of particles in phase-space cells. This fluctuation theorem can be extended to gaseous mixtures.

François Golse. The Mean-Field Limit for a Regularized Vlasov-Maxwell Dynamics

In 1977, Braun and Hepp established the mean-field limit for a system of particles leading to a variant of the Vlasov-Poisson system where the Coulomb potential is replaced by a smooth kernel. Their result was strengthened by Dobrushin in 1979, who estimated the rate of convergence in that limit. An analogue of their results for the Vlasov-Maxwell system will be presented.

Joel L. Lebowitz. Microscopic Origin of Macroscopic Laws: Transport Processes and Lyapunov Functionals

The equations describing the time evolution of macroscopic quantities, i.e. macrostates M(t), such as the heat , Boltzmann, and Enskog eqs., have their origin in the dynamics of the microscopic constituents of matter. The derivation of such equations from realistic models of microscopic dynamics is the central open mathematical problem of nonequilibrium statistical mechanics. Even in the absence of such derivations the microscopic dynamics suggest and put constraints on the form of the macroscopic equations.
Thus the Boltzmann entropy of a macrostate M, given by S(M), cannot decrease for an isolated system described by a macroscopic equation for M(t). This is an example of the large deviation function for M ,obtained from the stationary measure of the microstate of the system, serving as a Lyapunov function for the macroscopic equations. This is true also for cases where the microscopic dynamics are unrealistic but give rise to realistic macroscopic equations. I will describe some old and some recent work on this subject ( done jointly with many collaborators) to which Carlo made many contributions.

Stéphane Mischler. Statistical solutions to the Boltzmann equation and chaos

Jaroslaw Piasecki. Approach to field induced stationary states

The stochastic dynamics of a particle coupled to an external field and propagating through a scattering thermal bath will be discussed within the Boltzmann kinetic theory. Apart from an exponential approach to the stationary velocity distribution the study of a number of one-dimensional models reveals diffusion in the position space in the reference system moving with average particle velocity. It turns out that the diffusion coefficient is correctly given by the Green-Kubo formula generalized to stationary states for any value of the accelerating field.

Mario Pulvirenti. The Boltzmann-Grad limit for a soft-ball system in a stationary state

Let us consider a stochastic particle system, introduced by C. Cercignani in 1983, yielding the Povzner equation in the appropriate (Boltzmann-Grad) scaling limit. The physical setting is that of a stationary nonequilibrium situation. Under suitable assumptions it is proven the convergence of the stationary solution of the particle system to the unique stationary solution to the Povzner equation.

Sauro Succi. Lattice Boltzmann across scales: from fluid turbulence to quantum mechanics

Since its inception, over two decades ago, the lattice Boltzmann (LB) method has made proof of great versatility in computing a broad variety of complex flows across scales, from large-scale turbulence, all the way down to micro and nanoflows of biological interest, including quantum mechanical fluids. In this lecture, we shall present a brief survey of these developments and point out challenging directions for future research in the field.

Giuseppe Toscani. Kinetic models with randomly perturbed binary collisions

In this talk, we introduce and discuss a class of Kac-like kinetic equations on the real line, with general random collisional rules, which include as particular cases models for wealth redistribution in an agent-based market, or models for granular gases in a background [1, 2]. This last problem is related to the dissipative Boltzmann equation in presence of a heat bath, studied by Cercignani and coworkers [3]. The main results are concerned with the characterization of the stationary solutions. Explicit steady solutions are found in some simple but relevant case.
[1] F. Bassetti, L. Ladelli and G. Toscani. Kinetic models with randomly perturbed binary collisions. J. Stat. Phys. (in press) (2011)
[2] F. Bassetti and G. Toscani. Explicit equilibria in a kinetic model of gam- bling. Phys. Rev. E, 81, 066115 (2010)
[3] C. Cercignani, R. Illner and C. Stoica. On diffusive equilibria in generalized kinetic theory. J. Stat. Phys., 105, 337Ð352.

Bernt Wennberg. On the propagation of chaos in pair interaction driven master equations

Work in collaboration with Eric Carlen and Pierre Degond.
Maybe the first rigorous result on the propagation of chaos in kinetic models is that of Mark Kac, who first of all gave a strict definition of the concept, and then proved that it holds for a "caricature" of the Boltzmann equation.
We make a generalization of his result and apply that to two models of schooling fish, which are also discussed in detail. One particular difference between these models and the one considered by Kac is that the stationary measures are not chaotic, but this does not prevent propagation of chaos for any finite time interval.

Front page of the conference
Program of the conference