Giuseppe Toscani

ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS

It is well known that the analysis of the large-time asymptotics of Fokker-Planck type equations by the entropy method is closely related to proving the validity of convex Sobolev inequalities. Here we highlight this connection from an applied PDE point of view. In our unified presentation of the theory we present new results to the following topics: an elementary derivation of Bakry-Emery type conditions, results concerning perturbations of invariant measures with general admissible entropies, sharpness of convex Sobolev inequalities, applications to non-symmetric linear and certain non-linear Fokker-Planck type equations (Desai-Zwanzig model, drift-diffusion-Poisson model).

ASYMPTOTIC FLOCKING DYNAMICS FOR THE KINETIC CUCKER-SMALE MODEL

In this paper, we analyze the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [IEEE Trans. Automat. Control, 52 (2007), pp. 852–862], which describes the collective behavior of an ensemble of organisms, animals, or devices. This kinetic version introduced in [S.-Y. Ha and E. Tadmor, Kinet. Relat. Models, 1 (2008), pp. 415–435] is here obtained starting from a Boltzmann-type equation. The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [F. Cucker and S. Smale, IEEE Trans. Automat. Control, 52 (2007), pp. 852–862] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast in velocity to the mean velocity of the initial condition, while in space they will converge towards a translational flocking solution.

Particle, kinetic, and hydrodynamic models of swarming

We review the state-of-the-art in the modelling of the aggregation and collective behavior of interacting agents of similar size and body type, typically called swarming. Starting with individual-based models based on “particle”-like assumptions, we connect to hydrodynamic/macroscopic descriptions of collective motion via kinetic theory. We emphasize the role of the kinetic viewpoint in the modelling, in the derivation of continuum models and in the understanding of the complex behavior of the system.

On a kinetic model for a simple market economy

In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasi-stationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker–Planck equation for the distribution of wealth among individuals. For this equation the stationary state can be easily derived and shows a Pareto power law tail. Numerical results confirm the previous analysis.

Probability Metrics and Uniqueness of the Solution to the Boltzmann Equation for a Maxwell Gas

We consider a metric for probability densities with finite variance on ℝd, and compare it with other metrics. We use it for several applications both in probability and in kinetic theory. The main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous Boltzmann equation for a gas of true Maxwell molecules.

Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations

Many transport equations, such as the neutron transport, radiative transfer, and transport equations for waves in random media, have a diffusive scaling that leads to the diffusion equations. In many physical applications, the scaling parameter (mean free path) may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic (diffusive) regimes within one problem, and it is desirable to develop a class of robust numerical schemes that can work uniformly with respect to this relaxation parameter. In an earlier work [Jin, Pareschi, and Toscani, SIAM J. Numer. Anal., 35 (1998), pp. 2405--2439] we handled this numerical problem for discrete-velocity kinetic models by reformulating the system into a form commonly used for a relaxation scheme for conservation laws [Jin and Xin, Comm. Pure Appl. Math., 48 (1995), pp. 235--277]. Such a reformulation allows us to use the splitting technique for relaxation schemes to design a class of implicit, yet explicitly implementable, schemes that work with high resolution uniformly with respect to the relaxation parameter. In this paper we show that such a numerical technique can be applied to a large class of transport equations with continuous velocities, when one uses the even and odd parities of the transport equation.

Metrics for Probability Distributions and the Trend to Equilibrium for Solutions of the Boltzmann Equation

This paper deals with the trend to equilibrium of solutions to the spacehomogeneous Boltzmann equation for Maxwellian molecules with angular cutoff as well as with infinite-range forces. The solutions are considered as densities of probability distributions. The Tanaka functional is a metric for the space of probability distributions, which has previously been used in connection with the Boltzmann equation. Our main result is that, if the initial distribution possesses moments of order 2+ε, then the convergence to equilibrium in his metric is exponential in time. In the proof, we study the relation between several metrics for spaces of probability distributions, and relate this to the Boltzmann equation, by proving that the Fourier-transformed solutions are at least as regular as the Fourier transform of the initial data. This is also used to prove that even if the initial data only possess a second moment, then ∫∣v∣>Rf(v, t) ∣v∣2dv→0 asR→∞, and this convergence is uniform in time.

Diffusive Relaxation Schemes for Multiscale Discrete-Velocity Kinetic Equations

Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the Navier--Stokes-type parabolic equations, such as the heat equation, the porous media equations, the advection-diffusion equation, and the viscous Burgers equation. In such problems the diffusive relaxation parameter may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic (diffusive) regimes, and it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter. Earlier approaches that work from the rarefied regimes to the Euler regimes do not directly apply to these problems since here, in addition to the stiff relaxation term, the convection term is also stiff. Our idea is to reformulate the problem in the form commonly used for the relaxation schemes to conservation laws by properly combining the stiff component of the convection terms into the relaxation term. This, however, introduces new difficulties due to the dependence of the stiff source term on the gradient. We show how to overcome this new difficulty with an adequately designed, economical discretization procedure for the relaxation term. These schemes are shown to have the correct diffusion limit. Several numerical results in one and two dimensions are presented, which show the robustness, as well as the uniform accuracy, of our schemes.

Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences

Economic modelling and financial markets.- Agent-based models of economic interactions.- On kinetic asset exchange models and beyond: microeconomic formulation,trade network, and all that.- Microscopic and kinetic models in financial markets.- A mathematical theory for wealth distribution.- Tolstoys dream and the quest for statistical equilibrium in economics and the social sciences.- Social modelling and opinion formation.- New perspectives in the equilibrium statistical mechanics approach to social and economic sciences.- Kinetic modelling of complex socio-economic systems.- Mathematics and physics applications in sociodynamics simulation: the case of opinion formation and diffusion.- Global dynamics in adaptive models of collective choice with social influence.- Modelling opinion formation by means of kinetic equations.- Human behavior and swarming.- On the modelling of vehicular traffic and crowds by kinetic theory of active particles.- Particle, kinetic, and hydrodynamic models of swarming.- Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints.- Statistical physics and modern human warfare.- Diffusive and nondiffusive population models.

An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations

Abstract We propose here a well-balanced numerical scheme for the one-dimensional Goldstein–Taylor system which is endowed with all the stability properties inherent to the continuous problem and works in both rarefied and diffusive regimes. To cite this article: L. Gosse, G. Toscani, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 337–342.