The dissipative linear boltzmann equation

AbstractWe introduce and discuss a linear Boltzmann equation describing dissipative interactions of a gas of test particles with a fixed background. For a pseudo-Maxwellian collision kernel, it is shown that, if the initial distribution has finite temperature, the solution converges exponentially for large time to a Maxwellian profile drifting at the same velocity as field particles and with a universal nonzero temperature which is lower than the given background temperature

The dissipative linear Boltzmann equation for hard spheres

We prove the existence and uniqueness of an equilibrium state with unit mass to the dissipative linear Boltzmann equation with hard--spheres collision kernel describing inelastic interactions of a gas particles with a fixed background. The equilibrium state is a universal Maxwellian distribution function with the same velocity as field particles and with a non--zero temperature lower than the background one, which depends on the details of the binary collision. Thanks to the H--theorem we then prove strong convergence of the solution to the Boltzmann equation towards the equilibrium.Comment: 17 pages, submitted to Journal of Statistical Physic

Dissipative hydrodynamic models for the diffusion of impurities in a gas

AbstractRecently, linear dissipative models of the Boltzmann equation have been introduced in [B. Loods, G. Toscani, The dissipative linear Boltzmann equation for hard spheres, J. Statist. Phys. 117 (2004) 635–664] and [G. Spiga, G. Toscani, The dissipative linear Boltzmann equation, Appl. Math. Lett. 17 (2004) 255–301]. In this note, we consider the problem of constructing suitable hydrodynamic approximations for such models

Dissipative hydrodynamic models for the diffusion of impurities in a gas.

Recently linear dissipative models of the Boltzmann equation have been introduced. In this work, we consider the problem of constructiing suitable hydrodynamic approximations for such models where the mean velocity and the temperature of inelastic particles appear as independent variables

Long time behavior of non-autonomous Fokker-Planck equations and the cooling of granular gases

We analyze the asymptotic behavior of linear Fokker-Planck equations with time-dependent coefficients. Relaxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly computable conditions. We apply this result to the study of Brownian motion in granular gases as introduced J. J. Brey, J. Dufty and A. Santos (1999), by showing that the Homogenous Cooling State attracts any solution at an algebraic rate.Comment: 15 pages, submitted to Ukrainian Math.

Integral representation of the linear Boltzmann operator for granular gas dynamics with applications

We investigate the properties of the collision operator associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain collision operator is an integral operator whose kernel is made explicit. One deduces from this result a complete picture of the spectrum of the collision operator in an Hilbert space setting, generalizing results from T. Carleman to granular gases. In the same way, we obtain from this integral representation of the gain operator that the semigroup in L^1(\R \times \R,\d \x \otimes \d\v) associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from the first author.Comment: 19 pages, to appear in Journal of Statistical Physic

Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models

We consider the linear dissipative Boltzmann equation describing inelastic interactions of particles with a fixed background. For the simplified model of Maxwell molecules first, we give a complete spectral analysis, and deduce from it the optimal rate of exponential convergence to equilibrium. Moreover we show the convergence to the heat equation in the diffusive limit and compute explicitely the diffusivity. Then for the physical model of hard spheres we use a suitable entropy functional for which we prove explicit inequality between the relative entropy and the production of entropy to get exponential convergence to equilibrium with explicit rate. The proof is based on inequalities between the entropy production functional for hard spheres and Maxwell molecules. Mathematical proof of the convergence to some heat equation in the diffusive limit is also given. From the last two points we deduce the first explicit estimates on the diffusive coefficient in the Fick's law for (inelastic hard-spheres) dissipative gases.Comment: 25 page

Mesoscopic modelling of financial markets

We derive a mesoscopic description of the behavior of a simple financial market where the agents can create their own portfolio between two investment alternatives: a stock and a bond. The model is derived starting from the Levy-Levy-Solomon microscopic model (Econ. Lett., 45, (1994), 103--111) using the methods of kinetic theory and consists of a linear Boltzmann equation for the wealth distribution of the agents coupled with an equation for the price of the stock. From this model, under a suitable scaling, we derive a Fokker-Planck equation and show that the equation admits a self-similar lognormal behavior. Several numerical examples are also reported to validate our analysis

EQUILIBRIUM SOLUTION TO THE INELASTIC BOLTZMANN EQUATION DRIVEN BY A PARTICLES THERMAL BATH

International audienceWe show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a host-medium with a fixed distribution. This is achieved by controlling the Lp-norms, the moments and the regularity of the solutions for the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states

Mesoscopic modelling of financial markets

We derive a mesoscopic description of the behavior of a simple financial market where the agents can create their own portfolio between two investments alternatives: a stock and a bond. The model is derived starting from the Levy-Levy-Solomon microscopic model using the methods of kinetic theory and consists of a linear Boltzmann equation for the wealth distribution of the agents coupled with an equation for the price of the stock. From this model under a suitable scaling we derive a Fokker-Planck equation and show that the equation admits a self-similar lognormal behavior. Several numerical examples are also reported to validate our analysis.wealth distribution, power-law tails, stock market, self-similarity, kinetic equations.