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The study of the fluctuations in the steady state of a heated granular system is
reviewed. A Boltzmann-Langevin description can be built requiring consistency with the
equations for the one- and two-particle correlation functions. From the Boltzmann-Langevin
equation, Langevin equations for the total energy and the transverse velocity field are
derived. The existence of a fluctuation-dissipation relation for the transverse velocity
field is also...
A computer aided method using symbolic computations that enables the calculation of the
source terms (Boltzmann) in Grad’s method of moments is presented. The method is extremely
powerful, easy to program and allows the derivation of balance equations to very high
moments (limited only by computer resources). For sake of demonstration the method is
applied to a simple case: the one-dimensional stationary granular gas under gravity. The
method should...
This paper deals with explicit spectral gap estimates for the linearized Boltzmann operator with hard potentials (and hard spheres). We prove that it can be reduced to the Maxwellian case, for which explicit estimates are already known. Such a method is constructive, does not rely on Weyl's Theorem and thus does not require Grad's splitting. The more physical idea of the proof is to use geometrical properties of the whole collision operator. In a second part, we use the fact that the Landau operator...
We investigate a model describing the dynamics of a gas of self-gravitating Brownian particles. This model can also have applications for the chemotaxis of bacterial populations. We focus here on the collapse phase obtained at sufficiently low temperature/energy and on the post-collapse regime following the singular time where the central density diverges. Several analytical results are illustrated by numerical simulations.
An overview of recent results pertaining to the hydrodynamic description (both Newtonian
and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic
Maxwell models is presented. The use of this mathematical model allows us to get exact
results for different problems. First, the Navier–Stokes constitutive equations with
explicit expressions for the corresponding transport coefficients are derived by applying
the Chapman–Enskog...
We prove global stability results of DiPerna-Lionsrenormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann,...
The model for an ultra-relativistic gas is here considered in the framework of Extended Thermodynamics. The closure, satisfying exactly the principles of relativity and of entropy, is obtained by following the approach «at a macroscopic level». Our results are compared with the ones of the kinetic approach.
Se expone la geometría diferencial del espacio de las velocidades relativistas y se obtiene la función de distribución de velocidades de un gas de partículas relativistas, que modifica la función de Maxwell de Mecánica Estadística Clásica. Se introducen los espacios de Hilbert-Lobatschewsky.
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