Boltzmann Equation For Hard Spheres Research Articles

Vanishing Angular Singularity Limit to the Hard-Sphere Boltzmann Equation

In this note we study Boltzmann’s collision kernel for inverse power law interactions U_s(r)=1/r^{s-1} for s>2 in dimension d=3 . We prove the limit of the non-cutoff kernel to the hard-sphere kernel and give precise asymptotic formulas of the singular layer near theta simeq 0 in the limit srightarrow infty . Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions.

Factorization of Non-Symmetric Operators and Exponential H-Theorem

We present an abstract method for deriving decay estimates on the resolvents and semigroups of non-symmetric operators in Banach spaces in terms of estimates in another smaller reference Banach space. This applies to a class of operators writing as a regularizing part, plus a dissipative part. The core of the method is a high-order quantitative factorization argument on the resolvents and semigroups. We then apply this approach to the Fokker-Planck equation, to the kinetic Fokker- Planck equation in the torus, and to the linearized Boltzmann equation in the torus. We finally use this information on the linearized Boltzmann semi- group to study perturbative solutions for the nonlinear Boltzmann equation. We introduce a non-symmetric energy method to prove nonlinear stability in this context in $L^1_v L^\infty _x (1 + |v|^k)$, $k > 2$, with sharp rate of decay in time. As a consequence of these results we obtain the first constructive proof of exponential decay, with sharp rate, towards global equilibrium for the full nonlinear Boltzmann equation for hard spheres, conditionally to some smoothness and (polynomial) moment estimates. This improves the result in [32] where polynomial rates at any order were obtained, and solves the conjecture raised in [91, 29, 86] about the optimal decay rate of the relative entropy in the H-theorem.

Comment on “Velocity slip coefficients based on the hard-sphere Boltzmann equation” [Phys. Fluids 24, 022001 (2012)

We provide a comment on "Velocity slip coefficients based on the hard-sphere Boltzmann equation" by Livio Gibelli published in Phys. Fluids 24, 022001 (2012).

Response to “Comment on ‘Velocity slip coefficients based on the hard-sphere Boltzmann equation’” [Phys Fluids 24, 079101 (2012)

Response to “Comment on ‘Velocity slip coefficients based on the hard-sphere Boltzmann equation’” [Phys Fluids 24, 079101 (2012)

Velocity slip coefficients based on the hard-sphere Boltzmann equation

We present a kinetic theory derivation of higher-order slip boundary conditions. The situation studied is that of a pressure driven isothermal gas flowing through a plane microchannel. The distribution function is expanded in terms of half-range Hermite polynomials and the system of moment equations in the expansion coefficients is analytically solved. The velocity slip coefficients, as well as their Knudsen-layer corrections, are obtained by evaluating the solution in the near continuum limit. The proposed approach is accurate and easy to implement. The results are presented for the hard-sphere Boltzmann equation and Maxwell's diffuse-specular boundary conditions, but can be extended to arbitrary intermolecular interactions and more general scattering kernels.

Blow Up Analysis for Anomalous Granular Gases

We investigate in this article the long-time behaviour of the solutions to the energy-dependant, spatially-homogeneous, inelastic Boltzmann equation for hard spheres. This model describes a diluted gas composed of hard spheres under statistical description, that dissipates energy during collisions. We assume that the gas is "anomalous", in the sense that energy dissipation increases when temperature decreases. This allows the gas to cool down in finite time. We study existence and uniqueness of blow up profiles for this model, together with the trend to equilibrium and the cooling law associated, generalizing the classical Haff's Law for granular gases. To this end, we investigate the asymptotic behaviour of the inelastic Boltzmann equation with and without drift term by introducing new strongly "nonlinear" self-similar variables.

Note on the relation between thermophoresis and slow uniform flow problems for a rarefied gas

A relation between the problem of thermophoresis of a sphere and that of a uniform flow past a sphere is discussed on the basis of the linearized Boltzmann equation. First pointed out is the disagreement between the relation predicted by the representation theorem recently developed by the author and that of the existing theory by Sharipov. The two contradicting predictions are assessed by the asymptotic theory for small Knudsen numbers, which results in showing the failure of the latter. The reason of this failure is also explained. Finally, new data of a slip coefficient, which is predominantly responsible for the thermal polarization in a slightly rarefied gas, are obtained for the hard-sphere Boltzmann equation by the use of the correct relation.

Modelling and numerical methods for the dynamics of impurities in a gas

AbstractThe dynamics of a mixture of impurities in a gas can be represented by a system of linear Boltzmann equations for hard spheres. We assume that the background is in thermodynamic equilibrium and that the polluting particles are sufficiently few (in comparison with the background molecules) to admit that there are no collisions among couples of them. In order to derive non‐trivial hydrodynamic models, we close the Euler system around local Maxwellian's which are not equilibrium states.The kinetic model is solved by using a Monte Carlo method, the hydrodynamic one by implicit–explicit Runge–Kutta schemes with weighted essentially non‐oscillatory reconstruction (J. Sci. Comput. 2005; 25(1–2):129–155). Several numerical tests are then computed in order to compare the results obtained with the kinetic and the hydrodynamic models. Copyright © 2007 John Wiley & Sons, Ltd.

Quantitative Linearized Study of the Boltzmann Collision Operator and Applications

We present recent results [4, 28, 29] about the quantitative study of the linearized Boltzmann collision operator, and its application to the study of the trend to equilibrium for the spatially homogeneous Boltzmann equation for hard spheres.

Cooling Process for Inelastic Boltzmann Equations for Hard Spheres, Part II: Self-Similar Solutions and Tail Behavior

We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of so-called constant normal restitution coefficients. We prove the existence of self-similar solutions, and we give pointwise estimates on their tail. We also give general estimates on the tail and the regularity of generic solutions. In particular we prove Haff's law on the rate of decay of temperature, as well as the algebraic decay of singularities. The proofs are based on the regularity study of a rescaled problem, with the help of the regularity properties of the gain part of the Boltzmann collision integral, well-known in the elastic case, and which are extended here in the context of granular gases.

Cooling Process for Inelastic Boltzmann Equations for Hard Spheres, Part I: The Cauchy Problem

We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity as well as the sticky particles model. We prove (local in time) non-concentration estimates in Orlicz spaces, from which we deduce weak stability and existence theorem. Strong stability together with uniqueness and instantaneous appearance of exponential moments are proved under additional smoothness assumption on the initial datum, for a restricted class of collision rates. Concerning the long-time behaviour, we give conditions for the cooling process to occur or not in finite time.

Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus

For a general class of linear collisional kinetic models in the torus, including in particular the linearized Boltzmann equation for hard spheres, the linearized Landau equation with hard and moderately soft potentials and the semi-classical linearized fermionic and bosonic relaxation models, we prove explicit coercivity estimates on the associated integro-differential operator for some modified Sobolev norms. We deduce the existence of classical solutions near equilibrium for the full nonlinear models associated with explicit regularity bounds, and we obtain explicit estimates on the rate of exponential convergence towards equilibrium in this perturbative setting. The proof is based on a linear energy method which combines the coercivity property of the collision operator in the velocity space with transport effects, in order to deduce coercivity estimates in the whole phase space.

Dissipative hydrodynamic models for the diffusion of impurities in a gas

Recently, 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.

The Newtonian limit of the relativistic Boltzmann equation

The relativistic Boltzmann equation for a constant differential cross section and with periodic boundary conditions is considered. The speed of light appears as a parameter c>c0 for a properly large and positive c0. A local existence and uniqueness theorem is proved in an interval of time independent of c>c0 and conditions are given such that in the limit c→+∞ the solutions converge, in a suitable norm, to the solutions of the nonrelativistic Boltzmann equation for hard spheres.

Fast deterministic method of solving the Boltzmann equation for hard spheres

A special form of the Boltzmann collision operator for the hard spheres model is introduced. The possibilities of fast numerical computation of the collision operator based on this form and the Fast Fourier Transform are discussed. A new difference scheme for the Boltzmann equation for the hard spheres model is developed. The results of some numerical tests and accuracy comparisons with the Direct Simulation Monte Carlo (DSMC) method are presented.

A note on the Boltzmann equation for hard spheres

A note on the Boltzmann equation for hard spheres

Large energy behavior of the velocity distribution for the hard-sphere gas

The initial-value problem for the Boltzmann-Lorentz equation for hard spheres at zero temperature is shown to be ill defined, the general solution depending on an arbitrary function. The uniqueness of the solution can be obtained by imposing the conservation of the number of particles (Carleman's type of condition does not suffice). The linearized Boltzmann equation for hard spheres is then analyzed, as it occurs in Enskog's method for calculating transport coefficients. It is demonstrated that in the case of viscosity and diffusion it is necessary to add supplementary conditions to obtain the uniqueness of the solution. The nonuniform character of Enskog's expansion and violation of positivity in the large velocity region are exhibited.

Enskog dense gas theory for soft potentials

Making use of equilibrium correlation functions and cluster functions, a soft potential analogue of the Enskog dense gas Boltzmann equation for hard spheres is derived by neglecting dynamical correlations due to multiple collisions. The equation is shown to reduce to the proper rigid sphere limit, and it is solved by the method of Chapman and Enskog. Expressions for the kinetic and collisional transfer contributions to shear viscosity, bulk viscosity, and thermal conductivity are developed in terms of multidimensional integrals.

Boltzmann Equation for Hard Spheres in Differential Form

It is shown that it is possible to recast the Boltzmann integro-differential equation for hard spheres into a pair of linear partial differential equations of first and eighth orders respectively. The eighth order equation is shown to be the nine-dimensional Laplacian four times repeated. The first order equation is the familiar df/dt with a different forcing function on the right hand side.