An Example of Nonuniqueness for Solutions to the Homogeneous Boltzmann Equation

The paper deals with the spatially homogeneous Boltzmann equation for hard potentials. An example is given which shows that, even though it is known that there is only one solution that conserves energy, there may be other solutions for which the energy is increasing; uniqueness holds if and only if energy is assumed to be conserved.     

Global Solution to the Relativistic Enskog Equation with Near-Vacuum Data

We give two hypotheses of the relativistic collision kernal and show the existence and uniqueness of the global mild solution to the relativistic Enskog equation with the initial data near the vacuum for a hard sphere gas. 2000 Mathematics Subject Classification. 76P05; 35Q75; 82-02.     

The Navier-Stokes limit of the stationary boltzmann equation for hard potentials

In this paper we extend recent results on the hydrodynamic Navier-Stokes limit of the stationary Boltzmann equation for the flow of a gas of hard spheres in a channel in the presence of an external force to the case of a hard intermolecular potential with Grad angular cutoff. We prove the convergence of the solution, for small Knudsen numbers, to the Maxwellian with parameters solving the corresponding Navier-Stokes equation. In the present case we only get polynomial decay of the solution for large velocities, instead of the exponential decay which holds for hard spheres.     

The Modified Group Expansions for Construction of Solutions to the BBGKY Hierarchy

A solution to the BBGKY hierarchy for nonequilibrium distribution functions is obtained within modified boundary conditions. The boundary conditions take into account explicitly both the nonequilibrium one-particle distribution function as well as local conservation laws. As a result, modified group expansions are proposed. On the basis of these expansions, a generalized kinetic equation for hard spheres and a generalized Bogolubov–Lenard–Balescu kinetic equation for a dense electron gas are derived within the polarization approximation.     

Critical behaviour of the lattice gas model in the Percus-Yevick approximation

The solution of the Percus-Yevick (P.Y.) equation for the lattice gas, obtained by Levesque and Verlet, is studied in the critical region. The behaviour of the equation ofstate is similar to the one found for a fluid of sticky hard spheres in the P.Y. approximation: the critical indices have the classical value but the scaling function is non-universal, is strongly asymmetric in density with respect to the critical value and there is a spinodal curve only for the liquid phase. This suggests that these features are generally valid for the P.Y. approximation and are not specific for sticky hard spheres.     

On the Boltzmann Equation for Diffusively Excited Granular Media

We study the Boltzmann equation for a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative L²(^N) function, with bounded mass and kinetic energy (second moment), we prove the existence of a solution to this model, which instantaneously becomes smooth and rapidly decaying. Under a weak additional assumption of bounded third moment, the solution is shown to be unique. We also establish the existence (but not uniqueness) of a stationary solution. In addition we show that the high-velocity tails of both the stationary and time-dependent particle distribution functions are overpopulated with respect to the Maxwellian distribution, as conjectured by previous authors, and we prove pointwise lower estimates for the solutions.     

Analytic solution of the Percus-Yevick equation for sticky hard sphere potential

It is shown that within the Percus-Yevick approximation the radial distribution function for sticky (i.e. with a surface adhesion) hard spheres satisfies a linear differential equation with retarded right-hand side. Using the theory of distributions and the Green's function technique the analytic solution of this equation is found and explicit formulas are given enabling one to evaluate the radial distribution function both for sticky and non-attractive hard spheres for any distance and any density.     

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. Mathematics Subject Classification (2000): 76P05 Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05].     

Self-Similar Solutions of the Boltzmann Equation for Non-Maxwell Molecules

We show that the method previously used by the authors to obtain self-similar, eternal solutions of the space-homogeneous Boltzmann equation for Maxwell molecules yields different results when extended to other power-law potentials (including hard spheres). In particular, self-similar solutions cease to exist for a positive time for hard potentials. In the case of soft potentials, the solutions exist for all potive times, but are not eternal.     

Self-Similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions

We consider some questions related to the self-similar asymptotics in the kinetic theory of both elastic and inelastic particles. In the second case we have in mind granular materials, when the model of hard spheres with inelastic collisions is replaced by a Maxwell model, characterized by a collision frequency independent of the relative speed of the colliding particles. We first discuss how to define the n-dimensional (n = 1,2,...) inelastic Maxwell model and its connection with the more basic Boltzmann equation for inelastic hard spheres. Then we consider both elastic and inelastic Maxwell models from a unified viewpoint. We prove the existence of (positive in the inelastic case) self-similar solutions with finite energy and investigate their role in large time asymptotics. It is proved that a recent conjecture by Ernst and Brito devoted to high energy tails for inelastic Maxwell particles is true for a certain class of initial data which includes Maxwellians. We also prove that the self-similar asymptotics for high energies is typical for some classes of solutions of the classical (elastic) Boltzmann equation for Maxwell molecules. New classes of (not necessarily positive) finite-energy eternal solutions of this equation are also studied.     

Universal cubic equation of state and contact values of the radial distribution functions for multi-component additive hard-sphere mixtures

A universal cubic equation of state (UC EOS) is proposed based on a modification of the virial Percus-Yevick (PY) integral equation EOS for hard-sphere fluid. The UC EOS is extended to multi-component hard-sphere mixtures based on a modification of Lebowitz solution of PY equation for hard-sphere mixtures. And expressions of the radial distribution functions at contact (RDFC) are improved with the form as simple as the original one. The numerical results for the compressibility factor and RDFC are in good agreement with the simulation results. The average errors of the compressibility factor relative to MC data are 3.40%, 1.84% and 0.92% for CP3P, BMCSL equations and UC EOS, respectively. The UC EOS is a unique cubic one with satisfactory precision among many EOSs in the literature both for pure and mixture fluids of hard spheres.     

Percus-Yevick theory for the system of hard spheres with a square-well attraction

Analytic solution of the Percus-Yevick equation for the system of hard spheres with a square-well attraction is proposed provided the range of attraction,γ, is much smaller than the hard sphere diameter. It is shown that forγ close to zero the system exhibits the first-order phase transition similar to that found for sticky hard spheres; for attraction ranges greater than a certainγ _m the triple point and the line of solidification appear as well.     

Equilibrium properties of hard non-sphere fluids

Derivation of the thermodynamic properties of fluids of hard non-spherical molecules of arbitrary symmetry is based on the decoupling approximation. Theoretical expressions are given and calculations made for the equation of state and virial coefficients for hard ellipsoids. These results are compared with Monte Carlo values and show fair agreement in all cases. The theoretical predictions for the equation of state for binary mixtures are compared with the Monte Carlo results for hard spheres and hard prolate spherocylinders. Theoretical expressions for the first order quantum correction to the free energy, pressure and virial coefficients are also given. The quantum effects increase with increase of density and with increase of anisotropy parameter.     

Steady state self-diffusion at low density

We prove that the motion of a test particle in a hard sphere fluid in thermal equilibrium converges, in the Boltzmann-Grad limit, to the stochastic process governed by the linear Boltzmann equation. The convergence is in the sense of weak convergence of the path measures. We use this result to study the steady state of a binary mixture of hard spheres of different colors (but equal masses and diameters) induced by color-changing boundary conditions. In the Boltzmann-Grad limit the steady state is determined by the stationary solution of the linear Boltzmann equation under appropriate boundary conditions.Supported in part by NSF Grant No. PHY 78-15920-02.Supported by a Heisenberg Fellowship of the Deutsche Forschungsgemeinschaft.     

A solution of radiative transfer in isotropic plane-parallel atmosphere by integrating Milne equation

In this paper we solve the inversion problem of the radiative transfer process in the isotropic plane-parallel atmosphere by iterative integrations of the Milne integral equation. As a result, we obtain the scattering function in the form of a cubic polynomial in optical thickness. The author has already solved the same problem by iterative integrations of Chandrasekhar's integral equation. In the Milne integral equation, both the cosines of the viewing angles and the optical thickness are integral variables, while in Chandrasekhar's integral equation the cosines of the viewing angles are variables but the optical thickness is not. We derive several series of exponential-like functions as intermediate derivations. Their convergences are evaluated by the author's previous work in the solution of Chandrasekhar's integral equation. The truncated scattering function up to the third order in optical thickness thus obtained is identical to that obtained from Chandrasekhar's integral equation, though their apparent forms are different. Chandrasekhar pointed out that the solution of Chandrasekhar's integral equation does not have a uniqueness of solution. The Milne equation, in contrast, has been proven to have a unique solution. We discuss the uniqueness of the solution by these two methods.     

Equilibrium time correlation functions in the low-density limit

We consider a system of hard spheres in thermal equilibrium. Using Lanford's result about the convergence of the solutions of the BBGKY hierarchy to the solutions of the Boltzmann hierarchy, we show that in the low-density limit (Boltzmann-Grad limit): (i) the total time correlation function is governed by the linearized Boltzmann equation (proved to be valid for short times), (ii) the self time correlation function, equivalently the distribution of a tagged particle in an equilibrium fluid, is governed by the Rayleigh-Boltzmann equation (proved to be valid for all times). In the latter case the fluid (not including the tagged particle) is to zeroth order in thermal equilibrium and to first order its distribution is governed by a combination of the Rayleigh-Boltzmann equation and the linearized Boltzmann equation (proved to be valid for short times).Supported in part by NSF Grant PHY 78-22302.     

Modified cell theory: Free volume distributions and the equation of state for D-dimensional hard sphere systems

A generalized cell model, using cells of different sizes, is applied to hard rods, disks and spheres. Structures is discussed in terms of free volumes. The derived equation of state is exact for rods. For disks and spheres it provides a good approximation in the dense fluid and solid state.     

A discrete velocity model of a gas: Global in time solutions of the BBGKY hierarchy

In the present paper we study the evolution of a system of hard disks moving in the plane with a finite number of velocities as in the framework of a discrete velocity model of the Enskog equation, proposed in previous papers. Starting from the BBGKY hierarchy of such a system we give existence and uniqueness results for the initial value problem in suitable Banach spaces. In particular, the main result presented is the global in time weak solution to the BBGKY hierarchy for local equilibrium initial data, in the thermodynamic limit.     

Generalized enskog theory for homogeneous systems

We propose a generalization of the Enskog equation for homogeneous dense systems including the complete three-particle dynamics. To this end the time derivative of the one-particle distribution is represented in the thermodynamic limit as the sum of three terms describing the effect of the initials-particle correlations, collisions withins-particle clusters, and coupling ofs-particle clusters to the surrounding gaseous medium, respectively. The analysis of casess=2 ands=3 is performed both for hard spheres and for a smooth, repulsive interaction. On assuming the equilibrium structure and spatial dependence of terms reflecting the effect of the medium, we obtain fors=2 the Enskog equation, and fors=3 a new equation, going beyond the Enskog theory. Apart from the Enskog collision term it contains additional contributions, and can be shown to reduce to the Choh-Uhlenbeck equation in the long-time, low-density limit.     

Phase Transitions in a Mixture of Hard-Sphere Dipoles and Neutral Hard Spheres

Using the reference hypernetted chain (RHNC) integral equation theory and a rigorous stability analysis method, we investigate the phase behavior of a mixture of hard-sphere dipoles and neutral hard spheres based on the correlations of the homogeneous isotropic phase. Lowering the temperature down to the points where the RHNC equations fail to have a solution, several fluctuations strongly increase. At low densities our results indicate the onset of chain formation, which is similar with the pure DHS system. At highdensities, the results indicate the appearance of isotropic-to-ferroelectric transitions (small neutral hard spheres concentrations) and demixing transitions (large neutral hard spheres concentrations).