[en] Transport of particles in media whose cross sections are random functions of space and time is considered. The linear transport equation, in the presence of this spacetime noise, is viewed as a Boltzmann-Langevin equation, the solution of which generates a stochastic process, the angular flux, for different realizations of the cross section. A Gaussian model of fluctuations is adopted with a prescribed mean, variance and correlation function. For white noise in time, but with otherwise arbitrary spatial correlation, an exactly closed equation for the ensemble averaged angular flux is obtained and seen to be identical to the transport equation but with renormalised cross sections. Similar exact closures are demonstrated for the second moment and the two-point space-angle correlation of the angular flux. Standard methods may be adapted to solve these averaged transport equations which are valid in the small correlation time limit. (Author)

[en] Boltzmann's equation is solved for the diffusion of nonabsorbing particles through a thick slab. The dimensionless thickness D=dμ_s, where d is the thickness and μ_s is the rate of scattering. Our solution neglects terms of order O(e^-D) and is exact in the limit that D much-gt 1. Simple expressions are obtained for the reflection and transmission coefficients, as well as for the angular distributions on the front and back sides of the slab. copyright 1995 American Institute of Physics

[en] The Uehling endash Uhlenbeck evolution equations for gases of identical quantum particles either fermions or bosons, in the case in which the collision kernel does not depend on the distribution function, are considered. The existence of solutions and their asymptotic relations with solutions of the hydrodynamic equations both at the level of the Euler system and at the level of the Navier endash Stokes system are proved. copyright 1997 American Institute of Physics

[en] Consider the Boltzmann equation ∂/∂_tu(t,x,v)=-v. ∇_xu(t,x,v)-σ_a(x,v)u(t,x,v)+∫_vk(x,v',v)u(t,x,v')dv' in Rⁿ x V such-that (x,v), V being an open subset of Rⁿ, n ≥ 2. Equation (1.1) describes the dynamics of a flow of particles in Rⁿ under the assumption that the interaction between them is neglectable (no non-linear terms). This is the case for example for a low-density flow of neutrons. The term involving σ_a describes the loss of particles from (x,v) ε Rⁿ x V due to absorption or scattering into another point (x,v'), while the last term in (1.1) involving k represents the production at x ε Rⁿ of particles with velocity v form particles with velocity v'. The total rate of this production at (x,v') is given by σ_p(x,v') = ∫_V k (x,v',v)dv. Following [RS] we say that the pair (σ_a, k) is admissible, if (i) O ≤ σ_a ε L^∞(Rⁿ x V), (ii) O ≤ k(x,v', ·) ε L¹(V) for a.e. (x,v') ε Rⁿ x V and σ_p ε L^∞(Rⁿ x V), (iii) There is an open bounded set X contained-in Rⁿ, such that k(x,v',v) and σ_a(x,v) vanish if x ε X. One can define the wave operators associated with T, T_O by W_- = s-lim/t→∞ U(t)U_O(-t), W₊ = s-lim/t→∞ U(O)(-t)U(t). if W_-, W₊ exist, then one can define the scattering operator S = W₊W_- as a bounded operator in L¹(Rⁿ x V). Scattering theory for (1.1) has been developed by other authors and we refer to these papers for sufficient conditions guaranteeing the existence of S. An abstract approach based on the Limiting Absorption Principle has been proposed. 22 refs

[en] The generalized local Boltzmann equation is derived for the distribution function of energetic particles interacting with the thermal vibrations of lattice. On using the Boltzmann collision term the particle energy losses A and the diffusion function B are obtained. The functions A and B have singularities connected with the difference of the contributions of the particles moving in crystals in the regimes of channeling, quasichanneling and random motion

No abstract available

[en] This paper considers the time- and space-dependent linear Boltzmann equation with general boundary conditions in the case of inelastic (granular) collisions. First, in the (angular) cut-off case, mild L¹-solutions are constructed as limits of the iterate functions and boundedness of higher velocity moments are discussed in the case of inverse power collisions forces. Then the problem of the weak solutions, as weak limit of a sequence of mild solutions, is studied for a bounded body, in the case of very soft interactions (including the Coulomb case). Furthermore, strong convergence of weak solutions to the equilibrium, when time goes to infinity, is discussed, using a generalized H-theorem, together with a translation continuity property.

No abstract available

[en] We discuss steady boundary value problems for the Boltzmann equation with inflow and diffusive boundary conditions in one, two, and three dimensions, with suitable truncations of the collision kernel. General existence and uniqueness results are obtained if the domain is sufficiently small. In one dimension, the existence of solutions on general intervals is obtained by abstract fixed-point theory

[en] Within about a year (1916–1917) Chapman and Enskog independently proposed an important expansion for solving the Boltzmann equation. However, the expansion is divergent or indeterminant in the case of relaxation time $\mathrm{\tau <!--\; ??\; -->}⩾<!--\; ???\; -->1$. Since then, this divergence problem has puzzled researchers for a century. Using a modified Möbius series inversion formula, we propose a modified Chapman–Enskog expansion with a variable upper limit of the summation. The new expansion can give not only a convergent summation but also the best-so-far explanation on some unbelievable scenarios occurring in previous practice. (paper)