Transport properties of the diluted Lorentz slab

Abstract

We study the behavior of a point particle incident on a slab of a randomly diluted triangular array of circular scatterers. Various scattering properties, such as the reflection and transmission probabilities and the scattering time are studied as a function of thickness and dilution. We show that a diffusion model satisfactorily describes the mentioned scattering properties. We also show how some of these quantities can be evaluated exactly and their agreement with numerical experiments. Our results exhibit the dependence of these scattering data on the mean free path. This dependence again shows excellent agreement with the predictions of a Brownian motion model.

Nonequilibrium Energy Profiles for a Class of 1-D Models

“…We describe in this subsection a family of models quite close to those studied numerically in [14,9]. The rules of interaction (though not the coupling to heat baths) are, in fact, used earlier in [17].…”

“…For more complete accounts, see the review papers [1,10,12]. The models which come closest to ours, and which to some degree inspired this work, are those in [14,9]. In these papers, the authors carried out a numerical study of a system comprised of an array of disks similar to those in Sect.…”

“…4.1, we introduce a family of Hamiltonian models generalizing those studied numerically in [14,9]. A single-cell analysis similar to that in Section 3 is carried out for this family in Sect.…”

“…The system below describes the free motion of k point particles (i.e., tracers) in Γ = Γ 0 \ D. When a tracer runs into ∂Γ 0 , the boundary of Γ 0 , the reflection is specular. When it hits the rotating disk, the energy exchange is according to the rules introduced in [17,14,9]. A more precise description of the system follows.…”

“…Our main stochastic example, dubbed the "random-halves model", is particularly simple: A clock rings with rate proportional to √ x where x is the (kinetic) energy of the tracer; at the clock, energy exchange between tracer and tank takes place; and the rule of exchange consists simply of pooling the two energies together and randomly dividing -in an unbiased way -the total energy into two parts. Our main Hamiltonian example is a variant of the model studied in [14,9]. Here the role of the "tank" is played by a rotating disk nailed down at its center, and stored energy is ω 2 where ω is the angular velocity of the disk.…”