The free path in a high velocity random flight process associated to a Lorentz gas in an external field (Q2809290)

The paper under review studies the asymptotic behavior of the free path of a variable density random flight model in an external field as the initial velocity of the particle goes to infinity. In 1905, Lorentz first introduced the gas model for the motion of electrons in metallic bodies, and the random scatterer Lorentz gas is difficult to study directly even in the absence of an external field. The Boltzmann-Grad limit, which is a low density limit in which the number of scatterers goes to zero, in such a way that the total volume of the scatterers in the box goes to zero, is a tractable approach. The Markovian nature of the Boltzmann-Grad limit follows from the fact that the re-collisions with scatterers become unlikely as the size of each scatterer goes to zero and the Poisson nature of the scatterer locations.NEWLINENEWLINEThe paper focuses on the regime where the particle's velocity is large: (i) an external field which accelerates the particle towards infinity (as the periodic Lorentz gas in two-dimension), (ii) a particle motion in a centered mean zero isotropic force field (as the soft core Poisson Lorentz gas). A remarkable contribution of the paper is to have fields which do not, in general, have mean zero, and the primary focus is to study the free path of the particle between two reflection times.NEWLINENEWLINESection 1.1 sets up the model in dimension three, and the random flight process \((X(t), V(t))\) is given recursively in (1.3)--(1.4). The authors further look for the spherically symmetric case to simplify the situation, to have the motion in polar coordinates for a particle of mass \(m\) in the potential \(\mathbf{U}\). The authors give a heuristic argument in Section 1.2, by Brownian scaling, to get the diffusive limit. There are approximately \(v_0^4\) reflections per unit time. The general assumptions (A1)--(A5) given in Section 1.3 allow \(g\) to be a small perturbation of the constant density and \(\mathbf{U}\) to produce a small perturbation of a constant field. The main result (Theorem 1.3) is given in Section 1.4. Theorem 1.3 shows that both \(\mathbf{U}_n\) and \(g_n\) impart a drift towards the area where the corresponding function has a smaller value. Theorem 1.4 presents the fact that the family of continuous-time processes converges in distribution on the Skorokhod space to the diffusion, with joint convergence on the stopping times. Theorem 1.7 shows the trajectory of the particle, and fixed \(l<u\) limits to a specified time change process, whose generator acts on functions with compact support, are given explicitly.NEWLINENEWLINE Section 2 analyzes the jumps of the free flight chain. Lemma 2.1 gives local control over how the radical part of the path behaves, and Lemma 2.2 illustrates that the scaled trajectory converges to the starting point uniformly over a fixed time interval. Lemma 2.4 gives the \(L^p\)-bounds for the moments in terms of polar coordinates for \(1\leq p <\infty\). For the Markovian chain with transition operator, Lemma 2.10 and Lemma 2.11 give the result of Theorem 1.3, to keep track of the time between collisions.NEWLINENEWLINE Section 3 proves the convergence of the free path process with transition operator in Theorem 3.1, by the method of Theorem IX 4.21 of [\textit{J. Jacod} and \textit{A. N. Shiryaev}, Limit theorems for stochastic processes. 2nd ed. Berlin: Springer (2003; Zbl 1018.60002)].NEWLINENEWLINE Section 4 studies the convergence of the full trajectory of the particle in Theorem 4.1, by first inverting the time process and examining the stopped process with cutoffs.NEWLINENEWLINE Section 5 lists conditions under which the boundary points of the domain are inaccessible, i.\,e., the boundary points cannot be reached in finite time (see Proposition 5.2). If the potential function \(\mathbf{U}(x) = Cx\) is from a constant force field, then both 0 and \(h\) in \([0, h]\) are inaccessible boundary points. If the potential function is given by \(\mathbf{U}(x) = - \frac{k}{x}\) for some constant \(k>0\), then the boundary points 0 and \(\infty\) are inaccessible.NEWLINENEWLINE Section 6 tries to extend the results from 3-dimensional to higher dimensional Lorentz gases (Theorem 6.1 and Theorem 6.2) by similar arguments without proofs. It is necessary to have the cutoffs, due to the generality for the potential and the scattering densities, such that the limit behavior can be very different at the boundaries. Therefore, it is still a challenging problem to characterize the inaccessible boundaries for general potentials and scattering densities.