The periodic Lorentz gas describes an ensemble of non-interacting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (Boltzmann-Grad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic transport equation is not the linear Boltzmann equation--in contrast to the Lorentz gas with a disordered scatterer configuration. The present paper focuses on the two-dimensional set-up, and reports an explicit, elementary formula for the collision kernel of the transport equation.
The periodic Lorentz gas is the dynamical system corresponding to the free motion of a point particle in a periodic system of fixed spherical obstacles of radius $r$ centered at the integer points, assuming all collisions of the particle with the obstacles to be elastic. In this Note, we study this motion on time intervals of order $1/r$ and in the limit as $r\to 0^+$, in the case of two space dimensions.
We study the dynamics of a point particle in a periodic array of spherical scatterers, and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (Boltzmann-Grad limit). A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two.
We provide sufficient conditions for the existence of periodic solutions of the of the Lorentz force equation, which models the motion of a charged particle under the action of an electromagnetic fields. The basic assumptions cover relevant models with singularities like Coulomb-like electric potentials or the magnetic dipole.
The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the Boltzmann-Grad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length of the periodic Lorentz gas. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.
The dynamics of a point particle in a periodic array of spherical scatterers converges, in the limit of small scatterer size, to a random flight process, whose paths are piecewise linear curves generated by a Markov process with memory two. The corresponding transport equation is distinctly different from the linear Boltzmann equation observed in the case of a random configuration of scatterers. In the present paper we provide asymptotic estimates for the transition probabilities of this Markov process. Our results in particular sharpen previous upper and lower bounds on the distribution of fr…
In earlier work we showed that the particle displacement for the multidimensional periodic Lorentz gas, in the limit of low scatterer density (Boltzmann-Grad limit), satisfies a central limit theorem with superdiffusive scaling. The present paper extends this result to a functional central limit theorem, i.e., the weak convergence of the particle trajectory to Brownian motion.
In a flat 2-torus with a disk of diameter $r$ removed, let $Φ_r(t)$ be the distribution of free-path lengths (the probability that a segment of length larger than $t$ with uniformly distributed origin and direction does not meet the disk). We prove that $Φ_r(t/r)$ behaves like $\frac{2}{π^2 t}$ for each $t>2$ and in the limit as $r\to 0^+$, in some appropriate sense. We then discuss the implications of this result in the context of kinetic theory.
We study the free path length and the geometric free path length in the model of the periodic two-dimensional Lorentz gas (Sinai billiard). We give a complete and rigorous proof for the existence of their distributions in the small-scatterer limit and explicitly compute them. As a corollary one gets a complete proof for the existence of the constant term $c=2-3\ln 2+\frac{27ζ(3)}{2π^2}$ in the asymptotic formula $h(T)=-2\ln \eps+c+o(1)$ of the KS entropy of the billiard map in this model, as conjectured by P. Dahlqvist.
We prove a superdiffusive central limit theorem for the displacement of a test particle in the periodic Lorentz gas in the limit of large times $t$ and low scatterer densities (Boltzmann-Grad limit). The normalization factor is $\sqrt{t\log t}$, where $t$ is measured in units of the mean collision time. This result holds in any dimension and for a general class of finite-range scattering potentials. We also establish the corresponding invariance principle, i.e., the weak convergence of the particle dynamics to Brownian motion.
The Lorentz gas is one of the simplest, most widely used models to study the transport properties of rarified gases in matter. It describes the dynamics of a cloud of non-interacting point particles in an infinite array of fixed spherical scatterers. More than one hundred years after its conception, it is still a major challenge to understand the nature of the kinetic transport equation that governs the macroscopic particle dynamics in the limit of low scatterer density (the Boltzmann-Grad limit). Lorentz suggested that this equation should be the linear Boltzmann equation. This was confirmed …
The Lorentz gas is one of the simplest and most widely-studied models for particle transport in matter. It describes a cloud of non-interacting gas particles in an infinitely extended array of identical spherical scatterers. The model was introduced by Lorentz in 1905 who, following the pioneering ideas of Maxwell and Boltzmann, postulated that in the limit of low scatterer density, the macroscopic transport properties of the model should be governed by a linear Boltzmann equation. The linear Boltzmann equation has since proved a useful tool in the description of various phenomena, including s…
Recently, stretched exponential decay of multiple correlations in the periodic Lorentz gas has been used to show the convergence of a series of correlations which has the physical interpretation as the fourth order Burnett coefficient, a generalisation of the diffusion coefficient. Here the result is extended to include all higher order Burnett coefficients, and give a plausible argument that the expansion constructed from the Burnett coefficients has a finite radius of convergence.
The Drude-Lorentz model for the motion of electrons in a solid is a classical model in statistical mechanics, where electrons are represented as point particles bouncing on a fixed system of obstacles (the atoms in the solid). Under some appropriate scaling assumption -- known as the Boltzmann-Grad scaling by analogy with the kinetic theory of rarefied gases -- this system can be described in some limit by a linear Boltzmann equation, assuming that the configuration of obstacles is random [G. Gallavotti, [Phys. Rev. (2) vol. 185 (1969), 308]). The case of a periodic configuration of obstacles …
We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external electric field E as well as a Gaussian thermostat which keeps the speed |v| constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E. Some of them are computed numerically. We compare these r…
We consider a model of a dynamical Lorentz gaz : a single particle is moving in $\mathbb{R}^d$ through an array of fixed an soft scatterers each possessing an internal degree of freedom coupled to the particle. Assuming the initial velocity is sufficiently high and modelling the parameters of the scatterers as random variables, we describe the evolution of the kinetic energy of the particle by a Markov chain for which each step corresponds to a collision. We show that the momentum distribution of the particle approaches a Maxwell-Boltzmann distribution with effective temperature $T$ such that …
We study the long time evolution and stationary speed distribution of N point particles in 2D moving under the action of an external field E, and undergoing elastic collisions with either a fixed periodic array of convex scatterers, or with virtual random scatterers. The total kinetic energy of the N particles is kept fixed by a Gaussian thermostat which induces an interaction between the particles. We show analytically and numerically that for weak fields this distribution is universal, i.e. independent of the position or shape of the obstacles or the nature of the stochastic scattering. Our …
We study numerically and theoretically the $d$-dimensional Hamiltonian motion of fast particles through a field of scatterers, modeled by bounded, localized, (time-dependent) potentials, that we refer to as (in)elastic non-dissipative Lorentz gases. We illustrate the wide applicability of a random walk picture previously developed for a field of scatterers with random spatial and/or time-dependence by applying it to four other models. First, for a periodic array of spherical scatterers in $d\geq2$, with a smooth (quasi)periodic time-dependence, we show Fermi acceleration: the ensemble averaged…
We review some recent results concerning the derivation of the diffusion equation and the validation of Fick's law for the microscopic model given by the random Lorentz Gas. These results are achieved by using a linear kinetic equation as an intermediate level of description between our original mechanical system and the diffusion equation.
The two-dimensional, periodic Lorentz gas, is the dynamical system corresponding with the free motion of a point particle in a planar system of fixed circular obstacles centered at the vertices of a square lattice in the Euclidian plane. Assuming elastic collisions between the particle and the obstacles, this dynamical system is studied in the Boltzmann-Grad limit, assuming that the obstacle radius $r$ and the reciprocal mean free path are asymptotically equivalent small quantities, and that the particle's distribution function is slowly varying in the space variable. In this limit, the period…
We report on the stationary dynamics in classical Sinai billiard (SB) corresponding to the unit cell of the periodic Lorentz gas (LG) formed by square lattice of length $L$ and dispersing circles of radius $R$ placed in the center of unit cell. Dynamic correlation effects for classical particles, initially distributed by random way, are considered within the scope of deterministic and stochastic descriptions. A temporal analysis of elastic reflections from the SB square walls and circle obstacles is given for distinct geometries in terms of the wall-collision and the circle-collision distribut…
The kinetic energy term of Hamiltonian systems with balanced loss and gain is not semi-positive-definite, leading to instabilities at the classical as well quantum level. It is shown that an additional Lorentz interaction in the Hamiltonian allows the kinetic energy term to be semi-positive-definite and thereby, improving the stability properties of the system. Further, a consistent quantum theory admitting bound states may be obtained on the real line instead of Stoke wedges on the complex plane. The Landau Hamiltonian in presence of balanced loss and gain is considered for elucidating the ge…
A kinetic theory is developed to describe radiating electrons whose motion is governed by the Lorentz-Dirac equation. This gives rise to a generalized Vlasov equation coupled to an equation for the evolution of the physical submanifold of phase space. The pathological solutions of the 1-particle theory may be removed by expanding the latter equation in powers of $τ:= q^2/ 6πm$. The radiation-induced change in entropy is explored, and its physical origin is discussed. As a simple demonstration of the theory, the radiative damping rate of longitudinal plasma waves is calculated.
We prove the invariance principle for a \emph{random Lorentz-gas} particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius $r$, placed according to a Poisson point process of density $\varrho$, in the limit $\varrho\to\infty$, $r\to0$, $\varrho r^{2}\to1$ up to time scales of order $T=o(r^{-2}{|\log r|}^{-2})$. To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibri…
We study a system of charged, noninteracting classical particles moving in a Poisson distribution of hard-disk scatterers in two dimensions, under the effect of a magnetic field perpendicular to the plane. We prove that, in the low-density (Boltzmann-Grad) limit, the particle distribution evolves according to a generalized linear Boltzmann equation, previously derived and solved by Bobylev et al. [4, 5, 6]. In this model, Boltzmann's chaos fails, and the kinetic equation includes non-Markovian terms. The ideas of [13] can be however adapted to prove convergence of the process with memory.
We develop quantum electrodynamics into a kinetic-theory-like evolution equation for electrons, positrons and photons. To keep the "collision rules" simple, we make use of longitudinal and temporal photons in addition to the usual transverse photons. For our explicitly time-dependent approach, we introduce proper time-correlation functions. We then develop a stochastic simulation technique for solving the resulting kinetic equation and we comment on the subtleties associated with gauge invariance, Lorentz invariance and divergent integrals. To illustrate the validity and power of the proposed …
