62 pages. Course at the conference "Topics in PDEs and applications 2008" held in Granada, April 7-11 2008; figure 13 and a misprint in Theorem 4.6 corrected in the new version
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 (like atoms in a crystal) leads to a completely different limiting dynamics. These lecture notes review several results on this problem obtained in the past decade as joint work with J. Bourgain, E. Caglioti and B. Wennberg.
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.
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 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 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.
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.
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.
It is a fundamental problem in mathematical physics to derive macroscopic transport equations from microscopic models. In this paper we derive the linear Boltzmann equation in the low-density limit of a damped quantum Lorentz gas for a large class of deterministic and random scatterer configurations. Previously this result was known only for the single-scatterer problem on the flat torus, and for uniformly random scatterer configurations where no damping is required. The damping is critical in establishing convergence -- in the absence of damping the limiting behaviour depends on the exact con…
We consider a quantum particle interacting with $N$ obstacles, whose positions are independently chosen according to a given probability density, through a two-body potential of the form $N^2 V(Nx)$ (Gross-Pitaevskii potential). We show convergence of the $N$ dependent one-particle Hamiltonian to a limiting Hamiltonian where the quantum particle experiences an effective potential depending only on the scattering length of the unscaled potential and the density of the obstacles. In this sense our Lorentz gas model exhibits a universal behavior for $N$ large. Moreover we explicitely characterize…
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.
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 study the motion of a particle on the two-dimensional honeycomb lattice, whose sites are occupied by either flipping rotators or flipping mirrors, which scatter the particle according to a deterministic rule. For both types of scatterers we find a new type of motion that has not been observed in a Lorentz Lattice gas, where the particle's trajectory is a self-avoiding walk between returns to its initial position. We show that this behavior is a consequence of the deterministic scattering rule and the particular class of initial scatterer configurations we consider. Since self-avoiding walks…
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 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…
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 consider transport properties for a non-homogeneous persistent random walk, that may be viewed as a mean-field version of the Lévy-Lorentz gas, namely a 1-d model characterized by a fat polynomial tail of the distribution of scatterers' distance, with parameter $α$. By varying the value of $α$ we have a transition from normal transport to superdiffusion, which we characterize by appropriate continuum limits.
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.
We prove that any generic (i.e., possibly aperiodic) Lorenz gas in two dimensions, with finite horizon and non-degenerate geometrical features, is ergodic if it is recurrent. We also give examples of aperiodic recurrent gases.
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.
Linear motion of a rigid body in a special kind of Lorentz gas is mathematically analyzed. The rigid body moves against gas drag according to Newton's equation. The gas model is a special Lorentz gas consisting of gas molecules and background obstacles, which was introduced in~(Tsuji and Aoki: J. Stat. Phys. \textbf{146}, 620--645, 2012). The specular boundary condition is imposed on the resulting kinetic equation. This study complements the numerical study by Tsuji and Aoki cited above --- although the setting in this paper is slightly different from theirs, qualitatively the same asymptotic …
We investigate the evolution of a particle in a Lorentz gas where the background scatters move and collide with each other. As in the standard Lorentz gas, we assume that the particle is negligibly light in comparison with scatters. We show that the average particle speed grows in time as t^{lambda/(4+lambda)} in three dimensions when the particle-scatter potential diverges as r^{-lambda} in the small separation limit. The typical displacement of the particle exhibits a universal linear growth in time independently on the density of the background gas and the particle-scatter interaction. The …
It is well known that, in the Boltzmann-Grad limit, the distribution of the free path length in the Lorentz gas with disordered scatterer configuration has an exponential density. If, on the other hand, the scatterers are located at the vertices of a Euclidean lattice, the density has a power-law tail proportional to xi^{-3}. In the present paper we construct scatterer configurations whose free path lengths have a distribution with tail xi^{-N-2} for any positive integer N. We also discuss the properties of the random flight process that describes the Lorentz gas in the Boltzmann-Grad limit. T…
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 …
In this paper we deal with further recent developments, strictly connected to the recent result obtained by Basile, Nota, Pezzotti and Pulvirenti. We consider the Lorentz gas out of equilibrium in a weak coupling regime. Each obstacle of the Lorentz gas generates a smooth radially symmetric potential with compact support. We prove that the macroscopic current in the stationary state is given by the Fick's law of diffusion. The diffusion coefficient is given by the Green-Kubo formula associated to the generator of the diffusion process dictated by the linear Landau equation.
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 …
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 …
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…
We provide a proof that the stationary macroscopic current of particles in a random lattice Lorentz gas satisfies Fick's law when connected to particles reservoirs. We consider a box on a d+1 dimensional lattice and when $d\geq7$, we show that under a diffusive rescaling of space and time, the probability to find a current different from its stationary value is exponentially small in time. Its stationary value is given by the conductivity times the difference of chemical potentials of the reservoirs. The proof is based on the fact that in high dimension, random walks have a small probability o…
In this paper we study the physical and statistical properties of the periodic Lorentz gas with finite horizon driven to a non-equilibrium steady state by the combination of non-conservative external forces and deterministic thermostats. A version of this model was introduced by Chernov, Eyink, Lebowitz, and Sinai and subsequently generalized by Chernov and the third author. Non-equilibrium steady states for these models are SRB measures and they are characterized by the positivity of the steady state entropy production rate. Our main result is to establish that the entropy production, in this…
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…
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…
Possibility of the diverging gradient of the macroscopic quantity near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect of the grazing collision for the infinite-range potential on the velocity distribution function (VDF) and its influence on the macroscopic quantity. By careful numerical analyses of the steady one-dimensional boundary-value problem, it is confirmed that the grazing collision suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However, as the price for that regularization, the collisio…
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…
Computing Nonequilibrium Transport from Short-Time Transients: From Lorentz Gas to Heat Conduction in One Dimensional Chains
Laboratoire de Physique de l’Ecole Normale Supérieure, ENS Université PSL, CNRS, Sorbonne Université, Université de Paris（巴黎高等师范学院物理实验室，ENS巴黎文理研究大学，法国国家科学研究中心，索邦大学，巴黎大学）；Politecnico di Milano（米兰理工大学）；Fondazione Istituto Italiano di Tecnologia（意大利技术研究院基金会）；Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi（国家研究委员会，复杂系统研究所）；INFN, Sezione di Firenze（意大利国家核物理研究所，佛罗伦萨分部）；INFN, Sezione di Torino（意大利国家核物理研究所，都灵分部）；Politecnico di Torino（都灵理工大学）
The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor $\sqrt{t}$ are normally distributed, as shown by Bunimovich and Sinai in 1981. In the infinite horizon case, motion is superdiffusive, however the normal distribution is recovered when scaling by $\sqrt{t\ln t}$, with an explicit formula for its variance. Here we explore the infinite horizon case in arbit…
