The Distribution of the Free Path Lengths in the Periodic Two-Dimensional Lorentz Gas in the Small-Scatterer Limit

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 in the asymptotic formula of the KS entropy of the billiard map in this model, as conjectured by P. Dahlqvist.In memory of Walter Philipp     

Free Path Lengths in Quasicrystals

Previous studies of kinetic transport in the Lorentz gas have been limited to cases where the scatterers are distributed at random (e.g., at the points of a spatial Poisson process) or at the vertices of a Euclidean lattice. In the present paper we investigate quasicrystalline scatterer configurations, which are non-periodic, yet strongly correlated. A famous example is the vertex set of a Penrose tiling. Our main result proves the existence of a limit distribution for the free path length, which answers a question of Wennberg. The limit distribution is characterised by a certain random variable on the space of higher dimensional lattices, and is distinctly different from the exponential distribution observed for random scatterer configurations. The key ingredients in the proofs are equidistribution theorems on homogeneous spaces, which follow from Ratner’s measure classification.     

Superdiffusion in the Periodic Lorentz Gas

We introduce the dynamical sine-Gordon equation in two space dimensions with parameter \({\beta}\), which is the natural dynamic associated to the usual quantum sine-Gordon model. It is shown that when \({\beta^{2} \in (0, \frac{16\pi}{3})}\) the Wick renormalised equation is well-posed. In the regime \({\beta^{2} \in (0, 4\pi)}\), the Da Prato–Debussche method [J Funct Anal 196(1):180–210, 2002; Ann Probab 31(4):1900–1916, 2003] applies, while for \({\beta^{2} \in [4\pi, \frac{16\pi}{3})}\), the solution theory is provided via the theory of regularity structures [Hairer, Invent Math 198(2):269–504, 2014]. We also show that this model arises naturally from a class of \({2 + 1}\) -dimensional equilibrium interface fluctuation models with periodic nonlinearities. The main mathematical difficulty arises in the construction of the model for the associated regularity structure where the role of the noise is played by a non-Gaussian random distribution similar to the complex multiplicative Gaussian chaos recently analysed in Lacoin et al. [Commun Math Phys 337(2):569–632, 2015].     

Free Path Lengths in Quasi Crystals

The Lorentz gas is a model for a cloud of point particles (electrons) in a distribution of scatterers in space. The scatterers are often assumed to be spherical with a fixed diameter d, and the point particles move with constant velocity between the scatterers, and are specularly reflected when hitting a scatterer. There is no interaction between point particles. An interesting question concerns the distribution of free path lengths, i.e. the distance a point particle moves between the scattering events, and how this distribution scales with scatterer diameter, scatterer density and the distribution of the scatterers. It is by now well known that in the so-called Boltzmann–Grad limit, a Poisson distribution of scatterers leads to an exponential distribution of free path lengths, whereas if the scatterer distribution is periodic, the free path length distribution asymptotically behaves as a power law.     

Density-Dependent Diffusion in the Periodic Lorentz Gas

We study the deterministic diffusion coefficient of the two-dimensional periodic Lorentz gas as a function of the density of scatterers. Based on computer simulations, and by applying straightforward analytical arguments, we systematically improve the Machta–Zwanzig random walk approximation [Phys. Rev. Lett. 50:1959 (1983)] by including microscopic correlations. We furthermore, show that, on a fine scale, the diffusion coefficient is a non-trivial function of the density. On a coarse scale and for lower densities, the diffusion coefficient exhibits a Boltzmann-like behavior, whereas for very high densities it crosses over to a regime which can be understood qualitatively by the Machta–Zwanzig approximation.     

Power-Law Distributions for the Free Path Length in Lorentz Gases

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. The convergence of the distribution of the free path length follows from equidistribution of large spheres in products of certain homogeneous spaces, which in turn is a consequence of Ratner’s measure classification theorem.     

Microscopic Chaos and Reaction-Diffusion Processes in the Periodic Lorentz Gas

We apply the hypothesis of microscopic chaos to diffusion-controlled reaction which we study in a reactive periodic Lorentz gas. The relaxation rate of the reactive eigenmodes is obtained as eigenvalue of the Frobenius–Perron operator, which determines the reaction rate. The cumulative functions of the eigenstates of the Frobenius–Perron operator are shown to be generalizations of Lebesgue's singular continuous functions. For small enough densities of catalysts, the reaction is controlled by the diffusion. A random-walk model of this diffusion-controlled reaction process is presented, which is used to study the dependence of the reaction rate on the density of catalysts.Aspirant FNRS     

Thermostating by Deterministic Scattering: The Periodic Lorentz Gas

We present a novel mechanism for thermalizing a system of particles in equilibrium and nonequilibrium situations, based on specifically modeling energy transfer at the boundaries via a microscopic collision process. We apply our method to the periodic Lorentz gas, where a point particle moves diffusively through an ensemble of hard disks arranged on a triangular lattice. First, collision rules are defined for this system in thermal equilibrium. They determine the velocity of the moving particle such that the system is deterministic, time-reversible, and microcanonical. These collision rules can systematically be adapted to the case where one associates arbitrarily many degrees of freedom to the disk, which here acts as a boundary. Subsequently, the system is investigated in nonequilibrium situations by applying an external field. We show that in the limit where the disk is endowed by infinitely many degrees of freedom it acts as a thermal reservoir yielding a well-defined nonequilibrium steady state. The characteristic properties of this state, as obtained from computer simulations, are finally compared to those of the so-called Gaussian thermostated driven Lorentz gas.     

Spatial Structure of Stationary Nonequilibrium States in the Thermostatted Periodic Lorentz Gas

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 these projections are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions. Similarities and differences are noted.     

Periodic orbit expansions for the Lorentz gas

We apply the periodic orbit expansion to the calculation of transport, thermodynamic, and chaotic properties of the finite-horizon triangular Lorentz gas. We show numerically that the inverse of the normalized Lyapunov number is a good estimate of the probability of an individual periodic orbit. We investigate the convergence of the periodic orbit expansion and compare it with the convergence of the cycle expansions obtained from the Ruelle dynamical -function. For this system with severe pruning we find that applying standard convergence acceleration schemes to the periodic orbit expansion is superior to the dynamical -function approach. The averages obtained from the periodic orbit expansion are within 8% of the values obtained from direct numerical time and ensemble averaging. None of the periodic orbit expansions used here is computationally competitive with the standard simulation approaches for calculating averages. However, we believe that these expansion methods are of fundamental importance, because they give a direct route to the phase space distribution function.     

Diffusion in the Lorentz Gas

The Lorentz gas, a point particle making mirror-like reflections from an extended collection of scatterers, has been a useful model of deterministic diffusion and related statistical properties for over a century. This survey summarises recent results, including periodic and aperiodic models, finite and infinite horizon, external fields, smooth or polygonal obstacles, and in the Boltzmann-Grad limit. New results are given for several moving particles and for obstacles with flat points. Finally, a variety of applications are presented.     

The Equilibrium Distribution for Free Electrons in a Real Gas

The equilibrium distribution for free electrons in a real gas is derived based on the existence of an interaction potential between a free electron and its nearest neighbor atom (or ion). The form of the interaction potential is inversely proportional to rl where l is a constant; and the derived distribution reduces to the Maxwellian form when the interaction is neglected. It is shown that for most physical situations this interaction potential will only contribute a miniscule change to those electron properties which are dependent upon<|vj?>, j>-3 where is the electron speed.     

Fluctuation theorem for currents in the Spinning Lorentz Gas

We study the fluctuation theorem formulated in terms of the currents present in a Hamiltonian system with coupled mass and energy transport. To drive the system out of equilibrium, we assume it to be connected to two ideal thermodynamical baths. The fluctuation symmetry is, thus, expressed in terms of the joint probability distribution of energy and particle currents in the system. This relation is verified numerically for the stationary state in the Spinning Lorentz Gas (SLG), driven out of equilibrium by temperature and/or chemical potential differences between the baths, as well as in the presence of an applied field.     

Distinct Clusterings and Characteristic Path Lengths in Dynamic Small-World Networks with Identical Limit Degree Distribution

Many real-world networks belong to a particular class of structures, known as small-world networks, that display short distance between pair of nodes. In this paper, we introduce a simple family of growing small-world networks where both addition and deletion of edges are possible. By tuning the deletion probability q _t , the model undergoes a transition from large worlds to small worlds. By making use of analytical or numerical means we determine the degree distribution, clustering coefficient and average path length of our networks. Surprisingly, we find that two similar evolving mechanisms, which provide identical degree distribution under a reciprocal scaling as t goes to infinity, can lead to quite different clustering behaviors and characteristic path lengths. It is also worth noting that Farey graphs constitute the extreme case q _t ??0 of our random construction.     

Equilibration and Diffusion for a Dynamical Lorentz Gas

We consider a model of a dynamical Lorentz gaz: a single particle is moving in \({\mathbb {R}}^d\) through an array of fixed and 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 \(k_BT\) corresponds to an average of the scatterers’ kinetic energy.     

On the Boltzmann equation for the Lorentz gas

We consider the Boltzmann-Grad limit for the Lorentz, or wind-tree, model. We prove that if is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the Poisson distribution with parameter >0, then the evolution of an initial a.c. particle density tends in the Boltzmann-Grad limit to the solution of the Boltzmann equation for the model. As an intermediate step we prove that the process of the free path lengths and impact parameters induced by the Lebesgue measure on a small region tends to a limiting independent process.     

The Boltzmann Equation for a One-Dimensional Quantum Lorentz Gas

We study the macroscopic behavior of a quantum particle under the action of randomly distributed scatterers on the real line. Each scatterer generates a δ-potential. We prove that, in the low density limit, the Wigner function of the system converges to a probability distribution satisfying a classical linear Boltzmann equation, with a scattering cross section computed according to the Quantum Mechanical rules. Received: 2 April 1998 / Accepted: 12 February 1999     

On the quantum Lorentz group

The quantum analog of Pauli matrices are introduced and investigated. From these matrices and an appropriate trace over spinorial indices we construct a quantum Minkowski metric. In this framework we show explicitly the correspondence between the SL(2,C) and Lorentz quantum groups. Five matrices of the quantum Lorentz group are constructed in terms of the R matrix of SL(2,C) group. These matrices satisfy Yang–Baxter equations and two of which have adequate properties tied to the quantum Minkowski space structure as the reality conditions of the coordinates and the symmetrization of the metric. It is also shown that the Minkowski metric leads to invariant and central lengths of four-vectors.