Escape Rates and Physically Relevant Measures for Billiards with Small Holes

We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map.     

Dispersing Billiards with Moving Scatterers

We propose a model of Sinai billiards with moving scatterers, in which the locations and shapes of the scatterers may change by small amounts between collisions. Our main result is the exponential loss of memory of initial data at uniform rates, and our proof consists of a coupling argument for non-stationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on the statistical properties of time-dependent dynamical systems.     

Properties of Some Chaotic Billiards with Time-Dependent Boundaries

A dispersing billiard (Lorentz gas) and focusing billiards (in the form of a stadium) with time-dependent boundaries are considered. The problem of a particle acceleration in such billiards is studied. For the Lorentz gas two cases of the time-dependence are investigated: stochastic perturbations of the boundary and its periodic oscillations. Two types of focusing billiards with periodically forced boundaries are explored: stadium with strong chaotic properties and a near-rectangle stadium. It is shown that in all cases billiard particles can reach unbounded velocities. Average velocities of the particle ensemble as functions of time and the number of collisions are obtained.     

Heat conduction in caricature models of the Lorentz gas

Heat transport coefficients are calculated for various random walks with internal states (the Markov partition of the Sinai billiard connects these walks with the Lorentz gas among a periodic configuration of scatterers). Models with reflecting or absorbing barriers and also those without or with local thermal equilibrium are investigated. The method is unified and is based on the Keldysh expansion of the resolvent of a matrix polynomial.     

Current in Periodic Lorentz Gases with Twists

We study electrical current in two-dimensional periodic Lorentz gas in the presence of a twist force on the scatterers. In this deterministic system, billiard orbits are still geodesics between collisions, but do not reflect elastically when reaching the boundary. When the horizon is finite, i.e. the free flights between collisions are bounded, the resulting current J is proportional to the strength of the twist force measured by ε. We also prove the existence of a unique SRB measure, for which the Pesin entropy formula and Young’s expression for the fractal dimension are valid.     

Recurrence and Higher Ergodic Properties for Quenched Random Lorentz Tubes in Dimension Bigger than Two

We consider the billiard dynamics in a non-compact set of ℝ ^d that is constructed as a bi-infinite chain of translated copies of the same d-dimensional polytope. A random configuration of semi-dispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global realization of the scatterers, is called quenched random Lorentz tube. Under some fairly general conditions, we prove that every system in the ensemble is hyperbolic and almost every system is recurrent, ergodic, and enjoys some higher chaotic properties.     

New Horizons in Multidimensional Diffusion: The Lorentz Gas and the Riemann Hypothesis

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 arbitrary dimensions, giving explicit formulas for the mean square displacement, arguing that it differs from the variance of the limiting distribution, making connections with the Riemann Hypothesis in the small scatterer limit, and providing evidence for a critical dimension d=6 beyond which correlation decay exhibits fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions.     

Universality of level spacing distributions in classical chaos

We suggest that random matrix theory applied to a matrix of lengths of classical trajectories can be used in classical billiards to distinguish chaotic from non-chaotic behavior. We consider in 2D the integrable circular and rectangular billiard, the chaotic cardioid, Sinai and stadium billiard as well as mixed billiards from the Limaçon/Robnik family. From the spectrum of the length matrix we compute the level spacing distribution, the spectral auto-correlation and spectral rigidity. We observe non-generic (Dirac comb) behavior in the integrable case and Wignerian behavior in the chaotic case. For the Robnik billiard close to the circle the distribution approaches a Poissonian distribution. The length matrix elements of chaotic billiards display approximate GOE behavior. Our findings provide evidence for universality of level fluctuations—known from quantum chaos—to hold also in classical physics.     

Algebraic fidelity decay for local perturbations

From a reflection measurement in a rectangular microwave billiard with randomly distributed scatterers the scattering and the ordinary fidelity was studied. The position of one of the scatterers is the perturbation parameter. Such perturbations can be considered as local since wave functions are influenced only locally, in contrast to, e.g., the situation where the fidelity decay is caused by the shift of one billiard wall. Using the random-plane-wave conjecture, an analytic expression for the fidelity decay due to the shift of one scatterer has been obtained, yielding an algebraic 1/t decay for long times. A perfect agreement between experiment and theory has been found, including a predicted scaling behavior concerning the dependence of the fidelity decay on the shift distance. The only free parameter has been determined independently from the variance of the level velocities.     

One-Dimensional Lorentz Gas with Rotating Scatterers: Exact Solutions

The simplest solutions (orbits) to the recently introduced Lorentz gas with rotating scatterers are found by considering its one-dimensional one-particle reduction. This model has only one parameter which can be viewed as the amount of energy transfer between the scatterers and the particle during a collision. Exact solutions of the system are found for several values of this parameter. For some of these values, the dynamics is shown to be in many respects similar to the dynamics of the deterministic Lorentz lattice gases.     

Lorentz gas shear viscosity via nonequilibrium molecular dynamics and Boltzmann's equation

When nonequilibrium molecular dynamics is used to impose isothermal shear on a two-body periodic system of hard disks or spheres, the equations of motion reduce to those describing a Lorentz gas under shear. In this shearing Lorentz gas a single particle moves, isothermally, through a spatially periodic shearing crystal of infinitely massive scatterers. The curvilinear trajectories are calculated analytically and used to measure the dilute Lorentz gas viscosity at several strain rates. Simulations and solutions of Boltzmann's equation exhibit shear thinning resembling that found inN-body nonequilibrium simulations. For the three-dimensional Lorentz gas we obtained an exact expression for the viscosity which is valid at all strain rates. In two dimensions this is not possible due to the anisotropy of the scattering.     

Competition between unlimited and limited energy growth in a two-dimensional time-dependent billiard

Some dynamical properties for a dissipative time-dependent Lorentz gas are studied. We assume that the size of the scatterers change periodically in time. We show that for some combination of the control parameters the particles come to a complete stop between the scatterers, but for some other cases, the average velocity grows unbounded. This is the first time that the unlimited energy growth is observed in a dissipative system. Finally, we study the behavior of the average velocity as a function of the number of collisions and we show that the system is scaling invariant with scaling exponents well defined.     

Caustics for inner and outer billiards

With a plane closed convex curve,T, we associate two area preserving twist maps: the (classical) inner billiard inT and the outer billiard in the exterior ofT. The invariant circles of these twist maps correspond to certain plane curves: the inner and the outer caustics ofT. We investigate how the shape ofT determines the possible location of caustics, establish the existence of open regions which are free of caustics, and estimate fro below the size of these regions in terms of the geometry ofT.Partially supported by NSF.Partially supported by NSF Grant DMS 9017995.     

Using integrability to produce chaos: Billiards with positive entropy

A new open class of convex 2 dimensional planar billiards with positive Lyapunov exponent almost everywhere is constructed. We introduce the notion of a focusing arc and show that such arcs can be used to build billiard systems with positive Lyapunov exponents. We prove that under smallC ⁶ perturbations, focusing arcs remain focusing and thereby show that perturbations of the Bunimovich stadium billiard have positive Lyapunov exponents.Partially supported by NSF grant DMS 8806067     

Irregular scattering of particles confined to ring-bounded cavities

The classical motion of a free particle that scatters elastically from ring-bounded cavities is analyzed numerically. When the ring is a smooth circle the scattering follows a regular and periodic pattern. However, for rings composed ofN scatterers the flow is irregular, of Lyapunov type. The Lyapunov exponent is found to depend logarithmically withN, which is consistent with the theoretical derivation of Chernov for polygon-shaped billiard systems. The escape time from cavities bounded by a ring ofN separated scatterers is demonstrated to follow a geometric distribution as a function of the aperture size. An empirical scaling is proposed between the Lyapunov exponent, the escape time, andN.     

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.     

Central limit theorem for the Lorentz process via perturbation theory

The Markov partition of the Sinai billiard allows the following heuristic interpretation for the Lorentz process with a ²-periodic configuration of scatterers: while executing a (non-Markovian) random walk on ², and particle changes its internal state according to the symbolic dynamics defined by the Markov partition. This picture can be formalized and then the Lorentz process appears as the limit of a sequence of (Markovian!) random walks with a finite but increasing number of internal states and the central limit theorem can be proved for it by perturbational expansions with uniformly bounded — in a sence related to the Perron-Frobenius theorem — coefficients and uniform remainder terms.     

The Lorentz process converges to a random flight process

The Lorentz process is the stochastic process defined by a particle moving, according to Newton's law of motion, through static scatterers distributed according to some probability measure in space. We consider the Boltzmann-Grad limit: The density of scatterers increases to infinity and at the same time the diameter of the scatterers decreases to zero in such a way that the mean free path of the particle is kept constant. We show that the Lorentz process converges in the weak*-topology of regular Borel measures on the paths space to some stochastic process. The limit process is Markovian if and only if the rescaled density of scatterers converges in probability to its mean. In that case the limit process is a (spatially inhomogeneous) random flight process.On leave of absence of Fachbereich Physik der Universität München. Work supported by a DFG fellowship     

Diffusion and propagation in triangular Lorentz lattice gas cellular automata

The diffusion process of point particles moving on regular triangular and random lattices, randomly occupied with stationary scatterers (a Lorentz lattice gas cellular automaton), is studied, for strictly deterministic scattering rules, as a function of the concentration of the scatterers. In addition to the normal and various kinds of retarded diffusion found before on the regular square lattice, straight-line propagation through the scatterers is observed.