Exterior Billiards

Abstract

The book contains an account of results obtained by the author and his collaborators on billiards in the complement of bounded domains and their applications in aerodynamics and geometrical optics.We consider several problems related to aerodynamics of bodies in highly rarefied media. It is assumed that the medium particles do not interact with each other and are elastically reflected when colliding with the body boundary; these assumptions drastically simplify the aerodynamics and allow to reduce it to a number of purely mathematical problems.First we examine problems of minimal resistance in the case of translational motion of bodies. These problems generalize the Newton problem of least resistance; the difference is that the bodies are generally nonconvex in our case and therefore the particles can make multiple reflections from the body surface. It is proved that typically the infimum of resistance equals zero; thus, there exist 'almost perfectly streamlined' bodies.Next we consider the generalization of Newton's problem on minimal resistance of convex axisymmetric bodies to the case of media with thermal motion of particles. Two kinds of solutions are found: first, Newton-like bodies and second, shapes obtained by gluing together two Newton-like bodies along their rear ends.Further, we state results on characterization of billiard scattering by nonconvex and rough bodies; next we solve some special problems of optimal mass transportation. These two groups of results are applied to problems of minimal and maximal resistance for bodies that move forward and at the same time slowly rotate. It is found, in particular, that the resistance of a three-dimensional convex body can be increased at most twice and decreased at most by 3.05% by roughening its surface.Next, we consider a rapidly rotating rough disc moving in a rarefied medium on the plane. It is shown that the force acting on the disc is not generally parallel to the direction of the disc motion, that is, has a nonzero transversal component. This phenomenon is called Magnus effect (proper or inverse, depending on the direction of the transversal component). We show that the kind of Magnus effect depends on the kind of disc roughness, and study this dependence. The problem of finding all admissible values of the force acting on the disc is formulated in terms of a vector-valued problem of optimal mass transportation.Finally, we describe bodies that have zero resistance when translating through a medium, and state results on existence or non-existence of bodies with mirror surface invisible in one or several directions. We also consider the problem of constructing retroreflectors: bodies with specular surface that reverse the direction of any incident beam of light.

Fermi acceleration in rotating drums

“…In this subsection we will prove in Corollary 3.2 that the Knudsen reflection law (1.7) is the only random reflection distribution which can arise from classical specular reflections from a fractal surface consisting of small smooth crystals if we assume homogeneity and lack of memory, i.e., if we assume that the limiting distribution of the angle of reflection does not depend on the angle of incidence, and it is the same at all boundary points. Corollary 3.2 follows easily from results in [3,17,29,30,31], stated below as Propositions 3.1 and 3.3. The results in [3] rediscovered (independently) those in [30]; both articles were concerned with two-dimensional models.…”

“…The results in [3] rediscovered (independently) those in [30]; both articles were concerned with two-dimensional models. The full generality in an arbitrary number of dimensions was achieved in [29,31]. Our presentation will combine [3, Thms.…”

“…2]. For the multidimensional setup see [31]. We will be somewhat informal as the technical aspects of the model are tangential to our project.…”

“…We will limit our model to random reflections which are natural in two ways: (i) the microcanonical ensemble is an invariant measure for the model with random reflections and (ii) the random reflections are the limit of deterministic reflections from rough (fractal) surfaces, assuming that the size of small crystals forming the rough surface goes to zero. Reflection laws with these properties have been studied in [3,17,29,30,31]. We will recall the characterization of such laws in Section 3.1.…”

“…The relationship between rough billiard boundaries and random reflections (including Lambertian reflections; equivalently Knudsen law) was studied in [3,10,11,12,17,18,19,20,29,30,31].…”