Dynamics of two interacting particles in classical billiards

Abstract

The problem of two interacting particles moving in a ddimensional billiard is considered here. A suitable coordinate transformation leads to the problem of a particle in an unconventional hyperbilliard. A dynamical map can be readily constructed for this general system, which greatly simplifies calculations. As a particular example, we consider two identical particles interacting through a screened Coulomb potential in a one-dimensional billiard. We find that the screening plays an important role in the dynamical behavior of the system and only in the limit of vanishing screening length can the particles be considered as bouncing balls. For more general screening and energy values, the system presents strong non-integrability with resonant islands of stability.A system of two interacting bodies moving in an otherwise free space, is one of the few integrable problems known. The reduction to the one-body central force problem allows a solution by quadratures [1]. However, once the translational symmetries are broken, as when the system is placed inside a billiard, the center-of-mass (CM) and angular momenta are in general no longer constants of motion. In this case, the classical dynamics of the system may be chaotic even when the geometry yields an otherwise fully integrable one-particle case, as we shall see below.On the other hand, recent experimental realizations of billiards, such as suitably shaped resonators and quantum dots [2,3], have allowed the study of the quantum manifestations of well-known classical non-integrability in some billiards [4,5]. In the case of quantum dots, disagreement between theory and experiment has been attributed to geometrical factors [3]. A considerable amount of theoretical work exists on the effect that geometry has on the integrability of dynamics in billiards [4,[6][7][8], as well as on their quantum analogs [5,9]. However, the possibility of more than one particle in the quantum dot leaves the usual one-particle approach incomplete. In fact, some experiments have pointed out the importance of electron-electron interaction on various features observed in such mesoscopic systems [10]. In this article, we explore the role of the electrostatic interaction introduced when two particles are in the billiard. A formalism for billiards in any dimensions is developed, and as an example, we apply it to the one-dimensional case. Since we are interested in the role of the electrostatic interaction in mesoscopic systems, we consider particles interacting through a screened Coulomb potential.The hyperbilliard. The problem of two point masses moving along a finite line and suffering elastic impacts with the end walls and between themselves, can be transformed to the motion of one 'particle' moving in a triangular billiard. The coordinates of the particle in this billiard are the coordinates of the original masses. The ratio of the masses determines the integrability of the system [6], being regularizable for a particle mass ratio of 1 and 3 (or 1 3 ) [7]. We now introduce an interaction be...

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