Nonlinear dynamics and chaos in classical Coulomb-interacting many-body billiards

Chaos and nonlinear dynamics of single-particle Hamiltonian systems have been extensively studied in the past; however, less is known about interacting many-body systems in this respect even though all physical systems include particle-particle interactions in one way or another. To study Hamiltonian chaos, two-dimensional billiards are usually employed, and due to the realization of billiards in semiconductor quantum dots, the electrostatic Coulomb interaction is the natural choice for the interparticle interaction. Yet, surprisingly little is known about chaos and nonlinear dynamics of Coulomb-interacting many-body billiards. To address the challenging problems of interacting many-body billiards, we have developed a flexible and expandable code implementing methods previously used in molecular dynamics simulations. The code is \emph{generic} in sense that it is readily applicable to most two-dimensional billiards -- including periodic systems -- with different types of interparticle interactions. In this work, insights into Coulomb-interacting billiards are gained by applying the methods to two relevant systems: a two-particle circular billiards and a few-particle diffusion, the latter of which is studied only as a closed system. Also general implications of the results for other systems are discussed. The circular billiards is studied with the interaction strength varying from the weak to the strong-interaction limit. Bouncing maps show quasi-regular features in the weak and strong-interacting limits. In the strong-interaction regime an analytical model for the phase space trajectory is derived, and the model is found to agree with the simulated data. At intermediate interaction strengths the bouncing maps get filled. To obtain a quantitative view on the hyperbolicity and stickiness of the circular billiards, we calculate escape-time distributions of open circular billiards. At weak interactions the escape-time distributions show a power-law tail owing to the quasi-regular dynamics arising from the integrable non-interacting limit. At intermediate interaction strengths the distributions are exponential implying hyperbolicity within the studied time-scales. As the second application, the diffusion process between two square containers connected by a short channel is studied under a homogeneous magnetic field perpendicular to the table. During the propagation, over half of the particles -- all initially in the same container -- travel from one container to the other. The time this process takes is defined here as the relaxation time. The average relaxation times are calculated as a function of the effective Larmor radius, which describes the average effect of the magnetic field on the particles. The behavior of the average relaxation times as a function of the effective Larmor radius is studied thoroughly for different interaction strengths and channel widths. Interestingly, the graphs show a universal minimum for all interaction strengths, and in the weak-interaction limit also other extrema appear. The new extrema in the weak-interaction limit are explained by calculating properties of open single-particle magnetic square billiards for different Larmor radii.