Deterministic diffusion in smooth periodic potentials

Abstract

Understanding the macroscopic properties of matter, based on the microscopic interactions of the single particles requires to bring together the areas of statistical physics and dynamical systems. For deterministic diffusion one of the most prominent models is the Lorentz gas in which a point particle performs specular reflections with hard disks distributed in the plane. This model generates deterministic chaos and has led to many mathematical results revealing the origin of diffusion starting from chaotic dynamics. For the periodic Lorentz gas on a triangular lattice, it is possible to understand the diffusion coefficient, in the limit of high scatterer densities, in terms of random walk approximations. The key question addressed in this thesis is: What happens to the diffusion coefficient, as a function of control parameters, if the hard potential walls of the Lorentz gas scatterers are replaced by a soft potential? In this study we use a repulsive Fermi potential from which the hard limit can be recovered by varying a control parameter. We then performed computer simulations and analytical random walk approximations to understand the functional form of the diffusion coefficient as a function of the following parameters: the minimal distance between two scatters, the softness of the potential and the energy of a moving particle. Our main results is that the diffusion coefficient is a highly irregular function of each of these control parameters. Under certain assumptions one can construct analytical approximations that describe the coarse shape of the diffusion coeffi- cient when it exists: For high densities of scatterers we develop suitable random walk approximations, in the low density regime we apply a more elaborate argument that tests for memory effects. We find that diffusion in our soft Lorentz gas exhibits different random walk regimes, where either randomization characterizes the evolution of diffusion or spatio-temporal correlations take place. Via Poincare surfaces of section we show that the irregularities appearing in the diffusion coef- cient, as a function of parameters, which strongly deviate from simple random walk dynamics, come from non-trivial quasi-ballistic trajectories in con guration space.