Polygonal billiards and transport: diffusion and heat conduction.

Abstract

A detail study of the diffusive and heat conduction properties of a family of nonchaotic billiards is presented. For dynamical systems with dynamical instability the relation between transport properties and characteristic quantities of the chaotic dynamics naturally emerge. On the contrary, in dynamical systems without chaos (in the sense of exponential separation of nearby trajectories) much less is known. From numerical simulations we compute several quantities related to diffusion, such as the mean square displacement, the behavior of the hydrodynamic modes for long wavelengths, through the properties of the incoherent intermediate scattering function and the velocity autocorrelation function, in connection with the Green-Kubo formula. The analysis of all these quantities indicates that some systems among the family studied have normal diffusion and others anomalous diffusion. The spectral measure associated with the velocity autocorrelation function is also studied. The same analysis reveals that for all the systems treated there is not a well defined super Burnett coefficient. The heat conduction is also explored and found that, naturally, it is valid for the systems that behave diffusively.