Normal and Anomalous Diffusion in Soft Lorentz Gases

Engineered nanoscale structures known as artificial graphene exhibit the properties of real graphene but in a setup where it is easy to tune features such as the electronic density, lattice constant, geometry or coupling with the environment. The topology of such structures is the same as a paradigmatic model in dynamical systems theory – the periodic Lorentz gas depicting the motion of electrons in a purely classical metal. In the model, a point particle scatters elastically from a set of fixed hard spheres distributed either randomly or periodically in space. Particle dynamics in the Lorentz gas exhibit chaos and produce well-behaved transport behaviour, including Ohm’s law.
The growing power of nanoscience to design graphene-like systems encourages the study of closely related models. For example, in soft Lorentz gases, the hard spheres get replaced by more realistic potentials, such as the Fermi potential. In a new study, LML External Fellow Rainer Klages and colleagues systematically explore transport properties in a two-dimensional class of such models, and find quite striking differences from the classic Lorentz gas. In the classic model, the diffusion coefficient D characterising particle motion rises monotonically as one pushes the scattering centers apart, increasing the distance w between adjacent disks. In contrast, the authors find that the softer potentials lead to very different behaviour: D still increases with w but in a highly complicated way, rising and falling repeatedly. Moreover, for w having values within certain small intervals – most likely an infinite set of such intervals – the diffusion coefficient even fails to exist. This indicates regions where the particle motion shows anomalous diffusion – the mean square displacement of the particle grows faster than linearly in time.
As the authors demonstrate, the general rise of D with w can be understood on the basis of a simple random walk model, but the finer detail has a more interesting origin, being intimately linked to periodic orbits of particles within the structure. In some cases, these periodic orbits trap a particle in the vicinity of one or a few local disks. In other cases, periodic motion as a particle reflects repeatedly from a disk to one side or another effectively guides the particle down a narrow channel, making it cover a long distance quickly. Perhaps not surprisingly, they find that these two distinct kinds of orbits – one might call them trapping orbits and channeling orbits – appear near values of w where the diffusion coefficient is unusually small or unusually large. The former corresponds to diffusive motion along chaotic orbits that are nearly but not quite trapped, and the latter to chaotic orbits that are almost, but not quite, effectively channeled. Hence, the set of periodic orbits within the scattering topology exerts a direct influence on the fine features of how D grows with w.
The paper is available at

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