Negative and Nonlinear Response in an Exactly Solved Dynamical Model of Particle Transport

Abstract

We consider a simple model of particle transport on the line defined by a dynamical map F satisfying F(x+1) = 1 + F(x) for all x in R and F(x) = ax + b for |x| < 0.5. Its two parameters a (`slope') and b (`bias') are respectively symmetric and antisymmetric under reflection x -> R(x) = -x. Restricting ourselves to the chaotic regime |a| > 1 and therein mainly to the part a>1 we study not only the `diffusion coefficient' D(a,b), but also the `current' J(a,b). An important tool for such a study are the exact expressions for J and D as obtained recently by one of the authors. These expressions allow for a quite efficient numerical implementation, which is important, because the functions encountered typically have a fractal character. The main results are presented in several plots of these functions J(a,b) and D(a,b) and in an over-all `chart' displaying, in the parameter plane, all possibly relevant information on the system including, e.g., the dynamical phase diagram as well as invariants such as the values of topological invariants (kneading numbers) which, according to the formulas, determine the singularity structure of J and D. Our most significant findings are: 1) `Nonlinear Response': The parameter dependence of these transport properties is, throughout the `ergodic' part of the parameter plane (i.e. outside the infinitely many Arnol'd tongues) fractally nonlinear. 2) `Negative Response': Inside certain regions with an apparently fractal boundary the current J and the bias b have opposite signs.