Abstract
We consider a simple model of particle transport on the line ℝ defined by a dynamical map F satisfying F(x+1)=1+F(x) for all x∈ ℝ and F(x)=ax+b for |x|<1/2. 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, along the lines of previous investigations [R. Klages and J. R. Dorfman, Phys. Rev. Lett. 74:387 (1995)] on the restricted, symmetric (b=0) one-parameter version of the present model, the parameter dependence of the transport properties, i.e., not only of the “diffusion coefficient” D(a,b), but this time also of the “current” J(a,b). A major difference however is that this time an important tool for such a study has been available, in the form of exact expressions for J and D 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 of our present preliminary survey of the parameter plane of the model 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, in principle all possibly relevant information on the system including, e.g., the dynamical phase diagram as well as, by way of illustration, values of some topological invariants (kneading numbers) which, according to the formulas, determine the singularity structure of J(a,b) and D(a,b). What we regard as 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.
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Groeneveld, J., Klages, R. Negative and Nonlinear Response in an Exactly Solved Dynamical Model of Particle Transport. Journal of Statistical Physics 109, 821–861 (2002). https://doi.org/10.1023/A:1020487219455
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DOI: https://doi.org/10.1023/A:1020487219455