 02303 J. Groeneveld and R. Klages
 Negative and Nonlinear Response in an Exactly Solved Dynamical Model of Particle Transport
(3232K, 28 pages (latex/revtex) with 7 figures (postscript))
Jul 9, 02

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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
overall `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.
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