2,002 research outputs found
Negative and Nonlinear Response in an Exactly Solved Dynamical Model of Particle Transport
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.Comment: corrected typos and minor reformulations; 28 pages (revtex) with 7
figures (postscript); accepted for publication in JS
Capturing correlations in chaotic diffusion by approximation methods
We investigate three different methods for systematically approximating the
diffusion coefficient of a deterministic random walk on the line which contains
dynamical correlations that change irregularly under parameter variation.
Capturing these correlations by incorporating higher order terms, all schemes
converge to the analytically exact result. Two of these methods are based on
expanding the Taylor-Green-Kubo formula for diffusion, whilst the third method
approximates Markov partitions and transition matrices by using the escape rate
theory of chaotic diffusion. We check the practicability of the different
methods by working them out analytically and numerically for a simple
one-dimensional map, study their convergence and critically discuss their
usefulness in identifying a possible fractal instability of parameter-dependent
diffusion, in case of dynamics where exact results for the diffusion
coefficient are not available.Comment: 11 pages, 5 figure
Deterministic diffusion in flower shape billiards
We propose a flower shape billiard in order to study the irregular parameter
dependence of chaotic normal diffusion. Our model is an open system consisting
of periodically distributed obstacles of flower shape, and it is strongly
chaotic for almost all parameter values. We compute the parameter dependent
diffusion coefficient of this model from computer simulations and analyze its
functional form by different schemes all generalizing the simple random walk
approximation of Machta and Zwanzig. The improved methods we use are based
either on heuristic higher-order corrections to the simple random walk model,
on lattice gas simulation methods, or they start from a suitable Green-Kubo
formula for diffusion. We show that dynamical correlations, or memory effects,
are of crucial importance to reproduce the precise parameter dependence of the
diffusion coefficent.Comment: 8 pages (revtex) with 9 figures (encapsulated postscript
Fractal dimension of transport coefficients in a deterministic dynamical system
In many low-dimensional dynamical systems transport coefficients are very
irregular, perhaps even fractal functions of control parameters. To analyse
this phenomenon we study a dynamical system defined by a piece-wise linear map
and investigate the dependence of transport coefficients on the slope of the
map. We present analytical arguments, supported by numerical calculations,
showing that both the Minkowski-Bouligand and Hausdorff fractal dimension of
the graphs of these functions is 1 with a logarithmic correction, and find that
the exponent controlling this correction is bounded from above by 1 or
2, depending on some detailed properties of the system. Using numerical
techniques we show local self-similarity of the graphs. The local
self-similarity scaling transformations turn out to depend (irregularly) on the
values of the system control parameters.Comment: 17 pages, 6 figures; ver.2: 18 pages, 7 figures (added section 5.2,
corrected typos, etc.
Understanding deterministic diffusion by correlated random walks
Low-dimensional periodic arrays of scatterers with a moving point particle
are ideal models for studying deterministic diffusion. For such systems the
diffusion coefficient is typically an irregular function under variation of a
control parameter. Here we propose a systematic scheme of how to approximate
deterministic diffusion coefficients of this kind in terms of correlated random
walks. We apply this approach to two simple examples which are a
one-dimensional map on the line and the periodic Lorentz gas. Starting from
suitable Green-Kubo formulas we evaluate hierarchies of approximations for
their parameter-dependent diffusion coefficients. These approximations converge
exactly yielding a straightforward interpretation of the structure of these
irregular diffusion coeficients in terms of dynamical correlations.Comment: 13 pages (revtex) with 5 figures (postscript
Understanding Anomalous Transport in Intermittent Maps: From Continuous Time Random Walks to Fractals
We show that the generalized diffusion coefficient of a subdiffusive
intermittent map is a fractal function of control parameters. A modified
continuous time random walk theory yields its coarse functional form and
correctly describes a dynamical phase transition from normal to anomalous
diffusion marked by strong suppression of diffusion. Similarly, the probability
density of moving particles is governed by a time-fractional diffusion equation
on coarse scales while exhibiting a specific fine structure. Approximations
beyond stochastic theory are derived from a generalized Taylor-Green-Kubo
formula.Comment: 4 pages, 3 eps figure
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