Simple Maps with Fractal Diffusion Coefficients

Abstract

We consider chains of one-dimensional, piecewise linear, chaotic maps with uniform slope. We study the diffusive behaviour of an initially nonuniform distribution of points as a function of the slope of the map by solving Frobenius-Perron equation. For Markov partition values of the slope, we relate the diffusion coefficient to eigenvalues of the topological transition matrix. The diffusion coefficient obtained shows a fractal structure as a function of the slope of the map. This result may be typical for a wide class of maps, such as two dimensional sawtooth maps.Comment: 4 pages, LaTeX with REVTeX, and 3 figures, Postscript, in uuencoded tar file. to appear in Phys. Rev. Let

Fractality of deterministic diffusion in the nonhyperbolic climbing sine map

“…For simple one-dimensional hyperbolic maps it was shown that the diffusion coefficient is typically a fractal function of control parameters [37,38,39,40]. Subsequently an analogous behavior was detected for other transport coefficients [41,42], and in more complicated models [27,28,29].…”

“…For all three maps there is a very analogous oscillatory behavior of the parameter-dependent diffusion coefficient. These oscillations can be explained in terms of the changes of the microscopic dynamics under parameter variation, that is, whenever there is a local maximum there is an onset of strong backscattering in the dynamics yielding a local decrease of the diffusion coefficient in the parameter, and vice versa at local minima [37,38,39,40]. However, the five Markov partition series for the climbing sine diffusion coefficient already indicate that there are more irregularities on finer scales.…”

“…One can now identify an infinite series of parameter values corresponding to a certain type of Markov partition [37,38,39,40]. For parameter values which belong to such a Markov partition series the corresponding invariant densities ρ * (x) have a very similar functional form.…”

“…There exist various efficient numerical as well as, for some system parameters, analytical methods to exactly compute diffusion coefficients for piecewise linear hyperbolic maps, such as transition matrix methods based on Markov partitions [37,38,39,40], cycle expansion methods [15,16,17,18], and more recently a very powerful method related to kneading sequences [42].…”

Deterministic chaos and the foundations of the kinetic theory of gases

“…[53,54]. Here we will consider the shape of the diffusion coefficient, D, as a function of the slope a for a > 2.…”