Fractal fractal dimensions of deterministic transport coefficients

Abstract

If a point particle moves chaotically through a periodic array of scatterers the associated transport coefficients are typically irregular functions under variation of control parameters. For a piecewise linear two-parameter map we analyze the structure of the associated irregular diffusion coefficient and current by numerically computing dimensions from box-counting and from the autocorrelation function of these graphs. We find that both dimensions are fractal for large parameter intervals and that both quantities are themselves fractal functions if computed locally on a uniform grid of small but finite subintervals. We furthermore show that there is a simple functional relationship between the structure of fractal fractal dimensions and the difference quotient defined on these subintervals.

Continuity properties of transport coefficients in simple maps

“…Particularly, we numerically analyze local variations of these properties. Our work corrects and amends previous results reported in [30,35].…”

“…6 In [30] data sets were generated from these expressions and analyzed by standard numerical box counting [41]. This procedure relies on the assumption that…”

“…Analysing D(a) = D(a, 0) on 2 ≤ a ≤ 8, see Fig. 1 (a), based on 10 6 data points uniformly distributed in a yielded a box dimension of B ≃ 1.039 [30]. The inset in Fig.…”

Fractal dimension of transport coefficients in a deterministic dynamical system

“…We also proposed a conjecture that the exponent controlling this correction depends on the slope of the map and equals either 1 or 2, depending on existence and detailed properties of a Markov partition. That ∆ = 1 seems to be at odds with the above-cited results of [11], because the Minkowski-Bouligand dimension, if exists, is equivalent to the box-counting dimension [24]. However, one should note that while in our studies we focused on the point-wise dimension calculated only for those system control parameters that correspond to finite Markov partitions, Klages and Klauß computed the box-counting dimension on intervals of finite length, and these two values need not be the same.…”

“…Klages and Klauß [11] have recently used this exact solution to examine in detail irregular dependency of the drift velocity and diffusion coefficient on the control parameters of the system. Their aim was to verify an earlier hypothesis [12,13] that the graphs of these quantities as functions of the map slope are so irregular that actually they form fractals.…”