Distributional Response to Biases in Deterministic Superdiffusion

We report on a novel response to biases in deterministic superdiffusion. For its reduced map, we show using infinite ergodic theory that the time-averaged velocity (TAV) is intrinsically random and its distribution obeys the generalized arcsine distribution. A distributional limit theorem indicates that the TAV response to a bias appears in the distribution, which is an example of what we term a distributional response induced by a bias. Although this response in single trajectories is intrinsically random, the ensemble-averaged TAV response is linear.

Figure 1

(a) Asymmetric deterministic diffusion $T(x)$ ($z=3.0$ and $c=0.35$). (b) Its reduced map $R(x)$.

Figure 2

Mean square displacements ($z=1.9$ and $c=0.5$). Different symbols are the TAMSDs calculated by different initial points. Dashed and dotted lines are the EAMSDs based on equilibrium and ordinary ensembles, respectively, where an ordinary ensemble is a uniform ensemble on [0, 1] and an equilibrium ensemble is the points after ${10}^6$ times iterations. The slope of the solid line is the theoretical scaling (9).

Figure 3

Time-averaged drift ($z=3.0$, $c=0.45$, and $n={10}^5$). TADs are calculated by different initial points.

Figure 4

Probability density function of $\overline{V}$ ($z=2.5$ and $n={10}^7$). Numerical results are represented by the solid histograms with the theoretical PDFs (15) for $c=0.5$ and $c=0.4$, respectively. If there is no bias ($c=0.5$), the PDF is symmetric whereas it is not under bias.

Figure 5

Response to biases. Different curves are the response curves (16) for different $z$. Symbols are the results of numerical simulations in the inset figure.