Calculation of Superdiffusion for the Chirikov-Taylor Model

It is widely known that the paradigmatic Chirikov-Taylor model presents enhanced diffusion for specific intervals of its stochasticity parameter due to islands of stability, which are elliptic orbits surrounding accelerator mode fixed points. In contrast with normal diffusion, its effect has never been analytically calculated. Here, we introduce a differential form for the Perron-Frobenius evolution operator in which normal diffusion and superdiffusion are treated separately through phases formed by angular wave numbers. The superdiffusion coefficient is then calculated analytically resulting in a Schloemilch series with an exponent $\beta =3/2$ for the divergences. Numerical simulations support our results.

Figure 1

Measurements of $\theta$ ($n=100$) versus ${\theta }_0$ from numerical simulations performed for $N=5000$, $K=1.1\times 2\pi$, $I_0=2\pi$, and ${\theta }_0$ uniformly distributed on $[-\pi ,\pi )$. Accelerated orbits cluster along the diagonal $\theta ={\theta }_0$ on the gaps located inside the interval $\pi /2<|{\theta }_0|<\arccos (-4/K)\approx 2.188$.

Figure 2

Theoretical diffusion rate $D/D_{ql}$ (solid line) compared with normal rate $D_{\mathrm{nor}}/D_{ql}$ (dotted line) and measurements of $D/D_{ql}$ from numerical simulations ($△$) for $n=100$ and $N=5000$. First and second theoretical diffusion peaks occur for values 14.63 ($K=6.30$) and 6.07 ($K=12.58$), respectively.