Fractional-Power-Law Level Statistics Due to Dynamical Tunneling

For systems with a mixed phase space we demonstrate that dynamical tunneling universally leads to a fractional power law of the level-spacing distribution $P(s)$ over a wide range of small spacings $s$. Going beyond Berry-Robnik statistics, we take into account that dynamical tunneling rates between the regular and the chaotic region vary over many orders of magnitude. This results in a prediction of $P(s)$ which excellently describes the spectral data of the standard map. Moreover, we show that the power-law exponent is proportional to the effective Planck constant $h_{\mathrm{eff}}$.

Figure 1

Level-spacing distribution $P(s)$ at $h_{\mathrm{eff}}=1/100$ of the standard map [36], with the classical phase space shown in the inset. We compare the numerical data [(green) histogram] to the analytical prediction, Eq. (4), (black solid line) and the Berry-Robnik result [14] (black dashed line) (a) on a linear and (b) on a double-logarithmic scale. The typical couplings $2v_m$ (triangles) mark the power-law regime [straight (red) line]. Below the smallest typical coupling $2v_0$ one finds linear level repulsion.

Figure 2

Comparison of the level-spacing distribution $P(s)$ for the kicked system [37], with the classical phase space shown in the inset. The numerical data [(green) histograms], the analytical prediction, Eq. (4), (black solid lines), and the Berry-Robnik result [14] (black dashed line) are shown for $h_{\mathrm{eff}}=1/80$ (lower curves), $h_{\mathrm{eff}}=1/120$ (middle curves), and $h_{\mathrm{eff}}=1/160$ (upper curves). The corresponding typical couplings $2v_m$ (triangles, squares, and circles) mark the power-law regime [straight (red) lines]. The gray solid lines illustrate contributions from increasing $v_m$ by incrementing the upper summation index in Eq. (11) from 0 to $N_r-1$ for $h_{\mathrm{eff}}=1/80$. This gives rise to a staircase function.