Criticality and long-range correlations in time series in classical and quantum systems

We present arguments which indicate that a transitional state in between two different regimes implies the occurrence of 1/$f$ time series and that this property is generic in both classical and quantum systems. Our study focuses on two particular examples: the one-dimensional module-1 logistic map and nuclear excitation spectra obtained with a schematic shell-model Hamiltonian. We suggest that a transitional point is characterized by the long-range correlations implied by 1/$f$ time series. We apply a Fourier spectral analysis and the detrended fluctuation analysis method to study the fluctuations to each system.

Figure 1

The self-affine time series of the Module-1 Logistic Map of Eq. (3) with $k=1$, consists of bunches of peaks (upper panel) that contain more bunches on a smaller scale (bottom panel).

Figure 2

The correlation strength $\beta$ of time series $x_n\left(k\right)$ as a function of $k$ in the map of Eq. (3). The blue line is the spectral density exponent $\beta$ (in the figure as ${\beta }_{\mathit{PS}}$) from Eq. (1), whereas the red line is $\beta =2{\alpha }_{\mathit{DFA}}-1$(in the figure as ${\beta }_{\mathit{DFA}}$) from Eq. (2).

Figure 3

The values of $\beta$ calculated directly from the fit to the slope of the power spectral density (blue dots) to each ${\delta }_n$ in function of $\chi$ in the Hamiltonian of Eq. (6). The red squares are the ${\beta }^{\prime }s$ obtained using the relation with ${\alpha }_{\mathit{DFA}}$.