Signatures of chaos in time series generated by many-spin systems at high temperatures

Extracting reliable indicators of chaos from a single experimental time series is a challenging task, in particular, for systems with many degrees of freedom. The techniques available for this purpose often require unachievably long time series. In this paper, we explore a method of discriminating chaotic from multi-periodic integrable motion in many-particle systems. The applicability of this method is supported by our numerical simulations of the dynamics of classical spin lattices at high temperatures. We compared chaotic and nonchaotic regimes of these lattices and investigated the transition between the two. The method is based on analyzing higher-order time derivatives of the time series of a macroscopic observable—the total magnetization of the spin lattice. We exploit the fact that power spectra of the magnetization time series generated by chaotic spin lattices exhibit exponential high-frequency tails, while, for the integrable spin lattices, the power spectra are terminated in a non-exponential way. We have also demonstrated the applicability limits of the above method by investigating the high-frequency tails of the power spectra generated by quantum spin lattices and by the classical Toda lattice.

Figure 1

Numerical values of the maximum Lyapunov exponent ${\lambda }_{\max }$ for the chaotic (a) and the integrable (b) systems as a function of the reset time $\tau$ (dots). The lines represent the fits by a constant in (a) and by function $\frac{log(1+{\tau }^{\beta })}{\tau }$ in (b). The insets illustrate the growth of the distance between a pair of trajectories during five reset intervals.

Figure 2

Comparison between IE rates ${\mathit{h}}_{Sh}(\epsilon ,\tau ,N)$ produced by the chaotic system (solid lines) and by the integrable system (dashed lines) for $N=2$ and different values of $\tau$.

Figure 3

Comparison between ${\mathit{h}}_{\mathrm{CP}}(\epsilon ,\tau ,N)$ with $\tau =0.1$ and various values of $N$ for the chaotic time series (solid lines) and the integrable time series (dashed lines). Curves reaching higher maximum values of ${\mathit{h}}_{\mathrm{CP}}$ correspond to the smaller values of $N$.

Figure 4

Time series of $M_x\left(t\right)$ for $6\times 6\times 6$ spin clusters. Figures in the left column represent the chaotic cluster: (a) fragment of the time series $M_x\left(t\right)$; (c) the corresponding seventh time derivative $M_x^{\left(7\right)}\left(t\right)$; and (e) the power spectra $P\left(\omega \right)$ (blue) and $P^{\left(7\right)}\left(\omega \right)$ (red). (b), (d), and (f) in the right column represent the same quantities for the integrable Ising cluster. The black dashed line in (e) depicts fit to $P^{\left(7\right)}\left(\omega \right)$ of the form $P_0{\omega }^{14}{e}^{-\gamma \omega },$ where $P_0$ and $\gamma$ are fitting parameters. The power spectra were obtained as described in Appendix pp2. In (e), they were also smoothed out. The tails of the power spectra at $\omega /2\pi >4$ in (e) and $\omega /2\pi >0.6$ in (f) are affected by the spectral leakage from lower frequencies due to the finite length of the time series.

Figure 5

Statistical characteristics of the time series of $M_x^{\left(n\right)}\left(t\right)$ for the chaotic and the integrable systems: (a) ratios $R_n$ of the RMS value of $M_x^{\left(n\right)}\left(t\right)$ to the RMS value of $M_x^{(n-1)}\left(t\right)$; (b) relative width $W_n$ of the power spectrum $P^{\left(n\right)}\left(\omega \right)$ with respect to the width of $P\left(\omega \right)$.

Figure 6

${\mathit{h}}_{\mathrm{CP}}(\epsilon ,\tau ,10)$ calculated for both $M_x\left(t\right)$ (blue) and $M_x^{\left(7\right)}\left(t\right)$ (red) as described in the text. The values of $\varepsilon$ for ${\mathit{h}}_{\mathrm{CP}}(\epsilon ,\tau ,10)$ associated with the seventh derivative were rescaled by dividing the true values by the following factors: $1.13\times {10}^4$ in the chaotic case and $1.31\times {10}^3$ in the integrable case. These factors were chosen such to make the spread between the maximum and the minimum values of $M_x^{\left(7\right)}\left(t\right)$ equal to that of the corresponding $M_x\left(t\right)$, thereby, in effect, compensating for the different dimensions of $M_x\left(t\right)$ and $M_x^{\left(7\right)}\left(t\right)$.

Figure 7

(a) and (b) represent noisy versions of the time series shown in Figs. 4 and 4, respectively. The notations are also the same as in Fig. 4. Plots (c) and (d) show the same time series after being filtered at the cutoff frequency 1 Hz (solid blue lines) and compare them to the original noise-free time series (dashed gray lines). (e) and (f) show the fourth-order derivatives of the filtered time series (solid blue lines) and compare them to the derivatives of the original noise-free time series (dashed gray lines).

Figure 8

Time series of $M_x\left(t\right)$ for $4\times 4$ spin lattices. The notations and the interactions constants are the same as in Fig. 4. (a)–(d) now represent the entire time series used to obtain (e), (f).

Figure 9

Power spectra for 50 realizations of $J_x,J_y,J_z$ in the vicinity of the Ising limit subject to the constraint $J_x^2+J_y^2+J_z^2=1$ for a cubic spin lattice consisting of $16\times 16\times 16$ spins. The specific realizations of $J_x,J_y,J_z$ were selected from the data set used in Ref. [18]. The corresponding values of ${\lambda }_{\max }$ decrease monotonically by more than two orders of magnitude from the upper power spectra to the lower power spectra. The values of ${\lambda }_{\max }$ themselves are shown in the inset. They were computed in Ref. [18].

Figure 10

(a) The power spectra of a Toda lattice consisting of 32 particles with random initial conditions. (b) Same for the initial conditions chosen according to Eq. (11). (c) and (d) represent the integrals of motion $I_n$ as a function of time for the initial conditions of (a) and (b), respectively.

Figure 11

Power spectra for different quantum spin 1/2 lattices computed from the time series representing the expectation values for the operator $\widehat{M_x}$. The lengths of the time series are greater than 2000. The initial states are selected randomly from the respective infinite-temperature ensembles. The systems are (a) Nonintegrable 16-spin chain with nearest neighbor coupling coefficients $J_x=0.267,J_y=0.535$, and $J_z=-0.802$ and next-nearest-neighbor coupling of strength 30% of the nearest-neighbor coupling. (b) Square $5\times 4$ lattice with XXZ coupling coefficients $J_x=J_y=0.433$ and $J_z=-0.356$ (c) Integrable XXZ chain consisting of 16 spins with coupling coefficients $J_x=J_y=0.612$ and $J_z=-0.5$. (d) Integrable $XX$ chain consisting of 16 spins with coupling coefficients $J_x=J_y=0.707$ and $J_z=0$.

Figure 12

Plots of $M_x\left(t\right)$ and its nine derivatives for the chaotic system (left) and the integrable system (right). Note: the ninth derivative for the integrable case exhibits extrinsic noise due to numerical rounding errors.

Figure 13

Comparison between two power spectra for the same setting as in Fig. 4 obtained with single machine precision (solid red line) and double machine precision (dashed blue line).

Figure 14

(a) ${\lambda }_{\max }\left(k\tau \right)$ as defined in Eq. (2), computed for a Toda lattice consisting of 32 particles with exponential potential (blue), exponential potential truncated up to the third order (red) and exponential potential truncated up to the fifth order (brown). (b) The eigenvalues of the $L$ matrix defined in Eq. (12) for the third-order truncated potential.