Butterfly effect and spatial structure of information spreading in a chaotic cellular automaton

Inspired by recent developments in the study of chaos in many-body systems, we construct a measure of local information spreading for a stochastic cellular automaton in the form of a spatiotemporally resolved Hamming distance. This decorrelator is a classical version of an out-of-time-order correlator studied in the context of quantum many-body systems. Focusing on the one-dimensional Kauffman cellular automaton, we extract the scaling form of our decorrelator with an associated butterfly velocity $v_b$ and a velocity-dependent Lyapunov exponent $\lambda \left(v\right)$. The existence of the latter is not a given in a discrete classical system. Second, we account for the behavior of the decorrelator in a framework based solely on the boundary of the information spreading, including an effective boundary random walk model yielding the full functional form of the decorrelator. In particular, we obtain analytic results for $v_b$ and the exponent $\beta$ in the scaling ansatz $\lambda \left(v\right)\sim \mu {(v-v_b)}^{\beta }$, which is usually only obtained numerically. Finally, a full scaling collapse establishes the decorrelator as a unifying diagnostic of information spreading.

Figure 1

Light-cone structure of the decorrelator $D(x,t)$ with $N=2048$ and $K=4$, with a single spin flip at the origin $x=0$ when $t=0$. (a) The frozen phase at $p=0.11$ and (b) the chaotic phase at $p=0.40$. In the chaotic phase, the two dashed lines are $v_b$ and $v_{\max }$, respectively. Inset: Mean-field prediction (dashed line) of the long-time Hamming distance $H_{\infty }=H(t\to \infty )$ as a function of $p$, compared with numerical data. $p_c=0.13$. Close agreement is observed for $p>0.27$.

Figure 2

The decorrelator $D(x,t)$ along rays of different velocities for $p=0.4$, which is representative of all $p>p_c$. As $lnD=-\lambda \left(v\right)t=-\mu {(v-v_b)}^{\beta }t$, the slopes represent $-\mu {(v-v_b)}^{\beta }$. Thus, $v_b=2.9$ where the slope is zero. We compare the full $D(x,t)$ numerical data (solid lines) and the prediction (dashed lines) of the boundary random walk model.

Figure 3

Scaling collapse performed for a range of $p$ according to Eq. (14). At each specific $p$, the data are obtained by taking the value of $D$ in the $t\to \infty$ limit as permitted by computational power. The $x$ axis is rescaled to $\sqrt{\mu }(v-v_b)$, where both $\mu$ and $v_b$ are obtained by the slope method exemplified in Fig. 2, assuming $\beta =2$ in the scaling form. The black dashed line is obtained by numerically evaluating $D(x,t)$ from Eq. (11) for $p=0.5$. Inset: Comparison between $v_b$ obtained from the full $D(x,t)$ data (blue dots), from the boundary velocity (green line), and from the analytic calculation (8) (orange line).

Figure 4

Scaled probability density of boundary positions (defined in the text) of ${10}^4$ different initial configurations plotted against time. The black dotted line is the Gaussian profile predicted by the random walk model. The earliest two times ($t=50,100$) show slight deviations from the equilibrium/long-time limit. Top inset: Unscaled boundary probability density at $t=50,100,\cdots ,300$. Bottom inset: Raw data of boundary positions plotted against time. The red line is the average boundary, with the velocity $v=x/t=2.9$.