Localization of information driven by stochastic resetting

Abstract

The dynamics of extended many-body systems are generically chaotic. Classically, a hallmark of chaos is the exponential sensitivity to initial conditions captured by positive Lyapunov exponents. Supplementing chaotic dynamics with stochastic resetting drives a sharp dynamical phase transition: we show that the Lyapunov spectrum, i.e., the complete set of Lyapunov exponents, abruptly collapses to zero above a critical resetting rate. At criticality, we find a sudden loss of analyticity of the velocity-dependent Lyapunov exponent, which we relate to the transition from ballistic scrambling of information to an arrested regime where information becomes exponentially localized over a characteristic length diverging at criticality with an exponent $ν= 1/2$. We illustrate our analytical results on generic chaotic dynamics by numerical simulations of coupled map lattices.

Figures (2)

• Figure 1: Velocity-dependent Lyapunov exponent (VDLE), $\tilde{\lambda}(v)$, in the CLM model [Eq. (\ref{['eq:CLM']})] with $a = 4$, $c = 0.1$, for increasing resetting rates $r$ indicated in the key. The VDLE ansatzes in Eq. (\ref{['eq:VDLE_ansatz']}) and below Eq. (\ref{['eq:renewal_cont_lattice']}) are validated by comparing data for $L = 401$, $t=200$ (lines) with $L=801$, $t=400$ (crosses). When $r \leq r_{\rm c} \approx 0.37(2)$, $\tilde{\lambda}(v) = \lambda(v) + \log(1-r)$ as predicted in Eq. (\ref{['eq:lambda_v_shift']}). When $r > r_{\rm c}$, $\tilde{\lambda}(v)$ develops a non-analytic cusp at $v=0$, with linear branches predicted in Eq. (\ref{['eq:linear_branch']}). Inset: log-log plot of the $r=0$ data, consistent with the small-velocity expansion in Eq. (\ref{['eq:expansion']}).
• Figure 2: Variation of the Lyapunov spectrum across the dynamical phase transition driven by stochastic resetting. The analytical results in Eq. (\ref{['eq:spectrum_r']}) are tested in the CLM model [Eq. (\ref{['eq:CLM']})]. For resetting rates $r < r_{\rm c} \approx 0.37(2)$, $\Delta \Lambda(k) := \Lambda(k) - \tilde{\Lambda}(k)$ is independent of $k$ for all $k$ and falls on $-\log(1-r)$ shown in dashed red. In contrast, for $r > r_{\rm c}$, the collapse of $\tilde{\Lambda}(k)$ manifests itself by the dependence of $\Delta \Lambda(k)$ on $k$ and its independence on $r$. The blue dashed line is a horizontal guide for the eye. The spectra are computed for $L=401$ at time $t=200$, with a temporal average over $\Delta t = 10$. The transition between the two regimes sharpens when simultaneously increasing $L$ and $t$. The other parameters are as in Fig. \ref{['fig:VDLE']}.