Statistical prethermalization in randomly kicked many-body classical rotor system

TL;DR

The study addresses whether Floquet-like prethermalization persists under aperiodic driving in a classical many-body rotor system. It introduces three driving protocols with waiting-time distributions—Poisson, Binomial, and Thue-Morse—to quantify randomness via average surprisal $S$ and analyzes kinetic-energy growth, diffusion, and prethermal lifetimes. The results show a dichotomy: Poisson and Binomial driving produce a pseudo-prethermal regime with algebraic heating suppression and a diffusion constant renormalized by $S$, while Thue-Morse driving yields a robust regular prethermal regime with exponential heating suppression; lifetimes scale as $t^*\sim 1/(S K^2)$ for random drives and $t^*\sim e^{c/K}$ for TM drive, supported by energy-time uncertainty arguments. This framework demonstrates that aperiodic driving can sustain long-lived non-equilibrium states in classical many-body systems and connects heating times and diffusion to WTD surprisal and phase-space entropy changes, with potential implications for controlling heating in driven systems.

Abstract

We explore the phenomena of prethermalization in a many-body classical system of rotors under aperiodic drives characterised by waiting time distribution (WTD), where the waiting time is defined as the time between two consecutive kicks. We consider here two types of aperiodic drives: random and quasi-periodic. We observe a short-lived pseudo-thermal regime with algebraic suppression of heating for the random drive where WTD has an infinite tail, as observed for Poisson and binomial kick sequences. On the other hand, quasi-periodic drive characterised by a WTD with a sharp cut-off, observed for Thue-Morse sequence of kick, leads to prethermal region where heating is exponentially suppressed. The kinetic energy growth is analyzed using an average surprise associated with WTD quantifying the randomness of drive. In all of the aperiodic drives we obtain the chaotic heating regime for late time, however, the diffusion constant gets renormalized by the average surprise of WTD in comparison to the periodic case.

Figures (12)

• Figure 1: Schematic plot for the equivalent circuit model for a Trotterized schedule for global random kicks. The particles undergo free evolution denoted by the green nodes and interact with the light blue or grey nodes. The grey nodes represent discrete times with missing kicks. Each free evolution is connected in space with its nearest neighbour nodes. The minimum time gap between the kicks is $\tau$ and the frequency of the missing kicks is characterized by a single parameter waiting time distribution (WTD). For the generalized Chirkov map model considered here, the equation of motion representation of the nodes is given in Eq. \ref{['eq_motion']}.
• Figure 2: (a) WTD for Poisson distribution with $\gamma = 1$ having the surprisal $S_{\gamma}=S\approx1$. (b) Variation of the average kinetic energy per rotor with time for different values of $K$ and $\gamma$ with the exponential WTD. (c) Data collapse of the kinetic energy curves with rescaled time axis by $SK^2t$. (d) Scaling of the heating time $t^*$ with $1/K$ in log-log plot.
• Figure 3: (a) The blue bars indicate the normalized frequency of the integer waiting time $l$ corresponding to kicks from a Binomial distribution with probability of kicks, $p=0.63$ and surprisal $S=1.046$. The corresponding normalized distribution perfectly matches the analytically found expression indicated with a dotted line. (b) Variation of the average kinetic energy with time $t$ for different values of $K$ and $p$ for the binomial kicking sequence. (c) The collapse of the kinetic energy curves for different $K$ and $p$, except $p=0.5$, by rescaling the time axis as $K^2St$. It shows an excellent collapse in the chaotic regime. (d) Heating time $t^*$ as a function of $K$ for binomial kicking protocol with different values of $p$. It scales as $t^* \sim 1/K^{\alpha}$ with the driving frequency ($1/K$). Here $\alpha$ can be equated to $2$ within numerical precision.
• Figure 4: (a) The normalized WTD for this sequence shows three peaks at $l=1, 2$ and $3$ in contrast to the previous two cases in Figs. \ref{['fig:binomial_ke']}, and \ref{['fig:poisson_ke']}, where the integer waiting time can be extended up to $\infty$ with small but non-zero probability. (b) The average kinetic energy per rotor for different values of $K$, when the system is driven according to the Thue-Morse driving schedule. (c) Data collapse of the kinetic energy curves in the chaotic regime for different values of $K$ with rescaled time axis by the $e^{c/K}$, where $c=6.18$. (d) The heating time $t^*$ with $1/K$ in semi-log plot using the same scaling.
• Figure 5: We show the change in the phase space entropy $\Delta \mathbb{S}_n$ with the scaled surprisal $K\sqrt{S}$.