Macroscopic localization and collective memory in Poisson renewal resetting

TL;DR

The study shows that Poisson renewal processes in continuous time generate macroscopic localization through a discrete delta peak at reset locations, coexisting with a continuous density. By developing an age-structured hydrodynamic framework for CTRWs under renewal resets, it reveals that collective reset rules (extremal and rank-based) induce long-term memory and aging, with a first-order dynamical transition at a critical bias (ζ=1) in the infinite-N limit. Independent resetting yields a stationary mixed state, while extremal resetting yields nonstationary aging with a slow evolution of system size and condensation. Finite systems exhibit finite-size crossovers, with aging persisting up to times capping at $t_c \sim \log N$ (ζ=1) or $t_c \sim N^{ζ-1}$ (ζ>1); the framework also connects to fractional Fokker-Planck dynamics in the diffusion limit. These results highlight how collective interactions in renewal dynamics can imprint long-term memory and macroscopic structure, with ecological implications for localized site fidelity and cooperative organization.

Abstract

Stochastic renewal processes are ubiquitous across physics, biology, and the social sciences. Here, we show that continuous-time renewal dynamics can naturally produce a mixed discrete-continuous structure, with a macroscopic fraction of particles occupying a discrete state. For ensembles of continuous-time random walkers subject to Poissonian renewal resets, we develop an age-structured framework showing this discrete component corresponds to localization at the reset configuration. We next show that collective interactions can retain memory although all reset events are memoryless. Remarkably, the transition to collective memory is discontinuous, and we identify a first-order dynamical phase transition between weak collective bias, where the dynamics are stationary, to strong collective bias where the dynamics are nonstationary and display aging up to finite-size effects. We explicitly discuss ecological implications of our work, illustrating how continuous-time renewal dynamics shape macroscopic structure and collective organization with long-term memory.

Figures (5)

• Figure 1: Collective renewal and aging. Top: Simulation trajectory (1 in red overlaid on $10^3$ in gray) for reset protocols with different bias exponent $\zeta$. For $\zeta=0$, independent renewal suppresses aging; for $\zeta=1$, finite $N$ inhibits collective aging; and for strong bias $\zeta\gg1$, resets are dominated by extreme particles, leading to nonstationary system size (dashed line) and collective memory ($\beta=0.8$). Bottom: Dynamical regime diagram in the $(t,\zeta)$ plane. For $\zeta>1$, collective bias induces aging up to a finite-size crossover time $t_c\sim N^{\zeta-1}$.
• Figure 2: Mixed discrete-continuous steady state. (a) Simulations for the continuous density and delta-peak weight $m_0$ (blue dots and red dot, respectively) compared to Eq. \ref{['eq:IE_ind_main']} (dashed line and green x) for $\beta\!=\!0.8$, $\tau_0\!=\!0.1$. (b) Discrete weight $m$ versus $\tau_0$ for power-law [Eq. \ref{['psi_def_main']}] and exponential waiting times. Simulations (see legend), compared to Eq. \ref{['eq:IE_ind_main']} (dashed line). In both panels, $\tau_0\!=\!\ell^2$, $N\!=\!10^4$, and $t\!=\!10^3$.
• Figure 3: Extremal resets. (a) Steady state. Simulations for the continuous density and delta-peak weight (blue dots and red dot, respectively) compared to Eq. \ref{['eq:Csol_main']} (dashed line, green x) for $\beta=2$ and $\tau_0\!=\ell^2\!\!=\!0.5$. (b) Nonstationary regime. Simulations at different $t$ (legend) compared to Eq. \ref{['eq:C_adiabatic_main']} (dashed lines). Insets: system size $L(t)$ and condensation $m(t)$, Eq. \ref{['eq:L_t_main']} (dashed lines) compared to simulations (triangles), for $\beta\!=\!0.8$ and $\tau_0\!=\!\ell^2\!=\!0.001$. In both panels $N\!=\!10^5$ and $r\!=\!1$.
• Figure 4: Dynamical phase transition induced by collective bias. (a) Discrete weight $m(t)$ for different $N$ (see legend) and $\zeta=1.5$, showing an aging regime followed by saturation due to finite-size effects. Compared to the theoretical prediction $m(t)\sim t^{1-\beta}$ (dashed line). (b) Finite-size cutoff time $t_c$ versus $N$ for different $\zeta$ (see legend), with the scalings $t_c\sim\log N$ at $\zeta=1$ and $t_c\sim N^{\zeta-1}$ for $\zeta>1$ (dashed lines). (c) Aging exponent $\gamma\equiv d \log C(0, t)/d\log t$ as a function of $\zeta$, for different $N$ at $t=10^2$. (d) Map of $\gamma$ versus $\zeta$ and $t$ (see colorbar) for $N=5120$. Dashed lines denote $\zeta =1$ and the theoretical $t_c \sim N^{\zeta -1}$. In all panels, $\beta\!=\!0.8$, $\tau_0 = \ell^2 \!=\! 0.01$, and $r=1$.
• Figure S1: Comparison between simulations and theoretical predictions for rank-based resetting in the stationary regime ($\beta=1.5$, $\zeta=2$). Symbols show the stationary spatial density $C_s(x)$ obtained from $5\times10^{4}$ independent realizations of an ensemble of $N=10^{4}$ particles, which allows sampling of the far tail. Solid black lines indicate the theoretical bulk profile. Parameters are $\tau_0=\ell^2=0.01$. Inset: semi-log plot highlighting the exponential decay of the tail and the agreement with theory up to finite-size corrections around $x\simeq L_N$. The dashed red line marks the approximate system size $L_N$ predicted by the extremal-like bulk theory.