Collective behavior of independent scaled Brownian particles with renewal resetting

TL;DR

This work addresses the collective fluctuations of $N\gg 1$ independent particles performing renewal-reset scaled Brownian motion. By leveraging the known single-particle steady-state distribution $p_{\text{s}}(x)$ and extreme-value theory, the authors derive the full statistics of the system radius $\ell$, showing typical fluctuations fall into the Gumbel universality class for all $H>0$. They also analyze the center of mass, finding standard large-deviation scaling for $H\le 1/2$ and an anomalous, big-jump–driven scaling with a first-order dynamical phase transition for $H>1/2$, due to a single particle performing a large excursion. The results highlight how renewal resetting and the underlying diffusion exponent $H$ govern collective fluctuations in noninteracting ensembles and point to rich extensions to interacting systems and fluctuating hydrodynamics with resetting.

Abstract

We study collective dynamics of an ensemble of $N\gg 1$ independent particles undergoing anomalous diffusion with random renewal resetting. The anomalous diffusion is modeled by the scaled Brownian motion (sBm): a Gaussian process, characterized by a power-law time dependence of the diffusion coefficient, $D(t)\sim t^{2H}$, where $H>0$. The particles independently reset to the origin, and each particle's clock is set to zero upon spatial resetting. Employing the known steady-state position distribution of a \emph{single} particle undergoing the sBm with renewal resetting [Bodrova et al., Phys. Rev. E \textbf{100}, 012120 (2019)], we study the collective dynamics of $N$ particles. We determine the statistics of the system radius $\ell$. The typical fluctuations of $\ell$ fall under the Gumbel universality class, and we use extreme value statistics to calculate the moments of $\ell$. We also study the large-deviation statistics of the center of mass (COM), where for $H>1/2$ we uncover an anomalous scaling behavior of the COM distribution, and a singularity in the corresponding rate function, due to a ``big jump" effect.

Figures (5)

• Figure 1: Steady-state distribution of the system radius $\ell$ for different values of $H$ (see legend). Lines: Eq. \ref{['eq:f_N']} , evaluated numerically. Symbols: simulations, where $N = 100$, and the simulation time is $t=10^5$.
• Figure 2: (a) The average system radius $\bar{\ell}$, Eq. \ref{['eq:EL_final']} and (b) the variance of the system radius, Eq. \ref{['eq:Var_asymp']}, vs. $N$ for different values of $H$ (see legend). Lines: theoretical predictions. Symbols: simulation results, where the simulation time is $t=10^5$.
• Figure 3: Variance of the center of mass $X$ versus $N$, Eq. \ref{['varN']}, for different values of $H$ (see legend). Lines: theoretical predictions. Symbols: simulation results, where the simulation time is $t=10^5$.
• Figure 4: The rate function $f(a;H)$, see Eqs. (\ref{['saddlepointeq2']}) and (\ref{['ratef']}), which describes the full statistics of the center of mass of the system for $H=1/2$, $1/3$, and $1/8$, see legend.
• Figure 5: The anomalous rate function $\phi(y)$, see Eqs. (\ref{['anomalous']}) and (\ref{['munu']}), for $H=2/3$ (a), $H=1$ (b), and $H=2$ (c). The critical point is described by Eq. (\ref{['yc']}) and is marked in all panels by a dotted line. The red dashed line in panel c shows the large-$y$ asymptotic $b \,y^\frac{2}{2H+1}$.