Dynamically emergent correlations in Brownian particles subject to simultaneous non-Poissonian resetting protocols

TL;DR

This work analyzes a one-dimensional gas of $N$ independent Brownian particles subjected to simultaneous resetting with inter-reset times drawn from a general distribution $ψ(τ)$. Using renewal theory, the authors derive an exact time-dependent joint distribution and show that the nonequilibrium stationary state (NESS) exists under mild conditions and exhibits a CIID structure: conditioned on the last-reset interval $τ$, particle positions are IID Gaussian, and averaging over $τ$ induces strong all-to-all correlations. They compute a wide range of observables—density, extreme-value statistics, gaps, and full counting statistics—exactly for arbitrary $ψ(τ)$, and identify universal scaling forms governed by the tail of the resetting distribution. An explicit analysis of Poissonian, power-law, and bounded resetting protocols reveals three universality classes with distinct asymptotics, yet a common CIID framework that enables solvable descriptions of strongly correlated steady states. The results illustrate stochastic control of NESS in many-body systems and suggest experimental tests with cold-atom or colloidal setups, as well as extensions to interacting systems and first-passage problems.

Abstract

We consider a one-dimensional gas of $N$ independent Brownian particles subject to simultaneous stochastic resetting, with inter-reset times drawn from a general waiting-time distribution $ψ(τ)$. This includes the well-known Poissonian case, where $ψ(τ)=re^{-rτ}$, and extends to more general classes of resetting, such as heavy-tailed and bounded distributions. We show that the simultaneous resetting generates correlations between particles dynamically. These correlations grow with time and eventually drive the system into a strongly correlated non-equilibrium stationary state (NESS). Exploiting the renewal structure of the resetting dynamics, we derive explicit analytical expressions for the joint distribution of the positions of the particles in the NESS. We show that the NESS has a conditionally independent and identically distributed (CIID) structure that enables us to compute various physical observables exactly for arbitrary $ψ(τ)$. These observables include the average density, extreme value and order statistics, the spacing distribution between consecutive particles and the full counting statistics, i.e., the distribution of the number of particles in a given interval centered at the origin. We discuss the universal features of the large $N$ scaling behaviors of these observables for different choices of the resetting protocol $ψ(τ)$. Our results provide an interesting example of a stochastic control whereby, by tuning the inter-reset distribution $ψ(τ)$, one can generate a class of tunable, and yet solvable, strongly correlated NESS in a many-body system.

Figures (11)

• Figure 1: Schematic illustration of the process. The left panel shows the sketch of a trajectory where no reset occurs up to time $t$: the particles evolve purely by free diffusion. The right panel shows the sketch of a trajectory that undergoes at least one reset before time $t$. The last reset occurs at time $t_\ell$, and there are $n-1$ resets before it, with $n \geq 1$. We explicitly show the first, the second, and the last $(n^{th})$ reset. Since each reset completely deletes the system's past, the state of the system at time $t$ depends only on the time interval $\tau = t - t_{\ell}$ since the last reset.
• Figure 2: Plot of the function $f_C(z)$ vs $z$, as given in Eq. (\ref{['C2_exp']}). The horizontal dashed line indicates the asymptotic value $f_C(z \to \infty) = 1$ [see Eq. (\ref{['fC']})].
• Figure 3: Plot of the function $V(\sigma)$ vs $\sigma$, as given in Eq. (\ref{['V_gamma0']}).
• Figure 4: Plot of the density profile $\rho_N(x) = \frac{1}{l}\, R_\beta\left( \frac{x}{l} \right)$. The scaling function $R_\beta(z)$ is given in \ref{['eq:density_R(z)']}, and $l = \sqrt{\frac{D}{r}}$. The solid black curve shows the theoretical prediction for the resetting protocol $\psi(t) = \frac{r\beta}{(rt)^{1+\beta}}$ with $t \in [1/r, \infty)$ and $\beta = 4$. The symbols indicate simulation results for different values of $N$. Simulations were performed with $r = 1$ and $D = \frac{1}{2}$. The plots for different values of $N$ clearly show that $\rho_N(x)$ is independent of $N$.
• Figure 5: Plot of the distribution of the $k^{th}$ rightmost particle $M_k$ (order statistics). The distribution follows the scaling form $\text{Prob} [ M_k = \omega ]\approx \frac{1}{\Lambda(\alpha)}\, S_\beta\left( \frac{\omega}{\Lambda(\alpha)} \right)$ for large $N$, where the scaling function $S_\beta(z)$ is given in \ref{['eq:S_plaw']}, and the scaling parameter $\Lambda(\alpha)$ is defined in \ref{['eq:lambd_alpha_def']}. The solid black curve represents the theoretical prediction for the resetting protocol $\psi(t) = \frac{r\beta}{(rt)^{1+\beta}}$ with $t \in [1/r, \infty)$ and $\beta = 4$. The symbols represent results from simulations performed with $N = 10^6$ particles, $r = 1$, and $D = \frac{1}{2}$, for different values of $\alpha = \frac{k}{N}$. The scaling function $S_\beta(z)$ is clearly independent of $\alpha$.