Tuning the strength of emergent correlations in a Brownian gas via batch resetting

Abstract

We study a gas of $N$ diffusing particles on the line subject to batch resetting: at rate $r$, a uniformly random subset of $m$ particles is reset to the origin. Despite the absence of interactions, the dynamics generates a nonequilibrium stationary state (NESS) with long-range correlations. We obtain exact results, both for the NESS and for the time dependence of the correlations, which are valid for arbitrary $m$ and $N$. By varying $m$, the system interpolates between an uncorrelated regime ($m=1$) and the fully synchronous resetting case ($m=N$). For all $1<m<N$, correlations exhibit a non-monotonic time dependence due to the emergence of an intrinsic decorrelation mechanism. In the stationary state, the correlation strength can be tuned by varying $m$, and it displays a transition at a critical value $N_c=6$. Our predictions extend straightforwardly to any spatial dimension $d$ and the critical value $N_c=6$ remains the same in all dimensions. Our predictions are testable in existing experimental setups on optically trapped colloidal particles.

Figures (3)

• Figure 1: Schematic representation of batch resetting for $N=3$ and $m=2$. Four resetting events are shown (light-blue dots).
• Figure 2: Plot of the function $V(m,N)$ (Eq. \ref{['eq:C2stat']}) compared with numerical simulations, shown as a function of $m$ for different values of $N$. For $N<N_c=6$, $V(m,N)$ increases monotonically with $m$, whereas for $N>N_c$ it becomes non–monotonic: the global maximum shifts from $m=N$ to $m=2$ and a shallow local minimum develops at $m=N-1$. This minimum is clearly visible in the central panel but it is also present for all $N\geq N_c$. At the critical value $N=N_c$, the points $m=2,3,N$ all give the same maximal value $V(m,N_c)=1$. The rightmost point corresponds to fully synchronous resetting Biroli2023 for all values of $N$.
• Figure 3: Scaling function $A(z;\beta)$ defined in Eq. \ref{['eq:a_APP']} of the End Matter, shown for several values of $\beta=\frac{2N-m-1}{N-1}\in[1,2]$. For $1<m<N$ ($1<\beta<2$), $A(z;\beta)$ develops a maximum at a finite time $z$ (indicated by red dots), while for $m=N$ ($\beta=1$) it increases monotonically. For any $1<\beta<2$, $A(z;\beta)$ decays exponentially fast to its stationary value $(2/\beta-1)/5$.