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

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.