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Fluctuations and first-passage properties of systems of Brownian particles with reset

We study stationary fluctuations in two models involving $N$ Brownian particles undergoing stochastic resetting to the origin in 1d. We start with the basic reset model where the particles reset independently (model A). Then we introduce nonlocal interparticle correlations by postulating that only the particle farthest from the origin is reset (model B). At long times both models approach nonequilibrium steady states. In the limit of $N\to \infty$, the steady-state particle density in model A has an infinite support, whereas in model B it has a compact support. A finite system radius, which scales at large $N$ as $\ln N$, appears in model A when $N$ is finite. In both models we study stationary fluctuations of the center of mass of the system and of the system's radius due to the random character of the Brownian motion and of the resetting events. In model A we determine exact distributions of these two quantities. The variance of the center of mass for both models scales as $1/N$. The variance of the radius is independent of $N$ in model A and exhibits an unusual scaling $(\ln N)/N$ in model B. The latter scaling is intimately related to the $1/f$ noise in the radius autocorrelations. Finally, we evaluate the mean first-passage time (MFPT) to a distant target in model A, model B, and the BBM. For model A we obtain an exact asymptotic expression for the MFPT which scales as $1/N$. For model B, and for the "Brownian bees" model, we propose a sharp upper bound for the MFPT. The bound assumes an ``evaporation" scenario, where the first passage requires multiple attempts of a single particle, which breaks away from the rest of the particles, to reach the target. The resulting MFPT for model B and the Brownian bees model scales exponentially with $\sqrt{N}$. We verify this bound by performing highly efficient weighted-ensemble simulations of the first passage in model B.