Extreme statistics and spacing distribution in a Brownian gas correlated by resetting

Abstract

We study a one-dimensional gas of $N$ Brownian particles that diffuse independently, but are {\it simultaneously} reset to the origin at a constant rate $r$. The system approaches a non-equilibrium stationary state (NESS) with long-range interactions induced by the simultaneous resetting. Despite the presence of strong correlations, we show that several observables can be computed exactly, which include the global average density, the distribution of the position of the $k$-th rightmost particle and the spacing distribution between two successive particles. Our analytical results are confirmed by numerical simulations. We also discuss a possible experimental realisation of this resetting gas using optical traps.

Figures (3)

• Figure 1: Schematic trajectories of $N=3$ Brownian motions undergoing simultaneous resetting to the origin at random times. The observation time is marked by $t$ and the time of the last reset before $t$ is marked by $t-\tau$. During the last period $\tau$, the particles evolve independently as free Brownian motions.
• Figure 2: The solid blue line shows the average density $\rho(x,N) = \sqrt{\frac{r}{4D}}e^{- \sqrt{r/D}|x|}$. The positions of the particles in a typical sample are shown schematically on the line with most particles living over a distance $\sqrt{D/r}$ around the origin. The typical spacing in the bulk $\sim 1/N$, while it is of order $\sim 1/\sqrt{\ln N}$ near the extreme edges of the sample. The typical position of the rightmost particle $M_1 \sim \sqrt{\ln N}$ for large $N$.
• Figure 3: a) Scaled distribution of the position $M_k$ of the $k$-th particle from the right: $P(M_k) \approx \Lambda^{-1}(\alpha) f(M_k\Lambda^{-1}(\alpha))$ with $\Lambda(\alpha)$ given below Eq. (\ref{['scaling_kth_max']}). The symbols represent the results of simulations, while the solid curve shows the scaling function $f(z)$ in Eq. (\ref{['scaling_kth_max']}). b) Scaled distribution of the gap $d_k = M_k - M_{k+1}$ between the $k$-th and the $(k+1)$-th particle counted from the right: numerical simulations are in perfect agreement with the analytical scaling function $h(z)$ in Eq. (\ref{['h_of_z']}). We used the parameter values $D=0.5$ and $r=1$.