First Passage Resetting Gas

Abstract

We study a one-dimensional gas of $N$ Brownian particles that diffuse independently but are simultaneously reset whenever any of them reaches a fixed threshold located at $L > 0$. For any $N > 2$, the system reaches a non-equilibrium stationary state (NESS) at long-times with strong long-range correlations. These correlations emerge purely from the dynamics, and not from built-in interactions. Despite being strongly correlated, the NESS has a solvable structure that allows for an exact computation of several physical observables, both global and local. These include the average density profile, the distribution of the position of the $k$-th ordered particles, the distribution of the gap between two consecutive particles and the full counting statistics, i.e., the distribution of the number of particles in a finite interval around the origin.

Figures (2)

• Figure 1: a): A schematic representation of the trajectories of three particles (different colors) in the TR model. When any of them reaches $L$, all of them reset to the origin (shown by dashed lines). b): The scaling form of the average density $\rho(x,N)$ in Eq. (\ref{['eq:f-def']}) compared to simulations to $N=10^4$ particles with $D =1$, $L = 1$ and $\dd t = 10^{-5}$, averaged over 100 samples.
• Figure 2: a): The order statistics $M_k$ for $k = \alpha N$ with $\alpha = 0.2, 0.4, 0.6$ and 0.8 in purple (+), blue ($\times$), red ($\star$) and green ($\square$) compared to the theoretical prediction in Eq. (\ref{['eq:order']}). The numerical points were obtained by sampling directly $10^3$ samples of the distribution defined in Eq. (\ref{['eq:jpdf-NESS-dimensionless']}) with $N = 10^6$. b): The gap statistics $d_k$ for the same set of $k = \alpha N$ as in a) compared to the theoretical prediction in Eqs. (\ref{['eq:h']})-(\ref{['eq:gap-scaling']}).