Universality of order statistics for Brownian reshuffling

We discuss the order statistics of the particle positions of a gas of $N$ identical independent particles performing Brownian motion in one dimension in a potential that asymptotically behaves like $V(x)\sim x^{\gamma }$ for $x\to +\mathrm{\infty }$, with a positive power $\gamma >0$. We show that in the stationary state, the order statistics that describe how the leaders are reshuffled are universal and independent of $\gamma$. What depends on $\gamma$ is the timescale of the leaders’ reshuffling, which scales as a power of the logarithm of the population size: $t\sim (\ln N{)}^{\frac{2(1-\gamma )}{\gamma }}\tau$, where $\tau$ is of order one. We derive the probability that the particle which has the $k$th largest value of $x$ at some time $t_1$ will have the $j$th largest value at time $t_2=t_1+t$ in the form of an explicit expression for the generating function for the reshuffling probabilities for all $k\ge 1$ and $j\ge 1$. The generating function, expressed in scaled time $\tau$, is independent of $\gamma$. In particular, we show that the average percentage overlap coefficient of leader lists takes the universal, $\gamma$-independent form $\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(\sqrt{\tau })$ for long lists.