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)∼x^{γ} for x→+∞, with a positive power γ>0. We show that in the stationary state, the order statistics that describe how the leaders are reshuffled are universal and independent of γ. What depends on γ is the timescale of the leaders' reshuffling, which scales as a power of the logarithm of the population size: t∼(lnN)^{2(1-γ)/γ}τ, where τ 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≥1 and j≥1. The generating function, expressed in scaled time τ, is independent of γ. In particular, we show that the average percentage overlap coefficient of leader lists takes the universal, γ-independent form erfc(sqrt[τ]) for long lists.