Ageing single file motion

Abstract

The mean squared displacement of a tracer particle in a single file of identical particles with excluded volume interactions shows the famed Harris scaling 〈x ²(t)〉 ≃ K _1/2 t ^1/2 as function of time. Here we study what happens to this law when each particle of the single file interacts with the environment such that it is transiently immobilised for times τ with a power-law distribution ψ(τ) ≃ (τ^★)^α, and different ranges of the exponent α are considered. We find a dramatic slow-down of the motion of a tracer particle from Harris’ law to an ultraslow, logarithmic time evolution 〈x ²(t)〉 ≃ K ₀ log ^1/2(t) when 0 < α < 1. In the intermediate case 1 < α < 2, we observe a power-law form for the mean squared displacement, with a modified scaling exponent as compared to Harris’ law. Once α is larger than two, the Brownian single file behaviour and thus Harris’ law are restored. We also point out that this process is weakly non-ergodic in the sense that the time and ensemble averaged mean squared displacements are disparate.