Ageing single file motion

The mean squared displacement of a tracer particle in a single file of identical particles with excluded volume interactions shows the famed Harris scaling aEurox (2)(t)aEuro parts per thousand a parts per thousand integral 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 tau with a power-law distribution psi(tau) a parts per thousand integral (tau(a similar to...))(alpha), and different ranges of the exponent alpha are considered. We find a dramatic slow-down of the motion of a tracer particle from Harris' law to an ultraslow, logarithmic time evolution aEurox (2)(t)aEuro parts per thousand a parts per thousand integral K (0) log (1/2)(t) when 0 < alpha < 1. In the intermediate case 1 < alpha < 2, we observe a power-law form for the mean squared displacement, with a modified scaling exponent as compared to Harris' law. Once alpha 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.