Eur. Phys. J. Special Topics 223, 3287–3293 (2014)
https://doi.org/10.1140/epjst/e2014-02333-5

Regular Article

Ageing single file motion

¹ Institute of Physics & Astronomy, University of Potsdam, Potsdam-Golm, Germany
² Department of Physics, Tampere University of Technology, Tampere, Finland
³ Department of Astronomy and Theoretical Physics, Lund University, Lund, Sweden
⁴ MEMPHYS - Center for Biomembrane Physics, Department of Physics, Chemistry, and Pharmacy, University of Southern Denmark, Odense, Denmark
⁵ Integrated Science Lab, Department of Physics, Umeå University, Umeå, Sweden

^a e-mail: rmetzler@uni-potsdam.de

Received: 17 October 2014
Revised: 5 November 2014
Published online: 15 December 2014

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 〈x²(t)〉 ≃ K_1/2t^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 〈x²(t)〉 ≃ K₀ 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.