Single file dynamics of tethered random walkers

We consider the single file dynamics of N identical random walkers moving with diffusivity D in one dimension (walkers bounce off each other when attempting to overtake). In addition, we require that the separation between neighboring walkers does not exceed a threshold value Δ and therefore call them “tethered walkers” (they behave as if bounded by strings that fully tighten when reaching the maximum length Δ). For a finite Δ, we study the diffusional relaxation to the equilibrium state and characterize the latter [the long-time relaxation is exponential with a characteristic time that scales as (NΔ)2/D]. In particular, our approximate approach for the N-particle probability distribution yields the one-particle distribution function of the central and edge particles (the first two positional moments are given as power expansions in Δ/4Dt). For N = 2, we find an exact solution (both in the continuum and on-lattice case) and use it to test our approximations for one-particle distributions, positional moments, and correlations. For finite Δ and arbitrary N, edge particles move with an effective long-time diffusivity D/N, in sharp contrast with the 1/ln(N)-behavior observed when Δ = ∞. Finally, we compute the probability distribution of the equilibrium system length and associated entropy. We find that the force required to change this length by a given amount is linear in this quantity; the (entropic) spring constant is 6kBT/(NΔ2). In this respect, the system behaves as an ideal polymer. The main analytical results are confirmed using Monte Carlo simulations.