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). Additionally, we require that the separation between neighboring walkers cannot exceed a threshold value $\Delta$ and therefore call them ``tethered walkers'' (they behave as if bounded by strings which tighten fully when reaching the maximum length $\Delta$). For finite $\Delta$, 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 $(N\Delta)^2/D$]. 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 $\Delta/\sqrt{4Dt}$]. For $N=2$, we find an exact solution (both in the continuum case and on-lattice) and use it to test our approximations for one-particle distributions, positional moments, and correlations. For finite $\Delta$ 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 $\Delta=\infty$. Finally, we compute the probability distribution of the equilibrium system length and the 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 being $6k_BT/(N\Delta^2)$. In this respect, the system behaves like an ideal polymer. Our main analytical results are confirmed by Monte Carlo simulations.