Tagged-Particle Statistics in Single-File Motion with Random-Acceleration and Langevin Dynamics

In the simplest model of single-file diffusion, $N$ point particles wander on a segment of the $x$ axis of length $L$, with hard core interactions, which prevent passing, and with overdamped Brownian dynamics, $\lambda\dot{x}=\eta(t)$, where $\eta(t)$ has the form of Gaussian white noise with zero mean. In 1965 Harris showed that in the limit $N\to\infty$, $L\to\infty$ with constant $\rho=N/L$, the mean square displacement of a tagged particle grows subdiffusively, as $t^{1/2}$, for long times. Recently, it has been shown that the proportionality constants of the $t^{1/2}$ law for randomly-distributed initial positions of the particles and for equally-spaced initial positions are not the same, but have ratio $\sqrt{2}$. In this paper we consider point particles on the $x$ axis, which collide elastically, and which move according to (i) random-acceleration dynamics $\ddot{x}=\eta(t)$ and (ii) Langevin dynamics $\ddot{x}+\lambda\dot{x}=\eta(t)$. The mean square displacement and mean-square velocity of a tagged particle are analyzed for both types of dynamics and for random and equally-spaced initial positions and Gaussian-distributed initial velocities. We also study tagged particle statistics, for both types of dynamics, in the spreading of a compact cluster of particles, with all of the particles initially at the origin.