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In the simplest model of single-file diffusion, $N$ point particles wander ona segment of the $x$ axis of length $L$, with hard core interactions, whichprevent passing, and with overdamped Brownian dynamics,$\lambda\dot{x}=\eta(t)$, where $\eta(t)$ has the form of Gaussian white noisewith 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 taggedparticle grows subdiffusively, as $t^{1/2}$, for long times. Recently, it hasbeen shown that the proportionality constants of the $t^{1/2}$ law forrandomly-distributed initial positions of the particles and for equally-spacedinitial positions are not the same, but have ratio $\sqrt{2}$. In this paper weconsider point particles on the $x$ axis, which collide elastically, and whichmove according to (i) random-acceleration dynamics $\ddot{x}=\eta(t)$ and (ii)Langevin dynamics $\ddot{x}+\lambda\dot{x}=\eta(t)$. The mean squaredisplacement and mean-square velocity of a tagged particle are analyzed forboth types of dynamics and for random and equally-spaced initial positions andGaussian-distributed initial velocities. We also study tagged particlestatistics, for both types of dynamics, in the spreading of a compact clusterof particles, with all of the particles initially at the origin.

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