Journal of Statistical Physics

, Volume 177, Issue 5, pp 806–824 | Cite as

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

  • Theodore W. BurkhardtEmail author


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 \(N\rightarrow \infty \), \(L\rightarrow \infty \) 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.


Single-file Tracer diffusion Random acceleration Stochastic process 



I thank Ahmed Fouad and Edward Gawlinski for discussions about single-file diffusion and for sharing the results of their simulations.


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Authors and Affiliations

  1. 1.Department of PhysicsTemple UniversityPhiladelphiaUSA

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