Interacting Random Walkers in One-Dimansional Systems

Mar 17, 2010, 2:30 PM
Nordita Seminar Room 132:028 (Nordita)

Nordita Seminar Room 132:028



Tobias Ambjörnsson


The problem of a single random walker has received a lot of attention in the science community during the last century. There is now an increasing amount of interest in the problem of INTERACTING random walkers (due to the strong connection of this problem to the fields of, for instance, biophysics, nanofluidics, and cell biology). In particular, much attention has been on the behavior of the non-equilibrium problem of interacting walkers in (quasi)one dimensional systems, so called single-file diffusion. The quantities of main interest in such a system is the mean square displacement (MSD) of a (fluorescently) tagged particle. It has been found previously (theoretically and experimentally) that the MSD for a tagged particle in a single file system scales as t^(1/2) for long times (in the thermodynamic limit), rather than t as for unconstrained diffusion. In the talk three new single-file results will be presented: 1) The problem of hardcore interacting particles in a FINITE system (box) is solved analytically using a Bethe-ansatz, see Ref [1]. Analysis of our exact solution reveals three time regimes, where the t^(1/2)-behaviour appears as an intermediate regime. 2) We recently introduced a procedure, which we refer as to as Harmonization, which maps the diffusive motion of any type of 1d short-range single-file system onto that of chain of harmonically coupled beads; the effective spring constant in the system is connected to the details of the potential between particles. The Harmonization procedure reproduces all known long-time results in the single-file field with some back-of-the envelope calculations and allow us to analytically solve the long-time behavior of more complicated single-file systems. For instance, the tagged particle motion in a harmonic potential, in a time-varying force field and correlation functions between particles are calculated. 3) Finally, single-file diffusion in a system where the particles have different diffusion constants is considered. By combining the Harmonization procedure with effective medium theory we derive analytic results for the MSD, and find that for certain types of distributions for the diffusion constants, the dynamics becomes ultra-slow; the MSD scales as t^delta, with delta<1/2, [3]. [1] L. Lizana and T. Ambjornsson, Single-file diffusion in a box, Phys. Rev. Lett. 100, 200601 (2008); Phys. Rev. E 80, 051103 (2009). [2] T. Ambjornsson, L. Lizana, A. Taloni, E. Barkai and M.A. Lomholt, Foundation of fractional Langevin equations: Harmonization of a many-body problem, submitted, E-print: arXiv:0909.0881. [3]. M. A. Lomholt, L. Lizana and T. Ambjornsson, in preparation.

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