### Speaker

Tobias Ambjörnsson

### Description

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.