250 research outputs found
Ultra-Slow Vacancy-Mediated Tracer Diffusion in Two Dimensions: The Einstein Relation Verified
We study the dynamics of a charged tracer particle (TP) on a two-dimensional
lattice all sites of which except one (a vacancy) are filled with identical
neutral, hard-core particles. The particles move randomly by exchanging their
positions with the vacancy, subject to the hard-core exclusion. In case when
the charged TP experiences a bias due to external electric field ,
(which favors its jumps in the preferential direction), we determine exactly
the limiting probability distribution of the TP position in terms of
appropriate scaling variables and the leading large-N ( being the discrete
time) behavior of the TP mean displacement ; the latter is
shown to obey an anomalous, logarithmic law . On comparing our results with earlier predictions by Brummelhuis
and Hilhorst (J. Stat. Phys. {\bf 53}, 249 (1988)) for the TP diffusivity
in the unbiased case, we infer that the Einstein relation
between the TP diffusivity and the mobility holds in the leading in order, despite
the fact that both and are not constant but vanish as . We also generalize our approach to the situation with very small but
finite vacancy concentration , in which case we find a ballistic-type law
. We demonstrate that here,
again, both and , calculated in the linear in
approximation, do obey the Einstein relation.Comment: 25 pages, one figure, TeX, submitted to J. Stat. Phy
Kinetics of active surface-mediated diffusion in spherically symmetric domains
We present an exact calculation of the mean first-passage time to a target on
the surface of a 2D or 3D spherical domain, for a molecule alternating phases
of surface diffusion on the domain boundary and phases of bulk diffusion. We
generalize the results of [J. Stat. Phys. {\bf 142}, 657 (2011)] and consider a
biased diffusion in a general annulus with an arbitrary number of regularly
spaced targets on a partially reflecting surface. The presented approach is
based on an integral equation which can be solved analytically. Numerically
validated approximation schemes, which provide more tractable expressions of
the mean first-passage time are also proposed. In the framework of this minimal
model of surface-mediated reactions, we show analytically that the mean
reaction time can be minimized as a function of the desorption rate from the
surface.Comment: Published online in J. Stat. Phy
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