2,182 research outputs found
Diffusivity of a random walk on random walks
We consider a random walk with the constraint that each coordinate of the walk is at distance one from the following one. In this paper, we show that this random walk is slowed down by a variance factor with respect to the case of the classical simple random walk without constraint
Einstein relation for random walk in a one-dimensional percolation model
We consider random walks on the infinite cluster of a conditional bond
percolation model on the infinite ladder graph. In a companion paper, we have
shown that if the random walk is pulled to the right by a positive bias
, then its asymptotic linear speed is
continuous in the variable and differentiable for all
sufficiently small . In the paper at hand, we complement this
result by proving that is differentiable at . Further, we show the Einstein relation for the model, i.e., that the
derivative of the speed at equals the diffusivity of the unbiased
walk
Optimization and universality of Brownian search in quenched heterogeneous media
The kinetics of a variety of transport-controlled processes can be reduced to
the problem of determining the mean time needed to arrive at a given location
for the first time, the so called mean first passage time (MFPT) problem. The
occurrence of occasional large jumps or intermittent patterns combining various
types of motion are known to outperform the standard random walk with respect
to the MFPT, by reducing oversampling of space. Here we show that a regular but
spatially heterogeneous random walk can significantly and universally enhance
the search in any spatial dimension. In a generic minimal model we consider a
spherically symmetric system comprising two concentric regions with piece-wise
constant diffusivity. The MFPT is analyzed under the constraint of conserved
average dynamics, that is, the spatially averaged diffusivity is kept constant.
Our analytical calculations and extensive numerical simulations demonstrate the
existence of an {\em optimal heterogeneity} minimizing the MFPT to the target.
We prove that the MFPT for a random walk is completely dominated by what we
term direct trajectories towards the target and reveal a remarkable
universality of the spatially heterogeneous search with respect to target size
and system dimensionality. In contrast to intermittent strategies, which are
most profitable in low spatial dimensions, the spatially inhomogeneous search
performs best in higher dimensions. Discussing our results alongside recent
experiments on single particle tracking in living cells we argue that the
observed spatial heterogeneity may be beneficial for cellular signaling
processes.Comment: 19 pages, 11 figures, RevTe
Diffusivity in one-dimensional generalized Mott variable-range hopping models
We consider random walks in a random environment which are generalized
versions of well-known effective models for Mott variable-range hopping. We
study the homogenized diffusion constant of the random walk in the
one-dimensional case. We prove various estimates on the low-temperature
behavior which confirm and extend previous work by physicists.Comment: Published in at http://dx.doi.org/10.1214/08-AAP583 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Transport in quenched disorder: light diffusion in strongly heterogeneous turbid media
We present a theoretical and experimental study of light transport in
disordered media with strongly heterogeneous distribution of scatterers formed
via non-scattering regions. Step correlations induced by quenched disorder are
found to prevent diffusivity from diverging with increasing heterogeneity
scale, contrary to expectations from annealed models. Spectral diffusivity is
measured for a porous ceramic where nanopores act as scatterers and macropores
render their distribution heterogeneous. Results agree well with Monte Carlo
simulations and a proposed analytical model.Comment: 12 pages, 9 figures (significant amount of supplemental information
Crossing random walks and stretched polymers at weak disorder
We consider a model of a polymer in , constrained to join 0
and a hyperplane at distance . The polymer is subject to a quenched
nonnegative random environment. Alternatively, the model describes crossing
random walks in a random potential (see Zerner [Ann Appl. Probab. 8 (1998)
246--280] or Chapter 5 of Sznitman [Brownian Motion, Obstacles and Random Media
(1998) Springer] for the original Brownian motion formulation). It was recently
shown [Ann. Probab. 36 (2008) 1528--1583; Probab. Theory Related Fields 143
(2009) 615--642] that, in such a setting, the quenched and annealed free
energies coincide in the limit , when and the temperature
is sufficiently high. We first strengthen this result by proving that, under
somewhat weaker assumptions on the distribution of disorder which, in
particular, enable a small probability of traps, the ratio of quenched and
annealed partition functions actually converges. We then conclude that, in this
case, the polymer obeys a diffusive scaling, with the same diffusivity constant
as the annealed model.Comment: Published in at http://dx.doi.org/10.1214/10-AOP625 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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