172 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
Averaged residence times of stochastic motions in bounded domains
Two years ago, Blanco and Fournier (Blanco S. and Fournier R., Europhys.
Lett. 2003) calculated the mean first exit time of a domain of a particle
undergoing a randomly reoriented ballistic motion which starts from the
boundary. They showed that it is simply related to the ratio of the volume's
domain over its surface. This work was extended by Mazzolo (Mazzolo A.,
Europhys. Lett. 2004) who studied the case of trajectories which start inside
the volume. In this letter, we propose an alternative formulation of the
problem which allows us to calculate not only the mean exit time, but also the
mean residence time inside a sub-domain. The cases of any combinations of
reflecting and absorbing boundary conditions are considered. Lastly, we
generalize our results for a wide class of stochastic motions.Comment: 7 pages, 3 figure
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
Mean first-passage time of surface-mediated diffusion in spherical 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. 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: to appear in J. Stat. Phy
Facilitated diffusion of proteins on chromatin
We present a theoretical model of facilitated diffusion of proteins in the
cell nucleus. This model, which takes into account the successive
binding/unbinding events of proteins to DNA, relies on a fractal description of
the chromatin which has been recently evidenced experimentally. Facilitated
diffusion is shown quantitatively to be favorable for a fast localization of a
target locus by a transcription factor, and even to enable the minimization of
the search time by tuning the affinity of the transcription factor with DNA.
This study shows the robustness of the facilitated diffusion mechanism, invoked
so far only for linear conformations of DNA.Comment: 4 pages, 4 figures, accepted versio
Exact calculations of first-passage quantities on recursive networks
We present general methods to exactly calculate mean-first passage quantities
on self-similar networks defined recursively. In particular, we calculate the
mean first-passage time and the splitting probabilities associated to a source
and one or several targets; averaged quantities over a given set of sources
(e.g., same-connectivity nodes) are also derived. The exact estimate of such
quantities highlights the dependency of first-passage processes with respect to
the source-target distance, which has recently revealed to be a key parameter
to characterize transport in complex media. We explicitly perform calculations
for different classes of recursive networks (finitely ramified fractals,
scale-free (trans)fractals, non-fractals, mixtures between fractals and
non-fractals, non-decimable hierarchical graphs) of arbitrary size. Our
approach unifies and significantly extends the available results in the field.Comment: 16 pages, 10 figure
First exit times and residence times for discrete random walks on finite lattices
In this paper, we derive explicit formulas for the surface averaged first
exit time of a discrete random walk on a finite lattice. We consider a wide
class of random walks and lattices, including random walks in a non-trivial
potential landscape. We also compute quantities of interest for modelling
surface reactions and other dynamic processes, such as the residence time in a
subvolume, the joint residence time of several particles and the number of hits
on a reflecting surface.Comment: 19 pages, 2 figure
Enhanced reaction kinetics in biological cells
The cell cytoskeleton is a striking example of "active" medium driven
out-of-equilibrium by ATP hydrolysis. Such activity has been shown recently to
have a spectacular impact on the mechanical and rheological properties of the
cellular medium, as well as on its transport properties : a generic tracer
particle freely diffuses as in a standard equilibrium medium, but also
intermittently binds with random interaction times to motor proteins, which
perform active ballistic excursions along cytoskeletal filaments. Here, we
propose for the first time an analytical model of transport limited reactions
in active media, and show quantitatively how active transport can enhance
reactivity for large enough tracers like vesicles. We derive analytically the
average interaction time with motor proteins which optimizes the reaction rate,
and reveal remarkable universal features of the optimal configuration. We
discuss why active transport may be beneficial in various biological examples:
cell cytoskeleton, membranes and lamellipodia, and tubular structures like
axons.Comment: 10 pages, 2 figure
Residual mean first-passage time for jump processes: theory and applications to L\'evy flights and fractional Brownian motion
We derive a functional equation for the mean first-passage time (MFPT) of a
generic self-similar Markovian continuous process to a target in a
one-dimensional domain and obtain its exact solution. We show that the obtained
expression of the MFPT for continuous processes is actually different from the
large system size limit of the MFPT for discrete jump processes allowing
leapovers. In the case considered here, the asymptotic MFPT admits
non-vanishing corrections, which we call residual MFPT. The case of L/'evy
flights with diverging variance of jump lengths is investigated in detail, in
particular, with respect to the associated leapover behaviour. We also show
numerically that our results apply with good accuracy to fractional Brownian
motion, despite its non-Markovian nature.Comment: 13 pages, 8 figure
Windings of the 2D free Rouse chain
We study long time dynamical properties of a chain of harmonically bound
Brownian particles. This chain is allowed to wander everywhere in the plane. We
show that the scaling variables for the occupation times T_j, areas A_j and
winding angles \theta_j (j=1,...,n labels the particles) take the same general
form as in the usual Brownian motion. We also compute the asymptotic joint laws
P({T_j}), P({A_j}), P({\theta_j}) and discuss the correlations occuring in
those distributions.Comment: Latex, 17 pages, submitted to J. Phys.
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