94 research outputs found
Probing microscopic origins of confined subdiffusion by first-passage observables
Subdiffusive motion of tracer particles in complex crowded environments, such
as biological cells, has been shown to be widepsread. This deviation from
brownian motion is usually characterized by a sublinear time dependence of the
mean square displacement (MSD). However, subdiffusive behavior can stem from
different microscopic scenarios, which can not be identified solely by the MSD
data. In this paper we present a theoretical framework which permits to
calculate analytically first-passage observables (mean first-passage times,
splitting probabilities and occupation times distributions) in disordered media
in any dimensions. This analysis is applied to two representative microscopic
models of subdiffusion: continuous-time random walks with heavy tailed waiting
times, and diffusion on fractals. Our results show that first-passage
observables provide tools to unambiguously discriminate between the two
possible microscopic scenarios of subdiffusion. Moreover we suggest experiments
based on first-passage observables which could help in determining the origin
of subdiffusion in complex media such as living cells, and discuss the
implications of anomalous transport to reaction kinetics in cells.Comment: 21 pages, 3 figures. Submitted versio
Exact expressions of mean first-passage times and splitting probabilities for random walks in bounded rectangular domains
We give exact and explicit expressions of mean first-passage times for random
walks in a rectangular domain, in both cases of reflecting boundary conditions
and periodic boundary conditions. The situations with one or two absorbing
targets are considered.Comment: To appear as a note in J. Chem. Phy
Geometry-controlled kinetics
It has long been appreciated that transport properties can control reaction
kinetics. This effect can be characterized by the time it takes a diffusing
molecule to reach a target -- the first-passage time (FPT). Although essential
to quantify the kinetics of reactions on all time scales, determining the FPT
distribution was deemed so far intractable. Here, we calculate analytically
this FPT distribution and show that transport processes as various as regular
diffusion, anomalous diffusion, diffusion in disordered media and in fractals
fall into the same universality classes. Beyond this theoretical aspect, this
result changes the views on standard reaction kinetics. More precisely, we
argue that geometry can become a key parameter so far ignored in this context,
and introduce the concept of "geometry-controlled kinetics". These findings
could help understand the crucial role of spatial organization of genes in
transcription kinetics, and more generally the impact of geometry on
diffusion-limited reactions.Comment: Submitted versio
Diffusion-limited reactions and mortal random walkers in confined geometries
Motivated by the diffusion-reaction kinetics on interstellar dust grains, we
study a first-passage problem of mortal random walkers in a confined
two-dimensional geometry. We provide an exact expression for the encounter
probability of two walkers, which is evaluated in limiting cases and checked
against extensive kinetic Monte Carlo simulations. We analyze the continuum
limit which is approached very slowly, with corrections that vanish
logarithmically with the lattice size. We then examine the influence of the
shape of the lattice on the first-passage probability, where we focus on the
aspect ratio dependence: Distorting the lattice always reduces the encounter
probability of two walkers and can exhibit a crossover to the behavior of a
genuinely one-dimensional random walk. The nature of this transition is also
explained qualitatively.Comment: 18 pages, 16 figure
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
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
Occupation times of random walks in confined geometries: From random trap model to diffusion limited reactions
We consider a random walk in confined geometry, starting from a site and
eventually reaching a target site. We calculate analytically the distribution
of the occupation time on a third site, before reaching the target site. The
obtained distribution is exact, and completely explicit in the case or
parallepipedic confining domains. We discuss implications of these results in
two different fields: The mean first passage time for the random trap model is
computed in dimensions greater than 1, and is shown to display a non-trivial
dependence with the source and target positions ; The probability of reaction
with a given imperfect center before being trapped by another one is also
explicitly calculated, revealing a complex dependence both in geometrical and
chemical parameters
Scaling of mean first-passage time as efficiency measure of nodes sending information on scale-free Koch networks
A lot of previous work showed that the sectional mean first-passage time
(SMFPT), i.e., the average of mean first-passage time (MFPT) for random walks
to a given hub node (node with maximum degree) averaged over all starting
points in scale-free small-world networks exhibits a sublinear or linear
dependence on network order (number of nodes), which indicates that hub
nodes are very efficient in receiving information if one looks upon the random
walker as an information messenger. Thus far, the efficiency of a hub node
sending information on scale-free small-world networks has not been addressed
yet. In this paper, we study random walks on the class of Koch networks with
scale-free behavior and small-world effect. We derive some basic properties for
random walks on the Koch network family, based on which we calculate
analytically the partial mean first-passage time (PMFPT) defined as the average
of MFPTs from a hub node to all other nodes, excluding the hub itself. The
obtained closed-form expression displays that in large networks the PMFPT grows
with network order as , which is larger than the linear scaling of
SMFPT to the hub from other nodes. On the other hand, we also address the case
with the information sender distributed uniformly among the Koch networks, and
derive analytically the entire mean first-passage time (EMFPT), namely, the
average of MFPTs between all couples of nodes, the leading scaling of which is
identical to that of PMFPT. From the obtained results, we present that although
hub nodes are more efficient for receiving information than other nodes, they
display a qualitatively similar speed for sending information as non-hub nodes.
Moreover, we show that the location of information sender has little effect on
the transmission efficiency. The present findings are helpful for better
understanding random walks performed on scale-free small-world networks.Comment: Definitive version published in European Physical Journal
Non-Markovian polymer reaction kinetics
Describing the kinetics of polymer reactions, such as the formation of loops
and hairpins in nucleic acids or polypeptides, is complicated by the structural
dynamics of their chains. Although both intramolecular reactions, such as
cyclization, and intermolecular reactions have been studied extensively, both
experimentally and theoretically, there is to date no exact explicit analytical
treatment of transport-limited polymer reaction kinetics, even in the case of
the simplest (Rouse) model of monomers connected by linear springs. We
introduce a new analytical approach to calculate the mean reaction time of
polymer reactions that encompasses the non-Markovian dynamics of monomer
motion. This requires that the conformational statistics of the polymer at the
very instant of reaction be determined, which provides, as a by-product, new
information on the reaction path. We show that the typical reactive
conformation of the polymer is more extended than the equilibrium conformation,
which leads to reaction times significantly shorter than predicted by the
existing classical Markovian theory.Comment: Main text (7 pages, 5 figures) + Supplemantary Information (13 pages,
2 figures
Geometrical organization of solutions to random linear Boolean equations
The random XORSAT problem deals with large random linear systems of Boolean
variables. The difficulty of such problems is controlled by the ratio of number
of equations to number of variables. It is known that in some range of values
of this parameter, the space of solutions breaks into many disconnected
clusters. Here we study precisely the corresponding geometrical organization.
In particular, the distribution of distances between these clusters is computed
by the cavity method. This allows to study the `x-satisfiability' threshold,
the critical density of equations where there exist two solutions at a given
distance.Comment: 20 page
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