arXiv subjects

E. Abad

Publications and source records attributed to E. Abad.

At least 19 recordsLinked to original sources

arXiv2025

Single file dynamics of tethered random walkers

We consider the single-file dynamics of $N$ identical random walkers moving with diffusivity $D$ in one dimension (walkers bounce off each other when attempting to overtake). Additionally, we require that the separation between neighboring walkers cannot exceed a threshold value $\Delta$ and therefore call them ``tethered walkers'' (they behave as if bounded by strings which tighten fully when reaching the maximum length $\Delta$). For finite $\Delta$, we study the diffusional relaxation to the equilibrium state and characterize the latter [the long-time relaxation is exponential with a characteristic time that scales as $(N\Delta)^2/D$]. In particular, our approximate approach for the $N$-particle probability distribution yields the one-particle distribution function of the central and edge particles [the first two positional moments are given as power expansions in $\Delta/\sqrt{4Dt}$]. For $N=2$, we find an exact solution (both in the continuum case and on-lattice) and use it to test our approximations for one-particle distributions, positional moments, and correlations. For finite $\Delta$ and arbitrary $N$, edge particles move with an effective long-time diffusivity $D/N$, in sharp contrast with the $1/\ln(N)$-behavior observed when $\Delta=\infty$. Finally, we compute the probability distribution of the equilibrium system length and the associated entropy. We find that the force required to change this length by a given amount is linear in this quantity, the (entropic) spring constant being $6k_BT/(N\Delta^2)$. In this respect, the system behaves like an ideal polymer. Our main analytical results are confirmed by Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2024

Search and escape of mortal random walkers

We review some representative results for first-passage problems involving so-called mortal or evanescent walkers, i.e., walkers with a finite lifetime. The mortality constraint plays a key role in the modeling of many real scenarios, as it filters out long Brownian trajectories, thereby drastically modifying space exploration properties. Among such scenarios, we consider here different first-passage problems, including one or many searchers, resetting, anomalous diffusion, evolving domains, and the narrow escape problem. In spite of the different physics, the mathematical treatment draws strongly on the formalism for standard (i.e., immortal) walkers.

cond-mat.stat-mech↗

arXiv2022

First-encounter time of two diffusing particles in two- and three-dimensional confinement

The statistics of the first-encounter time of diffusing particles changes drastically when they are placed under confinement. In the present work, we make use of Monte Carlo simulations to study the behavior of a two-particle system in two- and three-dimensional domains with reflecting boundaries. Based on the outcome of the simulations, we give a comprehensive overview of the behavior of the survival probability $S(t)$ and the associated first-encounter time probability density $H(t)$ over a broad time range spanning several decades. In addition, we provide numerical estimates and empirical formulas for the mean first-encounter time $\langle \cal{T}\rangle $, as well as for the decay time $T$ characterizing the monoexponential long-time decay of the survival probability. Based on the distance between the boundary and the center of mass of two particles, we obtain an empirical lower bound $t_B$ for the time at which $S(t)$ starts to significantly deviate from its counterpart for the no boundary case. Surprisingly, for small-sized particles, the dominant contribution to $T$ depends only on the total diffusivity $D=D_1+D_2$, in sharp contrast to the one-dimensional case. This contribution can be related to the Wiener sausage generated by a fictitious Brownian particle with diffusivity $D$. In two dimensions, the first subleading contribution to $T$ is found to depend weakly on the ratio $D_1/D_2$. We also investigate the slow-diffusion limit when $D_2 \ll D_1$ and discuss the transition to the limit when one particle is a fixed target. Finally, we give some indications to anticipate when $T$ can be expected to be a good approximation for $\langle \cal{T}\rangle$.

cond-mat.stat-mech↗

arXiv2021

The pseudo-two-dimensional dynamics in a system of macroscopic rolling spheres

We study in this work the dynamics of a collection of identical hollow spheres (ping-pong balls) that rest on a horizontal metallic grid. Fluidization is achieved by means of a turbulent air current coming from below. The upflow is adjusted so that the particles do not levitate over the grid, resulting in quasi-two-dimensional dynamics. We show that the behavior of diffusion and correlations in this system is particularly rich. Noticeably as well (and related to the complex dynamical behavior), a variety of phases appear, with important peculiarities with respect to analogous setups. We observe gas, liquid, glass, and hexagonal crystal phases. Most notably, we show that the melting of the hexagonal crystal occurs in coexistence with a liquid phase. This strikingly differs from the corresponding transition in a purely two-dimensional systems of air-fluidized disks, for which no phase coexistence has been reported in the literature.

cond-mat.soft↗

arXiv2020

First-encounter time of two diffusing particles in confinement

We investigate how confinement may drastically change both the probability density of the first-encounter time and the related survival probability in the case of two diffusing particles. To obtain analytical insights into this problem, we focus on two one-dimensional settings: a half-line and an interval. We first consider the case with equal particle diffusivities, for which exact results can be obtained for the survival probability and the associated first-encounter time density over the full time domain. We also evaluate the moments of the first-encounter time when they exist. We then turn to the case when the diffusivities are not equal, and focus on the long-time behavior of the survival probability. Our results highlight the great impact of boundary effects in diffusion-controlled kinetics even for simple one-dimensional settings, as well as the difficulty of obtaining analytic results as soon as translational invariance of such systems is broken.

cond-mat.stat-mech↗

arXiv2020

Continuous Time Random Walk in a velocity field: Role of domain growth, Galilei-invariant advection-diffusion, and kinetics of particle mixing

We consider the dynamics of a separable Continuous Time Random Walk (CTRW) when the random walker is biased by a velocity field in a uniformly growing domain. Concrete examples for such domains include growing biological cells or lipid vesicles, biofilms and tissues, but also macroscopic systems such as expanding aquifers during rainy periods, or the expanding universe. The considered CTRW can be subdiffusive, normal diffusive or superdiffusive, including the particular case of a Lévy flight. We first consider the case of zero velocity field. In the subdiffusive case, we reveal an interesting time dependence of the kurtosis of the particle probability density function. In particular, for a suitable parameter choice, we find that the propagator, which is fat tailed at short times, may cross over to a Gaussian-like propagator. We subsequently incorporate the effect of the velocity field and derive a bi-fractional diffusion-advection equation encoding the time evolution of the particle distribution. We apply this equation to study the mixing kinetics of two diffusing pulses, whose peaks move towards each other under the action of velocity fields acting in opposite directions. This deterministic motion of the peaks, together with the diffusive spreading of each pulse, tends to increase particle mixing, thereby counteracting the peak separation induced by the domain growth. As a result of this competition, different regimes of mixing arise. In the case of Lévy flights, apart from the non-mixing regime, one has two different mixing regimes in the long-time limit, depending on the exact parameter choice: In one of these regimes, mixing is mainly driven by diffusive spreading, while in the other mixing is controlled by the velocity fields acting on each pulse. Possible implications for encounter-controlled reactions in real systems are discussed.

cond-mat.stat-mech↗

arXiv2020

Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains

Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a one-dimensional domain that is evolving. The model equations, which have been derived from generalized continuous time random walks, can incorporate complexities such as subdiffusive transport and inhomogeneous domain stretching and shrinking. A method for constructing analytic expressions for short time moments of the position of the particles is developed and moments calculated from this approach are shown to compare favourably with results from random walk simulations and numerical integration of the reaction transport equation. The results show the important role played by the initial condition. In particular, it strongly affects the time dependence of the moments in the short time regime by introducing additional drift and diffusion terms. We also discuss how our reaction transport equation could be applied to study the spreading of a population on an evolving interface.

cond-mat.stat-mech↗

arXiv2019

The standard and the fractional Ornstein-Uhlenbeck process on a growing domain

We study normal diffusive and subdiffusive processes in a harmonic potential (Ornstein-Uhlenbeck process) on a uniformly growing/contracting domain. Our starting point is a recently derived fractional Fokker-Planck equation, which covers both the case of Brownian diffusion and the case of a subdiffusive Continuous-Time Random Walk (CTRW). We find a high sensitivity of the random walk properties to the details of the domain growth rate, which gives rise to a variety of regimes with extremely different behaviors. At the origin of this rich phenomenology is the fact that the walkers still move while they wait to jump, since they are dragged by the deterministic drift arising from the domain growth. Thus, the increasingly long waiting times associated with the ageing of the subdiffusive CTRW imply that, in the time interval between two consecutive jumps, the walkers might travel over much longer distances than in the normal diffusive case. This gives rise to seemingly counterintuitive effects. For example, on a static domain, both Brownian diffusion and subdiffusive CTRWs yield a stationary particle distribution with finite width when a harmonic potential is at play, thus indicating a confinement of the diffusing particle. However, for a sufficiently fast growing/contracting domain, this qualitative behavior breaks down, and differences between the Brownian case and the subdiffusive case are found. In the case of Brownian particles, a sufficiently fast exponential domain growth is needed to break the confinement induced by the harmonic force; in contrast, for subdiffusive particles such a breakdown may already take place for a sufficiently fast power-law domain growth. Our analytic and numerical results for both types of diffusion are fully confirmed by random walk simulations.

cond-mat.stat-mech↗

arXiv2019

First-passage properties of mortal random walks: ballistic behavior, effective reduction of dimensionality, and scaling functions for hierarchical graphs

We consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites. The configuration comprising the graph, the starting point of the walk, and the locations of the trap sites is taken to be exactly self-similar as one goes from one generation of the family to the next. Under these circumstances, the total probability that the walker hits a trap is determined exactly as a function of the single-step survival probability $q$ of the mortal walker. On the $n^{\rm th}$ generation graph of the family, this probability is shown to be given by the $n^{\rm th}$ iterate of a certain scaling function or map $q \rightarrow f(q)$. The properties of the map then determine, in each case, the behavior of the trapping probability, the mean time to trapping, the temporal scaling factor governing the random walk dimension on the graph, and other related properties. The formalism is illustrated for the cases of a linear hierarchical lattice and the Sierpinski graphs in $2$ and $3$ Euclidean dimensions. We find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint. The relevance of this finding for experiments involving travel times of particles in diffusion-decay systems is discussed.

cond-mat.stat-mech↗

arXiv2018

Encounter-controlled coalescence and annihilation on a one-dimensional growing domain

The kinetics of encounter-controlled processes in growing domains is markedly different from that in a static domain. Here, we consider the specific example of diffusion limited coalescence and annihilation reactions in one-dimensional space. In the static case, such reactions are among the few systems amenable to exact solution, which can be obtained by means of a well-known method of intervals. In the case of a uniformly growing domain, we show that a double transformation in time and space allows one to extend this method to compute the main quantities characterizing the spatial and temporal behavior. We show that a sufficiently fast domain growth brings about drastic changes in the behavior. In this case, the reactions stop prematurely, as a result of which the survival probability of the reacting particles tends to a finite value at long times and their spatial distribution freezes before reaching the fully self-ordered state. We obtain exact results for the survival probability and for key properties characterizing the degree of self-ordering induced by the chemical reactions, i.e., the interparticle distribution function and the pair correlation function. These results are confirmed by numerical simulations.

cond-mat.stat-mech↗

arXiv2018

Random Walks on Lattices. Influence of Competing Reaction Centers on Diffusion-Controlled Processes

We study diffusion-reaction processes on periodic square planar lattices and simple cubic (sc) lattices. Considered first is a single diffusing reactant undergoing an irreversible reaction upon first encounter with a stationary co-reactant ["one-walker (1W) problem"]. We then generalize this scenario to allow for a competing reaction, i.e., instantaneous trapping of the diffusing reactant with probability $s$ at any vacant site before interacting with a (stationary) co-reactant at a target site. We determine the mean walklength of the diffusing reactant until irreversible reaction occurs. We use generating functions and the theory of finite Markov processes, as well as MC simulations. To investigate the dependence of walklength on lattice size we compute the first, finite size corrections to the Green function of the sc lattice, and provide a Padé approximation for this quantity. Finally, we consider the case where both reactant and co-reactant undergo synchronous nearest-neighbor displacements ["two-walker (2W) problem"]. In this case, reactant and co-reactant can individually be trapped with probability $s$ at any vacant lattice site, or can undergo an irreversible reaction on first encounter at any site. When $s=0$ we find that, both for the 1W and the 2W problem, for lattices with (approximately) the same number of sites, the mean walklength is smaller (and hence the reaction efficiency greater) in $d=3$ than in $d=2$. Increasing $s$ tends to reduce differences in system dimensionality, and distinctions between the 1W problem and the 2W problem. Our model provides a good starting point to develop studies on the efficiency of apparently diverse diffusion-reaction processes, such as diffusion on a partially poisoned catalytic substrate or photosynthetic trapping of excitations.

cond-mat.stat-mech↗

arXiv2018

Reaction-diffusion kinetics in growing domains

Reaction-diffusion models have been used over decades to study biological systems. In this context, evolution equations for probability distribution functions and the associated stochastic differential equations have nowadays become indispensable tools. In population dynamics, say, such approaches are utilized to study many systems, e.g., colonies of microorganisms or ecological systems. While the majority of studies focus on the case of a static domain, the time-dependent case is also important, as it allows one to deal with situations where the domain growth takes place over time scales that are relevant for the computation of reaction rates and of the associated reactant distributions. Such situations are indeed frequently encountered in the field of developmental biology, notably in connection with pattern formation, embryo growth or morphogen gradient formation. In this chapter, we review some recent advances in the study of pure diffusion processes in growing domains. These results are subsequently taken as a starting point to study the kinetics of a simple reaction-diffusion process, i.e., the encounter-controlled annihilation reaction. The outcome of the present work is expected to pave the way for the study of more complex reaction-diffusion systems of possible relevance in various fields of research.

cond-mat.stat-mech↗

arXiv2017

Continuous-time random-walk model for anomalous diffusion in expanding media

Expanding media are typical in many different fields, e.g. in Biology and Cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties. Here, we focus on such effects when the diffusion process is described by the Continuous Time Random Walk (CTRW) model. For the case where the jump length and the waiting time probability density functions (pdfs) are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Lévy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Green's function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. For a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This "Big Crunch" effect stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model. In the case of an exponential expansion, exact recurrence relations for the Laplace-transformed moments are obtained. Our results confirm the intuitive expectation that the medium expansion hinders the mixing of diffusive particles occupying separate regions. In the case of Lévy flights, we quantify this effect by means of the so-called "Lévy horizon".

cond-mat.stat-mech↗

arXiv2016

Diffusion in an expanding medium: Fokker-Planck equation, Green's function and first-passage properties

We present a classical, mesoscopic derivation of the Fokker-Planck equation for diffusion in an expanding medium. To this end, we take a conveniently generalized Chapman-Kolmogorov equation as the starting point. We obtain an analytical expression for the Green's function (propagator) and investigate both analytically and numerically how this function and the associated moments behave. We also study first-passage properties in expanding hyperspherical geometries. We show that in all cases the behavior is determined to a great extent by the so-called Brownian conformal time $τ(t)$, which we define via the relation $\dot τ=1/a^2$, where $a(t)$ is the expansion scale factor. If the medium expansion is driven by a power law [$a(t) \propto t^γ$ with $γ>0$], we find interesting crossover effects in the mixing effectiveness of the diffusion process when the characteristic exponent $γ$ is varied. Crossover effects are also found at the level of the survival probability and of the moments of the first passage-time distribution with two different regimes separated by the critical value $γ=1/2$. The case of an exponential scale factor is analyzed separately both for expanding and contracting media. In the latter situation, a stationary probability distribution arises in the long time limit.

cond-mat.stat-mech↗

arXiv2016

Anomalous diffusion and FRAP dynamics in the random comb model

We address the problem of diffusion on a comb whose teeth display a varying length. Specifically, the length $\ell$ of each tooth is drawn from a probability distribution displaying the large-$\ell$ behavior $P(\ell) \sim \ell^{-(1+α)}$ ($α>0$). Our method is based on the mean-field description provided by the well-tested CTRW approach for the random comb model, and the obtained analytical result for the diffusion coefficient is confirmed by numerical simulations. We subsequently incorporate retardation effects arising from binding/unbinding kinetics into our model and obtain a scaling law characterizing the corresponding change in the diffusion coefficient. Finally, our results for the diffusion coefficient are used as an input to compute concentration recovery curves mimicking FRAP experiments in comb-like geometries such as spiny dendrites. We show that such curves cannot be fitted perfectly by a model based on scaled Brownian motion, i.e., a standard diffusion equation with a time-dependent diffusion coefficient. However, differences between the exact curves and such fits are small, thereby providing justification for the practical use of models relying on scaled Brownian motion as a fitting procedure for recovery curves arising from particle diffusion in comb-like systems.

cond-mat.stat-mech↗

arXiv2015

Optimal search strategies of space-time coupled random walkers with finite lifetimes

We present a simple paradigm for detection of an immobile target by a space-time coupled random walker with a finite lifetime. The motion of the walker is characterized by linear displacements at a fixed speed and exponentially distributed duration, interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed. We call these walkers "mortal creepers". A mortal creeper may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate $ω_m$. While still alive, the creeper has a finite mean frequency $ω$ of change of the direction of motion. In particular, we consider the efficiency of the target search process, characterized by the probability that the creeper will eventually detect the target. Analytic results confirmed by numerical results show that there is an $ω_m$-dependent optimal frequency $ω=ω_{opt}$ that maximizes the probability of eventual target detection. We work primarily in one-dimensional ($d=1$) domains and examine the role of initial conditions and of finite domain sizes. Numerical results in $d=2$ domains confirm the existence of an optimal frequency of change of direction, thereby suggesting that the observed effects are robust to changes in dimensionality. In the $d=1$ case, explicit expressions for the probability of target detection in the long time limit are given. In the case of an infinite domain, we compute the detection probability for arbitrary times and study its early- and late-time behavior. We further consider the survival probability of the target in the presence of many independent creepers beginning their motion at the same location and at the same time. We also consider a version of the standard "target problem" in which many creepers start at random locations at the same time.

cond-mat.stat-mech↗

arXiv2014

A reaction-subdiffusion model of fluorescence recovery after photobleaching (FRAP)

Anomalous diffusion, in particular subdiffusion, is frequently invoked as a mechanism of motion in dense biological media, and may have a significant impact on the kinetics of binding/unbinding events at the cellular level. In this work we incorporate anomalous diffusion in a previously developed model for FRAP experiments. Our particular implementation of subdiffusive transport is based on a continuous time random walk (CTRW) description of the motion of fluorescent particles, as CTRWs lend themselves particularly well to the inclusion of binding/unbinding events. In order to model switching between bound and unbound states of fluorescent subdiffusive particles, we derive a fractional reaction-subdiffusion equation of rather general applicability. Using suitable initial and boundary conditions, this equation is then incorporated in the model describing two-dimensional kinetics of FRAP experiments. We find that this model can be used to obtain excellent fits to experimental data. Moreover, recovery curves corresponding to different radii of the circular bleach spot can be fitted by a single set of parameters. While not enough evidence has been collected to claim with certainty that CTRW is the underlying transport mechanism in FRAP experiments, the compatibility of our results with experimental data fuels the discussion as to whether normal diffusion or anomalous diffusion is the appropriate model, and as to whether anomalous diffusion effects are important to fully understand the outcomes of FRAP experiments. On a more technical side, we derive explicit analytic solutions of our model in certain limits.

cond-mat.stat-mech↗

arXiv2013

Quasi-resonant neutralization of He+ ions at a germanium surface

When low-energy He ions are scattered from a Ge surface, the fraction of positive ions exhibits characteristic oscillations as a function of ion energy. These oscillations are caused by quasi-resonant neutralization (qRN), a process which can be observed for materials with a narrow band nearly resonant with the unperturbed He-1s level. In the present manuscript, we employ recently developed theoretical methods to extract quantitative information on the efficiency of qRN. Our evaluation reveals that qRN is a highly efficient process leading to ion fractions two orders of magnitude lower than in systems for which neutralization is only due to Auger processes.

cond-mat.mtrl-sci↗