The mean first passage time, one of the important characteristics for a stochastic process, is often calculated assuming the observation time is infinite. However, in practice, the observation time, T, is always finite and the mean first passage time (MFPT) is dependent on the length of the observation time. In this work, we investigate the observation time dependence of the MFPT of a particle freely moving in the interval [-L,L] for a simple diffusion model and four different models of subdiffusion, the fractional diffusion equation (FDE), scaled Brown motion (SBM), fractional Brownian motion (FBM), and stationary Markovian approximation model of SBM and FBM. We find that the MFPT is linearly dependent on T in the small T limit for all the models investigated, while the large-T behavior of the MFPT is sensitive to stochastic properties of the transport model in question. We also discuss the relationship between the observation time, T, dependence and the travel-length, L, dependence of the MFPT. Our results suggest the observation time dependency of the MFPT can serve as an experimental measure that is far more sensitive to stochastic properties of transport processes than the mean square displacement.
We correct a previously erroneous calculation [Phys. Rev. E 62, 6065 (2000)] of the mean first passage time of a subdiffusive process to reach either end of a finite interval in one dimension. The mean first passage time is in fact infinite.
The approach the first-passage time (FPT) of a random process to a certain level is applied to the description of radiation-enhanced diffusion. This is an integral approach to describing the problem of radiation-enhanced diffusion, which does not specify the details of the process but takes into account the overall change in entropy during the process. Relationships are obtained to take into account the influence of radiation effects on the diffusion coefficient. The theoretical results obtained are compared with the experimental ones.
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 fun…
Consider a network embedded in the 2D plane, where a particle diffuses along the edges of the network. It is clear that over short length scales a particle moves along a single edge and thus undergoes one-dimensional diffusion. However, on larger length scales it is no longer immediately clear how the transport will behave. One could intuit that as the network is embedded in two dimensions for "large enough" length scales the transport will also appear two-dimensional. Is this true for all networks? Can we quantify the length scales upon which this transition occurs? What is the transport beha…
Evaluating the completion time of a random algorithm or a running stochastic process is a valuable tip not only from a purely theoretical, but also pragmatic point of view. In the formal sense, this kind of a task is specified in terms of the first-passage time statistics. Although first-passage properties of diffusive processes, usually modeled by different types of the linear differential equations, are permanently explored with unflagging intensity, there still exists noticeable niche in this subject concerning the study of the non-linear diffusive processes. Therefore, the objective of the…
We investigate non-equilibrium fluctuations of a solid surface governed by the stochastic Mullins-Herring equation with conserved noise. This equation describes surface diffusion of adatoms accompanied by their exchange between the surface and the bulk of the solid, when desorption of adatoms is negligible. Previous works dealt with dynamic scaling behavior of the fluctuating interface. Here we determine the probability that the interface first reaches a large given height at a specified time. We also find the optimal time history of the interface conditional on this non-equilibrium fluctuatio…
We propose a generalized diffusion equation for a flat Euclidean space subjected to a continuous infinitesimal scale transform. For the special cases of an algebraic or exponential expansion/contraction, governed by time-dependent scale factors $a(t)\sim t^λ$ and $a(t)\sim \exp(μt)$, the partial differential equation is solved analytically and the asymptotic scaling behavior, as well as the dynamical exponents, are derived. Whereas in the algebraic case the two processes (diffusion and expansion) compete and a crossover is observed, we find that for exponential dynamics the expansion dominates…
We study the diffusive motion of a test particle in a two-dimensional comb structure consisting of a main backbone channel with continuously distributed side branches, in the presence of stochastic Markovian resetting to the initial position of the particle. We assume that the motion along the infinitely long branches is biased by a confining potential. The crossover to the steady state is quantified in terms of a large deviation function, which is derived for the first time for comb structures in present paper. We show that the relaxation region is demarcated by a nonlinear "light-cone" beyon…
The diffusion equation is the primary tool to study the movement dynamics of a free Brownian particle, but when spatial heterogeneities in the form of permeable interfaces are present, no fundamental equation has been derived. Here we obtain such an equation from a microscopic description using a lattice random walk model. The sought after Fokker-Planck description and the corresponding backward Kolmogorov equation are employed to investigate first-passage and local time statistics and gain new insights. Among them a surprising phenomenon, in the case of a semibounded domain, is the appearance…
Measurements of spatial diffusion of cold atoms in optical lattices have revealed anomalous super-diffusion, which is controlled by the depth of the optical lattice. We use first passage time statistics to derive the diffusion front of the atoms. In particular, the distributions of areas swept under the first passage curve till its first arrival, and of areas under the Bessel excursion are shown to be powerful tools in the analysis of the atomic cloud. A rather general relation between first passage time statistics and diffusivity is discussed, showing that first passage time analysis is a use…
We explore the effect of stochastic resetting on the first-passage properties of space-dependent diffusion in presence of a constant bias. In our analytically tractable model system, a particle diffusing in a linear potential $U(x)\proptoμ|x|$ with a spatially varying diffusion coefficient $D(x)=D_0|x|$ undergoes stochastic resetting, i.e., returns to its initial position $x_0$ at random intervals of time, with a constant rate $r$. Considering an absorbing boundary placed at $x_a<x_0$, we first derive an exact expression of the survival probability of the diffusing particle in the Laplace spac…
We calculate the first passage time distribution for diffusion through a cylindrical pore with sticky walls. A particle diffusively explores the interior of the pore through a series of binding and unbinding events with the cylinder wall. Through a diagrammatic expansion we obtain first passage time statistics for the particle's exit from the pore. Connections between the model and nucleocytoplasmic transport in cells are discussed.
We investigate the long-time behavior of the survival probability of a tagged particle in a single-file diffusion in a finite interval. The boundary conditions are of two types: 1) one boundary is absorbing the second is reflecting, 2) both boundaries are absorbing. For each type of the boundary conditions we consider two types of initial conditions: a) initial number of particles N is given, b) initial concentration of particles is given (N is random). In all four cases the tagged-particle survival probability exhibits different asymptotic behavior. When the both boundaries are absorbing we a…
The standard setup for single-file diffusion is diffusing particles in one dimension which cannot overtake each other, where the dynamics of a tracer (tagged) particle is of main interest. In this article we generalise this system and investigate first-passage properties of a tracer particle when flanked by crowder particles which may, besides diffuse, unbind (rebind) from (to) the one-dimensional lattice with rates $k_{\rm off}$ ($k_{\rm on}$). The tracer particle is restricted to diffuse with rate $k_D$ on the lattice. Such a model is relevant for the understanding of gene regulation where r…
We study the first passage time processes of anomalous diffusion on self similar curves in two dimensions. The scaling properties of the mean square displacement and mean first passage time of the ballistic motion, fractional Brownian motion and subordinated walk on different fractal curves (loop erased random walk, harmonic explorer and percolation front) are derived. We also define natural parametrized subordinated Schramm Loewner evolution (NS-SLE) as a mathematical tool that can model diffusion on fractal curves. The scaling properties of the mean square displacement and mean first passage…
We study the first passage time (FPT) problem in Levy type of anomalous diffusion. Using the recently formulated fractional Fokker-Planck equation, we obtain an analytic expression for the FPT distribution which, in the large passage time limit, is characterized by a universal power law. Contrasting this power law with the asymptotic FPT distribution from another type of anomalous diffusion exemplified by the fractional Brownian motion, we show that the two types of anomalous diffusions give rise to two distinct scaling behavior.
Many out of equilibrium phenomena, such as diffusion-limited reactions or target search processes, are controlled by first-passage events. So far the general determination of the mean first-passage time (FPT) to a target in confinement has left aside aging processes, involved in contexts as varied as glassy dynamics, tracer diffusion in biological membranes or transport of cold atoms in optical lattices. Here we consider general non-Markovian scale-invariant processes in arbitrary dimension, displaying aging, and demonstrate that all the moments of the FPT obey universal scalings with the conf…
It is shown that the single-step periodic signal (periodic telegraph signal) can not produce coherent stochastic resonance for diffusion on a segment with one absorbing and one reflecting end points while the multi-step periodic signal does. The general features of this process are exihibited. The resonant frequency is found to decrease and the mean first passage time at resonant frequency increases linearly, as we increase the length of the medium. The cycle variable is shown to be the proper argument to express the first passage time density function at resonance. A formula for first passage…
We present rigorous results for the mean first passage time and first passage time statistics for two-channel Markov additive diffusion in a 3-dimensional spherical domain. Inspired by biophysical examples we assume that the particle can only recognise the target in one of the modes, which is shown to effect a non-trivial first passage behaviour. We also address the scenario of intermittent immobilisation. In both cases we prove that despite the perfectly non-recurrent motion of two-channel Markov additive diffusion in 3 dimensions the first passage statistics at long times do not display Pois…
A rapidly increasing number of systems is identified in which the stochastic motion of tracer particles follows the Brownian law $\langle\mathbf{r}^2(t) \rangle\simeq Dt$ yet the distribution of particle displacements is strongly non-Gaussian. A central approach to describe this effect is the diffusing diffusivity (DD) model in which the diffusion coefficient itself is a stochastic quantity, mimicking heterogeneities of the environment encountered by the tracer particle on its path. We here quantify in terms of analytical and numerical approaches the first passage behaviour of the DD model. We…
Systems where resource availability approaches a critical threshold are common to many engineering and scientific applications and often necessitate the estimation of first passage time statistics of a Brownian motion (Bm) driven by time-dependent drift and diffusion coefficients. Modeling such systems requires solving the associated Fokker-Planck equation subject to an absorbing barrier. Transitional probabilities are derived via the method of images, whose applicability to time dependent problems is shown to be limited to state-independent drift and diffusion coefficients that only depend on…
We investigate a stochastic process consisting of a two-dimensional particle with anisotropic diffusion, mortality rate and a drift velocity, in the presence of an absorbing boundary. After the particle has encountered the boundary, the process is restarted with updated values of its diffusion constants and drift velocity. We then derive the joint probability that, after $M$ encounters, the particle is absorbed at a point of the boundary at a given time and show that, under certain conditions, the eventual hitting probability corresponds to a Bessel distribution. In the context of the man-mosq…
We obtain the first passage time density for a Lévy flight random process from a subordination scheme. By this method, we infer the asymptotic behavior directly from the Brownian solution and the Sparre Andersen theorem, avoiding explicit reference to the fractional diffusion equation. Our results corroborate recent findings for Markovian Lévy flights and generalize to broad waiting times.
We study the distribution of first passage time (FPT) in Levy type of anomalous diffusion. Using recently formulated fractional Fokker-Planck equation we obtain three results. (1) We derive an explicit expression for the FPT distribution in terms of Fox or H-functions when the diffusion has zero drift. (2) For the nonzero drift case we obtain an analytical expression for the Laplace transform of the FPT distribution. (3) We express the FPT distribution in terms of a power series for the case of two absorbing barriers. The known results for ordinary diffusion (Brownian motion) are obtained as s…
Based on the mean first passage time (MFPT) theory, we derive the expression of the MFPT in the energy-diffusion controlled regime with a power-law distribution. We discuss the finite barrier effect (i.e. thermal energy is not small with respect to the potential barrier) and compare it with the Kramers infinite barrier result both in the power-law distribution and in a Maxwell-Boltzmann distribution. It is shown that the MFPT with the power-law distribution extends the Kramers' low damping result to the relatively low barrier. We pay attention to the energy-diffusion controlled regime which is…
We propose a unifying theoretical framework for the analysis of first-passage time distributions in two important classes of stochastic processes in which the diffusivity of a particle evolves randomly in time. In the first class of "diffusing diffusivity" models, the diffusivity changes continuously via a prescribed stochastic equation. In turn, the diffusivity switches randomly between discrete values in the second class of "switching diffusion" models. For both cases, we quantify the impact of the diffusivity dynamics onto the first-passage time distribution of a particle via the moment-gen…
Aging, the dependence of the dynamics of a physical process on the time $t_a$ since its original preparation, is observed in systems ranging from the motion of charge carriers in amorphous semiconductors over the blinking dynamics of quantum dots to the tracer dispersion in living biological cells. Here we study the effects of aging on one of the most fundamental properties of a stochastic process, the first passage dynamics. We find that for an aging continuous time random walk process the scaling exponent of the density of first passage times changes twice as the aging progresses and reveals…
Characterization of composite materials, whose properties vary in space over microscopic scales, has become a problem of broad interdisciplinary interest. In particular, estimation of the inhomogeneous transport coefficients, e.g. the diffusion coefficient or the heat conductivity which shape important processes in biology and engineering, is a challenging task. The analysis of such systems is further complicated, because two alternative formulations of the inhomogeneous transport equations exist in the literature -- the Smoluchowski and Fokker-Planck equations, which are also related to the s…
In this paper, we investigate a Brownian motion (BM) with purely time dependent drift and difusion by suggesting and examining several Brownian functionals which characterize the lifetime and reactivity of such stochastic processes. We introduce several probability distribution functions (PDFs) associated with such time dependent BMs. For instance, for a BM with initial starting point $x_0$, we derive analytical expressions for : (i) the PDF $P(t_f|x_0)$ of the first passage time $t_f$ which specify the lifetime of such stochastic process, (ii) the PDF $P(A|x_0)$ of the area A till the first p…
We develop a model to compute the first-passage time of a random walker in a crowded environment. Hard-core particles with the same size and diffusion coefficient than the tracer diffuse, and the model allows to compute the first passage time of the tracer on euclidian lattices. The result is compared to classical Nakazato-Kitahara model, and extends previous results obtained for persistent random walker. The crowding in a confined media acts as a memory effect, and thus lead to a persistent-like behavior.
We present a novel computational method of first-passage times between a starting site and a target site of regular bounded lattices. We derive accurate expressions for all the moments of this first-passage time, validated by numerical simulations. Their range of validity is discussed. We also consider the case of a starting site and two targets. In addition, we present the extension to continuous Brownian motion. These results are of great relevance to any system involving diffusion in confined media.
Brownian motion is a ubiquitous physical phenomenon across the sciences. After its discovery by Brown and intensive study since the first half of the 20th century, many different aspects of Brownian motion and stochastic processes in general have been addressed in Statistical Physics. In particular, there now exist a very large range of applications of stochastic processes in various disciplines. Here, we highlight some of the advances in stochastic processes prompted by novel experimental methods such as superresolution microscopy. Here we provide a summary of some of the recent developments …
We study the dynamics of protein folding via statistical energy-landscape theory. In particular, we concentrate on the local-connectivity case with the folding progress described by the fraction of native conformations. We obtain information for the first passage-time (FPT) distribution and its moments. The results show a dynamic transition temperature below which the FPT distribution develops a power-law tail, a signature of the intermittency phenomena of the folding dynamics. We also discuss the possible application of the results to single-molecule dynamics experiments.
For transport processes in geometrically restricted domains, the mean first-passage time (MFPT) admits a general scaling dependence on space parameters for diffusion, anomalous diffusion, and diffusion in disordered or fractal media. For transport in self-similar fractal structures, we obtain a new expression for the source-target distance dependence of the MFPT that exhibits both the leading power law behavior, depending on the Hausdorff and spectral dimension of the fractal, as well as small log periodic oscillations that are a clear and definitive signal of the underlying fractal structure.…
We study the statistics of last-passage time for linear diffusions. First we present an elementary derivation of the Laplace transform of the probability density of the last-passage time, thus recovering known results from the mathematical literature. We then illustrate them on several explicit examples. In a second step we study the spectral properties of the Schrödinger operator associated to such diffusions in an even potential $U(x) = U(-x)$, unveiling the role played by the so-called Weyl coefficient. Indeed, in this case, our approach allows us to relate the last-passage times for dual d…
Synaptic transmission between neurons is governed by a cascade of stochastic reaction-diffusion events that lead to calcium-induced vesicle release of neurotransmitter. Since experimental measurements of such systems are challenging due their nanometer and sub-millisecond scale, numerical simulations remain the principal tool for studying calcium dependent synaptic vesicle fusion, despite limitations of time-consuming calculations. In this paper we develop an analytical solution to rapidly explore dynamical stochastic reaction-diffusion problems, based on first-passage times. This is the first…
First passage under restart has recently emerged as a conceptual framework suitable for the description of a wide range of phenomena, but the endless variety of ways in which restart mechanisms and first passage processes mix and match hindered the identification of unifying principles and general truths. Hope that these exist came from a recently discovered universality displayed by processes under optimal, constant rate, restart---but extensions and generalizations proved challenging as they marry arbitrarily complex processes and restart mechanisms. To address this challenge, we develop a g…
Analytic and Monte Carlo Approximations to the Distribution of the First Passage Time of the Drifted Diffusion with Stochastic Resetting and Mixed Boundary Conditions
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 tractab…
The problems of escape from metastable state in randomly flipping potential and of diffusion in fast fluctuating periodic potentials are considered. For the overdamped Brownian particle moving in a piecewise linear dichotomously fluctuating metastable potential we obtain the mean first-passage time (MFPT) as a function of the potential parameters, the noise intensity and the mean rate of switchings of the dichotomous noise. We find noise enhanced stability (NES) phenomenon in the system investigated and the parameter region of the fluctuating potential where the effect can be observed. For the…
A recursive formula for the $n^\text{th}$ survival function and the $n^\text{th}$ first passage time distribution for jump and diffusion processes. Applications to the pricing of $n^\text{th}$-to-default CDS
We employ simulations of model proteins to study folding on rugged energy landscapes. We construct ``first-passage'' networks as the system transitions from unfolded to native states. The nodes and bonds in these networks correspond to basins and transitions between them in the energy landscape. We find power-laws between the folding time and number of nodes and bonds. We show that these scalings are determined by the fractal properties of first-passage networks. Reliable folding is possible in systems with rugged energy landscapes because first passage networks have small fractal dimension.
Fractional Brownian motion is a generalised Gaussian diffusive process that is found to describe numerous stochastic phenomena in physics and biology. Here we introduce a multi-dimensional fractional Brownian motion (FBM) defined as a superposition of conventional FBM for each coordinate in analogy to multi-dimensional Brownian motion, and study its first passage properties. Starting from the well-established first passage time statistics of one-dimensional FBM and the associated approximation schemes, we explore the first passage time behaviour of multi-dimensional FBM and compare these resul…
Diffusion and first passage in the presence of stochastic resetting and potential bias have been of recent interest. We study a few models, systematically progressing in their complexity, to understand the usefulness of resetting. In the parameter space of the models, there are multiple continuous and discontinuous transitions where the advantage of resetting vanishes. We show these results analytically exactly for a tent-potential, and numerically accurately for a quartic-potential relevant to a magnetic system at low temperatures. We find that the spatial asymmetry of the potential across th…
We study a Brownian particle diffusing under a time-modulated stochastic resetting mechanism to a fixed position. The rate of resetting r(t) is a function of the time t since the last reset event. We derive a sufficient condition on r(t) for a steady-state probability distribution of the position of the particle to exist. We derive the form of the steady-state distributions under some particular choices of r(t) and also consider the late time relaxation behavior of the probability distribution. Finally we consider first passage time properties for the Brownian particle to reach the origin and …
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 def…
We consider an anisotropic needle-like Brownian particle with nematic symmetry confined in a $2D$ domain. For this system, the coupling of translational and rotational diffusion makes the process ${\bf x} (t)$ of the positions of the particle non Markovian. Using scaling arguments, a Gaussian approximation and numerical methods, we determine the mean first passage time $\langle\mathbf{T}\rangle$ of the particle to a target of radius $a$ and show in particular that $\langle\mathbf{T}\rangle \sim a^{-1/2}$ for $a\to 0$, in contrast with the classical logarithmic divergence obtained in the case o…
