First Passage Time Distribution Research Articles

Active Brownian motion with directional reversals

Active Brownian motion with intermittent direction reversals is common in bacteria like Myxococcus xanthus and Pseudomonas putida. We show that, for such a motion in two dimensions, the presence of the two timescales set by the rotational diffusion constant D_{R} and the reversal rate γ gives rise to four distinct dynamical regimes: (I) t≪min(γ^{-1},D_{R}^{-1}), (II) γ^{-1}≪t≪D_{R}^{-1}, (III) D_{R}^{-1}≪t≪γ^{-1}, and (IV) t≫max(γ^{-1},D_{R}^{-1}), showing distinct behaviors. We characterize these behaviors by analytically computing the position distribution and persistence exponents. The position distribution shows a crossover from a strongly nondiffusive and anisotropic behavior at short times to a diffusive isotropic behavior via an intermediate regime, II or III. In regime II, we show that, the position distribution along the direction orthogonal to the initial orientation is a function of the scaled variable z∝x_{⊥}/t with a nontrivial scaling function, f(z)=(2π^{3})^{-1/2}Γ(1/4+iz)Γ(1/4-iz). Furthermore, by computing the exact first-passage time distribution, we show that a persistence exponent α=1 emerges due to the direction reversal in this regime.

Numerical analysis of first-passage processes in finite Markov chains exhibiting metastability.

We describe state-reduction algorithms for the analysis of first-passage processes in discrete- and continuous-time finite Markov chains. We present a formulation of the graph transformation algorithm that allows for the evaluation of exact mean first-passage times, stationary probabilities, and committor probabilities for all nonabsorbing nodes of a Markov chain in a single computation. Calculation of the committor probabilities within the state-reduction formalism is readily generalizable to the first hitting problem for any number of alternative target states. We then show that a state-reduction algorithm can be formulated to compute the expected number of times that each node is visited along a first-passage path. Hence, all properties required to analyze the first-passage path ensemble (FPPE) at both a microscopic and macroscopic level of detail, including the mean and variance of the first-passage time distribution, can be computed using state-reduction methods. In particular, we derive expressions for the probability that a node is visited along a direct transition path, which proceeds without returning to the initial state, considering both the nonequilibrium and equilibrium (steady-state) FPPEs. The reactive visitation probability provides a rigorous metric to quantify the dynamical importance of a node for the productive transition between two endpoint states and thus allows the local states that facilitate the dominant transition mechanisms to be readily identified. The state-reduction procedures remain numerically stable even for Markov chains exhibiting metastability, which can be severely ill-conditioned. The rare event regime is frequently encountered in realistic models of dynamical processes, and our methodology therefore provides valuable tools for the analysis of Markov chains in practical applications. We illustrate our approach with numerical results for a kinetic network representing a structural transition in an atomic cluster.

Diffusion and escape from polygonal channels: extreme values and geometric effects

Polygonal billiards are an example of pseudo-chaotic dynamics, a combination of integrable evolution and sudden jumps due to conical singular points that arise from the corners of the polygons. Such pseudo-chaotic behaviour, often characterised by an algebraic separation of nearby trajectories, is believed to be linked to the wild dependence that particle transport has on the fine details of the billiard table. Here, we address this relation through a detailed numerical study of the statistics of displacement in a family of polygonal channel billiards with parallel walls. We show that transport is characterised by strong anomalous diffusion, with a mean square displacement that scales in time faster than linear, and with a probability density of the displacement exhibiting exponential tails and ballistic fronts. In channels of finite length, the distribution of first-passage times is characterised by fat tails, with a mean first-passage time that diverges when the aperture angle is rational. These findings have non-trivial consequences for a variety of experiments.

Partial derivatives for the first-passage time distribution in Wiener diffusion models

The Wiener diffusion model with two absorbing barriers is frequently used in modeling decisions and decision latencies jointly. We derive representations of the partial derivatives of the density and cumulative distribution function of first-passage times. Different representations, based on infinite series, differ in convergence for small and large times, and we provide methods that control the approximation errors. These methods and their implementation in an R package are helpful whenever gradient information is required as is the case in many frequentist and Bayesian approaches to modeling.

Extreme first passage times of piecewise deterministic Markov processes

The time it takes the fastest searcher out of N ≫ 1 searchers to find a target determines the timescale of many physical, chemical, and biological processes. This time is called an extreme first passage time (FPT) and is typically much faster than the FPT of a single searcher. Extreme FPTs of diffusion have been studied for decades, but little is known for other types of stochastic processes. In this paper, we study the distribution of extreme FPTs of piecewise deterministic Markov processes (PDMPs). PDMPs are a broad class of stochastic processes that evolve deterministically between random events. Using classical extreme value theory, we prove general theorems which yield the distribution and moments of extreme FPTs in the limit of many searchers based on the short time distribution of the FPT of a single searcher. We then apply these theorems to some canonical PDMPs, including run and tumble searchers in one, two, and three space dimensions. We discuss our results in the context of some biological systems and show how our approach accounts for an unphysical property of diffusion which can be problematic for extreme statistics.

Reliability Analysis Based on a Gamma-Gaussian Deconvolution Degradation Modeling with Measurement Error

In most degradation tests, the measuring processes is affected by several conditions that may cause variation in the observed measures. As the measuring process is inherent to the degradation testing, it is important to establish schemes that define a certain level of permissible measurement error such that a robust reliability estimation can be obtained. In this article, an approach to deal with measurement error in degradation processes is proposed, the method focuses on studying the effect of such error in the reliability assessment. This approach considers that the true degradation is a function of the observed degradation and the measurement error. As the true degradation is not directly observed it is proposed to obtain an estimate based on a deconvolution operation, which considers the subtraction of random variables such as the observed degradation and the measurement error. Given that the true degradation is free of measurement error, the first-passage time distribution will be different from the observed degradation. For the establishment of a control mechanism, these two distributions are compared using different indices, which account to describe the differences between the observed and true degradation. By defining critical levels of these indices, the reliability assessment may be obtained under a known level of measurement error. An illustrative example based on a fatigue-crack growth dataset is presented to illustrate the applicability of the proposed scheme, the reliability assessment is developed, and some important insights are provided.

First passage time study of DNA strand displacement

DNA strand displacement, in which a single-stranded nucleic acid invades a DNA duplex, is pervasive in genomic processes and DNA engineering applications. The kinetics of strand displacement have been studied in bulk; however, the kinetics of the underlying strand exchange were obfuscated by a slow bimolecular association step. Here, we use a novel single-molecule fluorescence resonance energy transfer approach termed the “fission” assay to obtain the full distribution of first passage times of unimolecular strand displacement. At a frame time of 4.4 ms, the first passage time distribution for a 14-nucleotide displacement domain exhibited a nearly monotonic decay with little delay. Among the eight different sequences we tested, the mean displacement time was on average 35 ms and varied by up to a factor of 13. The measured displacement kinetics also varied between complementary invaders and between RNA and DNA invaders of the same base sequence, except for T → U substitution. However, displacement times were largely insensitive to the monovalent salt concentration in the range of 0.25–1 M. Using a one-dimensional random walk model, we infer that the single-step displacement time is in the range of ∼30–300 μs, depending on the base identity. The framework presented here is broadly applicable to the kinetic analysis of multistep processes investigated at the single-molecule level.

Mean exit time for diffusion on irregular domains

Many problems in physics, biology, and economics depend upon the duration of time required for a diffusing particle to cross a boundary. As such, calculations of the distribution of first passage time, and in particular the mean first passage time, is an active area of research relevant to many disciplines. Exact results for the mean first passage time for diffusion on simple geometries, such as lines, discs and spheres, are well-known. In contrast, computational methods are often used to study the first passage time for diffusion on more realistic geometries where closed-form solutions of the governing elliptic boundary value problem are not available. Here, we develop a perturbation solution to calculate the mean first passage time on irregular domains formed by perturbing the boundary of a disc or an ellipse. Classical perturbation expansion solutions are then constructed using the exact solutions available on a disc and an ellipse. We apply the perturbation solutions to compute the mean first exit time on two naturally-occurring irregular domains: a map of Tasmania, an island state of Australia, and a map of Taiwan. Comparing the perturbation solutions with numerical solutions of the elliptic boundary value problem on these irregular domains confirms that we obtain a very accurate solution with a few terms in the series only. MATLAB software to implement all calculations is available at https://github.com/ProfMJSimpson/Exit_time.

Evaluation of mean-time-to-failure based on nonlinear degradation data with applications

In reliability engineering, obtaining lifetime information for highly reliable products is a challenging problem. When a product quality characteristic whose degradation over time can be related to lifetime, then the degradation data can be used to estimate the first-passage (failure) time distribution and the Mean-Time-To-Failure (MTTF) for a given threshold level. To model the degradation data, the commonly used Lévy process modeling approach assumes that the degradation measurements are linearly related to time throughout the lifetime of the product. However, the degradation data may not be linearly related to time in practice. For this reason, trend-renewal-process-type models can be considered for degradation modeling in which a proper trend function is used to transform the degradation data so that the Lévy process approach can be applied. In this article, we study several parametric and semiparametric models and approaches to estimate the first-passage time distribution and MTTF for degradation data that may be not linearly related to time. A Monte Carlo simulation study is used to demonstrate the performance of the proposed methods. In addition, a model selection procedure is proposed to select among different models. Two numerical examples of lithium-ion battery degradation data are applied to illustrate the proposed methodologies.

Roughness, inertia, and diffusion effects on anomalous transport in rough channel flows

We study how the complex interplay between roughness, inertia, and diffusion controls the tracer transport in rough channels. We first simulate the flow and tracer transport over wide ranges of channel roughness, Reynolds number ($Re$), and P\'eclet number ($Pe$) observable in nature. $Pe$ exerts a first-order control on first-passage time distributions, and the effect of roughness on the tracer transport becomes evident with the increase in $Re$. The interplay between the roughness and $Re$ causes eddy flows, which intensify or suppress anomalous transport depending on $Pe$. At infinite $Pe$, the late-time scaling follows a universal power-law scaling, which is explained by conducting a scaling analysis. With extensive numerical simulations and stochastic modeling, we show that the roughness, inertia, and diffusion effects are encoded in Lagrangian velocity statistics represented by velocity distribution and correlation. We finally predict the anomalous transport using a stochastic model that considers the Lagrangian velocity statistics.

First passage time distribution of active thermal particles in potentials

We introduce a perturbative method to calculate all moments of the first-passage time distribution in stochastic one-dimensional processes which are subject to both white and coloured noise. This class of non-Markovian processes is at the centre of the study of thermal active matter, that is self-propelled particles subject to diffusion. The perturbation theory about the Markov process considers the effect of self-propulsion to be small compared to that of thermal fluctuations. To illustrate our method, we apply it to the case of active thermal particles (i) in a harmonic trap (ii) on a ring. For both we calculate the first-order correction of the moment-generating function of first-passage times, and thus to all its moments. Our analytical results are compared to numerics.

Selfsimilarity of diffusions’ first passage times

Considering a general diffusion process that runs over the non-negative half-line, this paper addresses the first-passage time (FPT) to the origin: the time it takes the process to get from an arbitrary fixed positive level to the level zero. Inspired by the special features of Brownian motion, three types of FPT selfsimilarity are introduced: (i) stochastic, which holds in ‘real space’; (ii) Laplace, which holds in ‘Laplace space’; and (iii) joint, which is the combination of the stochastic and Laplace types. Analysis establishes that the three types of FPT selfsimilarity yield, respectively and universally, the following FPT distributions: inverse-gamma; inverse-Gauss; and Levy–Smirnov. Moreover, the analysis explicitly pinpoints the classes of diffusion processes that produce the three types of selfsimilar FPTs. Shifting from general diffusion dynamics to Langevin dynamics, it is shown that the three classes collapse, respectively, to the following specific processes: diffusion in a logarithmic potential; Brownian motion with drift; and Brownian motion. Also, the effect of the Girsanov transformation on the three types of selfsimilar FPTs is investigated, as well as the effect of initiating the diffusion process from its steady-state level (rather than from a fixed positive level). This paper presents a novel approach to the exploration of first-passage times.

Exact first-passage time distributions for three random diffusivity models

We study the extremal properties of a stochastic process x t defined by a Langevin equation , where ξ t is a Gaussian white noise with zero mean, D 0 is a constant scale factor, and V(B t ) is a stochastic ‘diffusivity’ (noise strength), which itself is a functional of independent Brownian motion B t . We derive exact, compact expressions in one and three dimensions for the probability density functions (PDFs) of the first passage time (FPT) t from a fixed location x 0 to the origin for three different realisations of the stochastic diffusivity: a cut-off case V(B t ) = Θ(B t ) (model I), where Θ(z) is the Heaviside theta function; a geometric Brownian motion V(B t ) = exp(B t ) (model II); and a case with (model III). We realise that, rather surprisingly, the FPT PDF has exactly the Lévy–Smirnov form (specific for standard Brownian motion) for model II, which concurrently exhibits a strongly anomalous diffusion. For models I and III either the left or right tails (or both) have a different functional dependence on time as compared to the Lévy–Smirnov density. In all cases, the PDFs are broad such that already the first moment does not exist. Similar results are obtained in three dimensions for the FPT PDF to an absorbing spherical target.

First-passage times in protein folding: exploring the native-like states vs. overcoming the free energy barrier.

Using a model β-hairpin protein as a representative example of simple two-state folders whose kinetics are uncomplicated by the presence of on- and off-pathway intermediates, it is studied how the search for the protein's native state among native-like states affects the folding kinetics. It is revealed that the first-passage time (FPT) distributions are essentially single-exponential not only for the times to overcome the free energy barrier between the unfolded and native-like states but also for the times to find the native state among the native-like ones. The FPT distributions of this type are observed through all studied two-state-like regimes of protein folding, varying from a regime close to two-state folding to a regime close to downhill folding. If the protein explores native-like states for a time much longer than the time to overcome the free energy barrier, which is characteristic of high temperatures, the resulting FPT distribution to reach the native state remains close to exponential but the mean FPT (MFPT) is determined not by the height of the free energy barrier but by the time to explore native-like states. In particular, the mean time to overcome the free energy barrier is in reasonable agreement with the Kramers rate formula and generally far shorter than the overall MFPT to reach the native state. The observed increase of the overall MFPT, as a result of longer exploration of native-like states, may lead to an overestimate of the height of the free energy barrier between the unfolded and folded states when it is calculated from the overall MFPT.

The Effects of Fast Inactivation on Conditional First Passage Times of Mortal Diffusive Searchers

The first time a searcher finds a target is called a first passage time (FPT). In many physical, chemical, and biological processes, the searcher is “mortal,” which means that the searcher might become inactivated (degrade, die, etc.) before finding the target. In the context of intracellular signaling, an important recent work discovered that fast inactivation can drastically alter the conditional FPT distribution of a mortal diffusive searcher, if the searcher is conditioned to find the target before inactivation. In this paper, we prove a general theorem that yields an explicit formula for all the moments of such conditional FPTs in the fast inactivation limit. This formula is quite universal, as it holds under very general conditions on the diffusive searcher dynamics, the target, and the spatial domain. These results prove in significant generality that if inactivation is fast, then the conditional FPT compared to the FPT without inactivation is (i) much faster, (ii) much less affected by spatial heterogeneity, and (iii) much less variable. Our results agree with recent computational and theoretical analysis of a certain discrete intracellular diffusion model and confirm a conjecture related to the effect of spatial heterogeneity on intracellular signaling.

Space-dependent diffusion with stochastic resetting: A first-passage study.

We explore the effect of stochastic resetting on the first-passage properties of space-dependent diffusion in the presence of a constant bias. In our analytically tractable model system, a particle diffusing in a linear potential U(x) ∝ μ|x| with a spatially varying diffusion coefficient D(x) = D0|x| undergoes stochastic resetting, i.e., returns to its initial position x0 at random intervals of time, with a constant rate r. Considering an absorbing boundary placed at xa < x0, we first derive an exact expression of the survival probability of the diffusing particle in the Laplace space and then explore its first-passage to the origin as a limiting case of that general result. In the limit xa → 0, we derive an exact analytic expression for the first-passage time distribution of the underlying process. Once resetting is introduced, the system is observed to exhibit a series of dynamical transitions in terms of a sole parameter, ν≔(1+μD0 -1), that captures the interplay of the drift and the diffusion. Constructing a full phase diagram in terms of ν, we show that for ν < 0, i.e., when the potential is strongly repulsive, the particle can never reach the origin. In contrast, for weakly repulsive or attractive potential (ν > 0), it eventually reaches the origin. Resetting accelerates such first-passage when ν < 3 but hinders its completion for ν > 3. A resetting transition is therefore observed at ν = 3, and we provide a comprehensive analysis of the same. The present study paves the way for an array of theoretical and experimental works that combine stochastic resetting with inhomogeneous diffusion in a conservative force field.

Protein hourglass: Exact first passage time distributions for protein thresholds.

Protein thresholds have been shown to act as an ancient timekeeping device, such as in the time to lysis of Escherichia coli infected with bacteriophage λ. The time taken for protein levels to reach a particular threshold for the first time is defined as the first passage time (FPT) of the protein synthesis system, which is a stochastic quantity. The first few moments of the distribution of first passage times were known earlier, but an analytical expression for the full distribution was not available. In this work, we derive an analytical expression for the first passage times for a long-lived protein. This expression allows us to calculate the full distribution not only for cases of no self-regulation, but also for both positive and negative self-regulation of the threshold protein. We show that the shape of the distribution matches previous experimental data on λ-phage lysis time distributions. We also provide analytical expressions for the FPT distribution with non-zero degradation in Laplace space. Furthermore, we study the noise in the precision of the first passage times described by coefficient of variation (CV) of the distribution as a function of the protein threshold value. We show that under conditions of positive self-regulation, the CV declines monotonically with increasing protein threshold, while under conditions of linear negative self-regulation, there is an optimal protein threshold that minimizes the noise in the first passage times.

Rare events and first passage time statistics from the energy landscape.

We analyze the probability distribution of rare first passage times corresponding to transitions between product and reactant states in a kinetic transition network. The mean first passage times and the corresponding rate constants are analyzed in detail for two model landscapes and the double funnel landscape corresponding to an atomic cluster. Evaluation schemes based on eigendecomposition and kinetic path sampling, which both allow access to the first passage time distribution, are benchmarked against mean first passage times calculated using graph transformation. Numerical precision issues severely limit the useful temperature range for eigendecomposition, but kinetic path sampling is capable of extending the first passage time analysis to lower temperatures, where the kinetics of interest constitute rare events. We then investigate the influence of free energy based state regrouping schemes for the underlying network. Alternative formulations of the effective transition rates for a given regrouping are compared in detail to determine their numerical stability and capability to reproduce the true kinetics, including recent coarse-graining approaches that preserve occupancy cross correlation functions. We find that appropriate regrouping of states under the simplest local equilibrium approximation can provide reduced transition networks with useful accuracy at somewhat lower temperatures. Finally, a method is provided to systematically interpolate between the local equilibrium approximation and exact intergroup dynamics. Spectral analysis is applied to each grouping of states, employing a moment-based mode selection criterion to produce a reduced state space, which does not require any spectral gap to exist, but reduces to gap-based coarse graining as a special case. Implementations of the developed methods are freely available online.

Target finding in fibrous biological environments

We use a lattice model to study first-passage time distributions of target finding events through complex environments with elongated fibers distributed with different anisotropies and volume occupation fractions. For isotropic systems and for low densities of aligned fibers, the three-dimensional search is a Poisson process with the first-passage time exponentially distributed with the most probable finding time at zero. At high enough densities of aligned fibers, elongated channels emerge, reducing the dynamics dimensionality to one dimension. We show how the shape and size of the channels modify the behavior of the first-passage time distribution and its short, intermediate, and long time scales. We develop an exactly solvable model for synthetic rectangular channels, which captures the effects of the tortuous local structure of the elongated channels that naturally emerge in our system. For arbitrary values of the nematic order parameter of fiber orientations, we develop a mapping to the simpler situation of fully aligned fibers at some other effective volume occupation fraction. Our results shed light on the molecular transport of biomolecules between biological cells in complex fibrous environments.

Surface hopping propagator: An alternative approach to diffusion-influenced reactions.

Dynamics of a particle diffusing in a confinement can be seen a sequence of bulk-diffusion-mediated hops on the confinement surface. Here, we investigate the surface hopping propagator that describes the position of the diffusing particle after a prescribed number of encounters with that surface. This quantity plays the central role in diffusion-influenced reactions and determines their most common characteristics such as the propagator, the first-passage time distribution, and the reaction rate. We derive explicit formulas for the surface hopping propagator and related quantities for several Euclidean domains: half-space, circular annuli, circular cylinders, and spherical shells. These results provide the theoretical ground for studying diffusion-mediated surface phenomena. The behavior of the surface hopping propagator is investigated for both "immortal" and "mortal" particles.