First Passage Time Research Articles

The Global Mean First-Passage Time for Degree-Dependent Random Walks in a Class of Fractal Scale-Free Trees

Fractal scale-free structures are widely observed across a range of natural and synthetic systems, such as biological networks, internet architectures, and social networks, providing broad applications in the management of complex systems and the facilitation of dynamic processes. The global mean first-passage time (GMFPT) for random walks on the underlying networks has attracted significant attention as it serves as an important quantitative indicator that can be used in many different fields, such as reaction kinetics, network transport, random search, pathway elaboration, etc. In this study, we first introduce two degree-dependent random walk strategies where the transition probability is depended on the degree of neighbors. Then, we evaluate analytically the GMFPT of two degree-dependent random walk strategies on fractal scale-free tree structures by exploring the connection between first-passage times in degree-dependent random walk strategies and biased random walks on the weighted network. The exact results of the GMFPT for the two degree-dependent random walk strategies are presented and are compared with the GMFPT of the classical unbiased random walk strategy. Our work not only presents a way to evaluate the GMFPT for degree-dependent biased random walk strategies on general networks but also offers valuable insights to enrich the controlling of chemical reactions, network transport, random search, and pathway elaboration.

A First-Passage Model of Intravitreal Drug Delivery and Residence Time-Influence of Ocular Geometry, Individual Variability, and Injection Location.

Standard of care for various retinal diseases involves recurrent intravitreal injections. This motivates mathematical modeling efforts to identify influential factors for ocular drug residence time, aiming to minimize administration frequency. We sought to describe the vitreal diffusion of therapeutics in nonclinical species frequently used during drug development assessments. In human eyes, we investigated the impact of variability in vitreous cavity size and eccentricity, and in injection location, on drug disposition. Using a first-passage time approach, we modeled the transport-controlled distribution of two standard therapeutic protein formats (Fab and IgG) and elimination through anterior and posterior pathways. Anatomical three-dimensional geometries of mouse, rat, rabbit, cynomolgus monkey, and human eyes were constructed using ocular images and biometry datasets. A scaling relationship was derived for comparison with experimental ocular half-lives. Model simulations revealed a dependence of residence time on ocular size and injection location. Delivery to the posterior vitreous resulted in increased vitreal half-life and retinal permeation. Interindividual variability in human eyes had a significant influence on residence time (half-life range of 5-7days), showing a strong correlation to axial length and vitreal volume. Anterior exit was the predominant route of drug elimination. Contribution of the posterior pathway displayed a 3% difference between protein formats but varied between species (10%-30%). The modeling results suggest that experimental variability in ocular half-life is partially attributed to anatomical differences and injection site location. Simulations further suggest a potential role of the posterior pathway permeability in determining species differences in ocular pharmacokinetics.

Lattice random walk dynamics with stochastic resetting in heterogeneous space

Abstract We examine the diffusive dynamics of a lattice random walk subject to resetting in a one-dimensional spatially heterogeneous environment composed of two media separated by an interface. At random times the walker may reset its position to the interface, but only when in the left medium. In addition the spatial heterogeneity results from having unequal diffusivities and biases in the two media. We construct the Master equation for the dynamics of the walker occupation probability in unbounded space, solve it exactly in terms of generating functions, and analyse the dynamics of the first and second moment. Making use of the closed form solution in the unbounded case, we build the analytic solution of the Master equation in finite and semi-infinite domains. By bounding the space on the right with a reflecting boundary we study the first-passage dynamics to a single fully absorbing target placed in the left medium away from the interface. As reset strongly increases the time to reach the target, we find that the first-passage dynamics enter the motion-limited regime even for relative small resetting probability. We also identify a surprising non-monotonic dependence of the first-passage probability mode as a function of the bias. By deriving an analytic expression for the mean first-passage time, we show when its value is independent of the diffusivity and bias in the left medium, uncovering another example of the so-called mean disorder indifference phenomenon.

Emerging cost-time Pareto front for diffusion with stochastic return

Abstract Resetting, in which a system is regularly returned to a given state after a fixed or random duration, has become a useful strategy to optimize the search performance of a system. While earlier theoretical frameworks focused on instantaneous resetting, wherein the system is directly teleported to a given state, there is a growing interest in physical resetting mechanisms that involve a finite return time. However employing such a mechanism involves cost and the effect of this cost on the search time remains largely unexplored. Yet answering this is important in order to design cost-efficient resetting strategies. Motivated from this, we present a thermodynamic analysis of a diffusing particle whose position is intermittently reset to a specific site by employing a stochastic return protocol with external confining trap. We show for a family of potentials U R ( x ) ∼ | x | m with m > 0, it is possible to find optimal potential shape that minimises the expected first-passage time for a given value of the thermodynamic cost, i.e. mean work. By varying this value, we then obtain the Pareto optimal front, and demonstrate a trade-off relation between the first-passage time and the work done.

Controlling and optimizing the transport (search) efficiency with local information on a class of scale-free trees.

The scale-free trees are fundamental dynamics networks with extensive applications in material and engineering fields owing to their high reliability and low power consumption characteristics. Controlling and optimizing transport (search) efficiency on scale-free trees has attracted much attention. In this paper, we first introduce degree-dependent weighted tree by assigning each edge (x,y) a weight wxy=(dxdy)θ, with dx and dy being the degree of nodes x and y, and θ being a controllable parameter. Then, we design a parameterized biased random walk strategy with the transition probability depending on the local information (the degree of neighboring nodes) and a parameter θ. Finally, we evaluate analytically the global mean first-passage time, which is an important indicator for measuring the transport (search) efficiency on the underlying networks, and find the interval for parameter θ where transport (search) efficiency can be improved on a class of scale-free trees. We also analyze the (transfinite) walk dimension for our biased random walk on the scale-free trees and find one can obtain arbitrary transfinite walk dimension in an interval by properly tuning the biased parameter θ. The results obtained here would shed light on controlling and optimizing transport (search) efficiency on objects with scale-free tree structures.

Predicting protracted binding kinetics of polymers: Integral of first-passage times

Predicting protracted binding kinetics of polymers: Integral of first-passage times

Laplacian operator and its square lattice discretization: Green function vs. Lattice Green function for the flat 2-torus and other related 2D manifolds

Abstract The paper investigates the truncation error between the Green function and the lattice Green function (LGF) for the Laplacian operator defined on the 2-torus and its discretization on a regular square lattice. Extensions to the cylinder and the rectangular domain with free (or Neumann) boundary conditions are also proposed. In each of these instances, the Green function and its discrete analog are given in exact analytical closed-form allowing to infer accurate estimates as the lattice spacing tends to zero. As expected, it is shown that the continuum limit of the LGF coincides well with the Green function in every case. In particular, the issue of logarithmic singularity regularization of the Green function by the lattice discretization is addressed through two related application examples regarding the rectangular domain, and devoted to the computation of corner-to-corner resistance of an electrical conducting square and the mean first-passage time between the diagonally opposite vertices of a square for a standard Brownian motion, both derived considering the continuum limit.

Rosetta stone for the population dynamics of spiking neuron networks.

Populations of spiking neuron models have densities of their microscopic variables (e.g., single-cell membrane potentials) whose evolution fully capture the collective dynamics of biological networks, even outside equilibrium. Despite its general applicability, the Fokker-Planck equationgoverning such evolution is mainly studied within the borders of the linear response theory, although alternative spectral expansion approaches offer some advantages in the study of the out-of-equilibrium dynamics. This is mainly due to the difficulty in computing the state-dependent coefficients of the expanded system of differential equations. Here, we address this issue by deriving analytic expressions for such coefficients by pairing perturbative solutions of the Fokker-Planck approach with their counterparts from the spectral expansion. A tight relationship emerges between several of these coefficients and the Laplace transform of the interspike interval density (i.e., the distribution of first-passage times). "Coefficients" like the current-to-rate gain function, the eigenvalues of the Fokker-Planck operator and its eigenfunctions at the boundaries are derived without resorting to integral expressions. For the leaky integrate-and-fire neurons, the coupling terms between stationary and nonstationary modes are also worked out paving the way to accurately characterize the critical points and the relaxation timescales in networks of interacting populations.

Workflow Trace Profiling and Execution Time Analysis in Quantitative Verification

Workflows orchestrate a collection of computing tasks to form a complex workflow logic. Different from the traditional monolithic workflow management systems, modern workflow systems often manifest high throughput, concurrency and scalability. As service-based systems, execution time monitoring is an important part of maintaining the performance for those systems. We developed a trace profiling approach that leverages quantitative verification (also known as probabilistic model checking) to analyse complex time metrics for workflow traces. The strength of probabilistic model checking lies in the ability of expressing various temporal properties for a stochastic system model and performing automated quantitative verification. We employ semi-Makrov chains (SMCs) as the formal model and consider the first passage times (FPT) measures in the SMCs. Our approach maintains simple mergeable data summaries of the workflow executions and computes the moment parameters for FPT efficiently. We describe an application of our approach to AWS Step Functions, a notable workflow web service. An empirical evaluation shows that our approach is efficient for computer high-order FPT moments for sizeable workflows in practice. It can compute up to the fourth moment for a large workflow model with 10,000 states within 70 s.

Two-dimensional Diffusion with a Transverse Gravitational Force: Analytical, Numerical and Simulations Results

Abstract Using the Kalinay and Percus projection method a new effective diffusion coefficient is found. This diffusivity generalizes the well-known results previously reported in the literature. Specifically, it is a coefficient that can be applied to two-dimensional asymmetrical channels under an external gravitational-like potential. Furthermore, Brownian Dynamics simulations are performed, and their agreement with the theoretical results is shown. Also, the Mean First-Passage Time is studied for two-dimensional straight conical channels, where the analytical results are analyzed to understand its applicability range using numerical methods, which are compared with Brownian Dynamics simulations. A remarkable result is also presented: the Mean First-Passage Time assumes a minimum at finite values of the external potential amplitude.

Designing variable flexible sampling systems for verifying harmful microorganisms in food

Foodborne illnesses resulting from the proliferation of excessive harmful microorganisms in foodstuffs pose a significant threat to human society. Sampling inspection for the quantity of pathogenic microorganisms within food products is a productive approach to mitigate the incidence of foodborne diseases. A recently developed generalised variable quick-switch sampling system (GVQSS) is an efficient sampling technology to verify harmful microorganisms in food. However, the rule-switching mechanism in the GVQSS is too simple, making it inflexible and may limit its applicability. To break through this limitation, we apply the Markov chain and mean first-passage time theory to develop two types of variable flexible sampling systems (VFSSs). Through a series of investigations, the various types of VFSSs exhibit different strengths with a higher sensitivity to detect quality degradation and cost efficiency. These findings can expand the applicability of the existing GVQSS and improve efficiency and flexibility when food practitioners conduct sampling inspections for food quality and safety in practice. Two real-world cases are illustrated to demonstrate the practicality of the proposed method.

First-passage and first-arrival problems in continuous-time random walks: Beyond the diffusion approximation.

Some exact solutions of the first-passage and first-arrival problems for the continuous-time random-walk model are obtained. On the basis of these exact solutions, the following has been revealed. First, for some jump-length distributions with a finite variance, the approximate solutions obtained in the diffusion approximation can differ significantly from the exact solutions. Second, for some waiting time distributions with a finite mean, the times of first passage and the times of first arrival can significantly depend on the ensemble under consideration. In particular, the mean first-passage time corresponding to the stationary ensemble can be significantly greater than the mean first-passage time corresponding to the nonaged ensemble. Third, for any continuous distribution of jump lengths, the probability of first arrival is zero for a point-like target. This last result is contrary to existing opinion, but it is consistent with the fact that a single point has a probability measure equal to zero in the probability space defined by a continuous distribution of jump lengths.

Universal distribution of the number of minima for random walks and Lévy flights.

We compute exactly the full distribution of the number m of local minima in a one-dimensional landscape generated by a random walk or a Lévy flight. We consider two different ensembles of landscapes, one with a fixed number of steps N and the other till the first-passage time of the random walk to the origin. We show that the distribution of m is drastically different in the two ensembles (Gaussian in the former case, while having a power-law tail m^{-3/2} in the latter case). However, the most striking aspect of our results is that, in each case, the distribution is completely universal for all m (and not just for large m), i.e., independent of the jump distribution in the random walk. This means that the distributions are exactly identical for Lévy flights and random walks with finite jump variance. Our analytical results are in excellent agreement with our numerical simulations.

Two-dimensional quad-stable Gaussian potential stochastic resonance model for enhanced bearing fault diagnosis

In this study, a two-dimensional quad-stable Gaussian potential stochastic resonance model is explored for the first time. First, the structure of the proposed model is analyzed to have a broader potential field and verified to break through the severe output saturation inherent in the classical two-dimensional quad-stable stochastic resonance model. Then, we analyze the relationship between the structure and parameters of the model and derive the steady-state probability density and the mean first-passage time using adiabatic approximation theory to describe the specific process of the Brownian particle transitions. By combining the adiabatic approximation theory and the probability flow equation, the spectral amplification factor of the model is derived, and the effects of different parameters on the model performance are investigated. Further, a fourth-order Runge-Kutta algorithm was applied to evaluate the model performance in multiple dimensions. Finally, the model parameters were optimized using an adaptive genetic algorithm and applied to complex practical engineering detection. The experimental results show that the proposed model is superior and universal in fault diagnosis. Overall, this study provides important mathematical support for solving various engineering problems and demonstrates a wide range of practical applications.

Optimizing leapover lengths of Lévy flights with resetting.

We consider a one-dimensional search process under stochastic resetting conditions. A target is located at b≥0 and a searcher, starting from the origin, performs a discrete-time random walk with independent jumps drawn from a heavy-tailed distribution. Before each jump, there is a given probability r of restarting the walk from the initial position. The efficiency of a "myopic search"-in which the search stops upon crossing the target for the first time-is usually characterized in terms of the first-passage time τ. On the other hand, great relevance is encapsulated by the leapover length l=x_{τ}-b, which measures how far from the target the search ends. For symmetric heavy-tailed jump distributions, in the absence of resetting the average leapover is always infinite. Here we show instead that resetting induces a finite average leapover ℓ_{b}(r) if the mean jump length is finite. We compute exactly ℓ_{b}(r) and determine the condition under which resetting allows for nontrivial optimization, i.e., for the existence of r^{*} such that ℓ_{b}(r^{*}) is minimal and smaller than the average leapover of the single jump.

Survival probability surfaces of hysteretic fractional order structures exposed to non-stationary code-compliant stochastic seismic excitation

A novel first-passage probability stochastic incremental dynamics analysis (SIDA) methodology tailored for hysteretic fractional order structural systems under a fully non-stationary seismic excitation vector consistently designated with contemporary aseismic codes provisions (e.g., Eurocode 8) is developed. Specifically, the vector of the imposed seismic excitations is characterised by evolutionary power spectra that stochastically align with aseismic codes elastic response acceleration spectra, defined for specified modal damping ratios and scaled ground accelerations. Leveraging the concepts of stochastic averaging and statistical linearization, the approximative non-stationary response displacement joint probability density function (PDF) is derived, retaining the particularly coveted attribute of computational efficacy. Subsequently, the coupling with the survival probability model allows for the efficient determination of the response first-passage time probability density surfaces and the survival probability surfaces across various limit-state rules and scalable intensity measures. The first-passage time probability serves as a robust engineering demand parameter, effectively monitoring structural behaviour by considering both intensity and timing information, while inherently aligned with pertinent limit-state requirements. Notably, the associated low computational cost and the ability to handle a wide range of complex nonlinear/hysteretic structural behaviours, coupled with its compliance with modern aseismic codes, underscore its potential for applications in the fields of structural and earthquake engineering. A nonlinear system endowed with fractional derivative elements is used to exemplify the method’s reliability. The accuracy of the proposed method is validated in a Monte Carlo-based context, conducting nonlinear response time–history analyses with an extensive ensemble of accelerograms compatible with Eurocode 8 response acceleration spectra.

Review on Some Boundary Value Problems Defining the Mean First-Passage Time in Cell Migration

Review on Some Boundary Value Problems Defining the Mean First-Passage Time in Cell Migration

Fractional Telegrapher's Equation under Resetting: Non-Equilibrium Stationary States and First-Passage Times.

We consider two different time fractional telegrapher's equations under stochastic resetting. Using the integral decomposition method, we found the probability density functions and the mean squared displacements. In the long-time limit, the system approaches non-equilibrium stationary states, while the mean squared displacement saturates due to the resetting mechanism. We also obtain the fractional telegraph process as a subordinated telegraph process by introducing operational time such that the physical time is considered as a Lévy stable process whose characteristic function is the Lévy stable distribution. We also analyzed the survival probability for the first-passage time problem and found the optimal resetting rate for which the corresponding mean first-passage time is minimal.

Clustering of primordial black holes from quantum diffusion during inflation

We study how large fluctuations are spatially correlated in the presence of quantum diffusion during inflation. This is done by computing real-space correlation functions in the stochastic-δ N formalism. We first derive an exact description of physical distances as measured by a local observer at the end of inflation, improving on previous works. Our approach is based on recursive algorithmic methods that consistently include volume-weighting effects. We then propose a “large-volume” approximation under which calculations can be done using first-passage time analysis only, and from which a new formula for the power spectrum in stochastic inflation is derived. We then study the full two-point statistics of the curvature perturbation. Due to the presence of exponential tails, we find that the joint distribution of large fluctuations is of the form P(ζ R 1, ζ R 2) = F(R 1,R 2, r) P(ζ R 1)P( ζ R 2), where ζ R 1 and ζ R 2 denote the curvature perturbation coarse-grained at radii R 1 and R 2, around two spatial points distant by r. This implies that, on the tail, the reduced correlation function, defined as P(ζ R 1 > ζc, ζ R 2 > ζc)/[P(ζ R 1 > ζc) P(ζ R 2 > ζc)]-1, is independent of the threshold value ζc. This contrasts with Gaussian statistics where the same quantity strongly decays with ζc, and shows the existence of a universal clustering profile for all structures forming in the exponential tails. Structures forming in the intermediate (i.e. not yet exponential) tails may feature different, model-dependent behaviours.

Vibrational Energy Landscapes and Energy Flow in GPCRs.

We construct and analyze disconnectivity graphs to provide the first graphical representation of the vibrational energy landscape of a protein, in this study β2AR, a G-protein coupled receptor (GPCR), in active and inactive states. The graphs, which indicate the relative free energy of each residue and the minimum free energy barriers for energy transfer between them, reveal important composition, structural and dynamic properties that mediate the flow of energy. Prolines and glycines, which contribute to GPCR plasticity and function, are identified as bottlenecks to energy transport along the backbone from which alternative pathways for energy transport via nearby noncovalent contacts emerge, seen also in the analysis of first passage time (FPT) distributions presented here. Striking differences between the disconnectivity graphs and FPT distributions for the inactive and active states of β2AR are found where structural and dynamic changes occur upon activation, contributing to allosteric regulation.