First Passage Time Research Articles

Non-renewable warranty cost analysis for dependent series configuration with distinct warranty periods

A warranty product often has an interval structure, resulting in different warranty periods for each component due to their distinct degradation paths. From the manufacturer’s perspective, it is reasonable to establish appropriate warranty periods for each part to achieve cost savings. This paper introduces a dependent series configuration consisting of a main component and an auxiliary component. The failure of the main component leads to system failure, while the failure of the auxiliary component only causes a random damage to the main component. These two components follow different Wiener degradation paths, and then their first passage times (FPTs) follow disparate inverse Gaussian distributions. Assuming a pre-deterministic warranty period, W, for the main component, and ηW,0<η<1, for the auxiliary component, the warranty period can be divided into two stages: [0,ηW] and [ηW,W]. In this context, the paper proposes warranty claim policies for replacement, objective-oriented imperfect repair (OOIR), and refund. The corresponding warranty costs sustained by manufacturer and consumer within these two stages are considered. Finally, a numerical example is presented to illustrate the warranty costs sustained by manufacturer and consumer under different scenarios, as well as the total warranty costs.

Being heterogeneous is disadvantageous: Brownian non-Gaussian searches.

Diffusing diffusivity models, polymers in the grand canonical ensemble and polydisperse, and continuous-time random walks all exhibit stages of non-Gaussian diffusion. Is non-Gaussian targeting more efficient than Gaussian? We address this question, central to, e.g., diffusion-limited reactions and some biological processes, through a general approach that makes use of Jensen's inequality and that encompasses all these systems. In terms of customary mean first-passage time, we show that Gaussian searches are more effective than non-Gaussian ones. A companion paper argues that non-Gaussianity becomes instead highly more efficient in applications where only a small fraction of tracers is required to reach the target.

First passage on disordered intervals.

We derive unexpected first-passage properties for nearest-neighbor hopping on finite intervals with disordered hopping rates, including (a) a highly variable spatial dependence of the first-passage time, (b) huge disparities in first-passage times for different realizations of hopping rates, (c) significant discrepancies between the first moment and the square root of the second moment of the first-passage time, and (d) bimodal first-passage time distributions. Our approach relies on the backward equation, in conjunction with probability generating functions, to obtain all moments, as well as the distribution of first-passage times. Our approach is simpler than previous approaches based on the forward equation, in which computing the mth moment of the first-passage time requires all preceding moments.

Work statistics at first-passage times

We investigate the work fluctuations in an overdamped non-equilibrium process that is stopped at a stochastic time. The latter is characterised by a first passage event that marks the completion of the non-equilibrium process. In particular, we consider a particle diffusing in one dimension in the presence of a time-dependent potential U(x,t)=k|x−vt|n/n , where k > 0 is the stiffness and n > 0 is the order of the potential. Moreover, the particle is confined between two absorbing walls, located at L±(t) , that move with a constant velocity v and are initially located at L±(0)=±L . As soon as the particle reaches any of the boundaries, the process is said to be completed and here, we compute the work done W by the particle in the modulated trap upto this random time. Employing the Feynman–Kac path integral approach, we find that the typical values of the work scale with L with a crucial dependence on the order n. While for n > 1, we show that ⟨W⟩∼L1−n expkLn/n−vL/D for large L, we get an algebraic scaling of the form ⟨W⟩∼Ln for the n < 1 case. The marginal case of n = 1 is exactly solvable and our analysis unravels three distinct scaling behaviours: (i) ⟨W⟩∼L for v > k, (ii) ⟨W⟩∼L2 for v = k and (iii) ⟨W⟩∼exp−(v−k)L for v < k. For all cases, we also obtain the probability distribution associated with the typical values of W. Finally, we observe an interesting set of relations between the relative fluctuations of the work done and the first-passage time for different n—which we argue physically. Our results are well supported by the numerical simulations.

Optimal mean first-passage time of a run-and-tumble particle in a class of one-dimensional confining potentials

We consider a run-and-tumble particle (RTP) in one dimension, subjected to a telegraphic noise with a constant rate γ, and in the presence of an external confining potential with . We compute the mean first-passage time (MFPT) at the origin for an RTP starting at x 0. We obtain a closed form expression for for all , which becomes fully explicit in the case and in the limit . For generic 1$ ?> we find that there exists an optimal rate that minimizes the MFPT and we characterize in detail its dependence on x 0. We find that as , while converges to a non-trivial constant as . In contrast, for p = 1, there is no finite optimum and in this case. These analytical results are confirmed by our numerical simulations.

Fractal and first-passage properties of a class of self-similar networks.

A class of self-similar networks, obtained by recursively replacing each edge of the current network with a well-designed structure (generator) and known as edge-iteration networks, has garnered considerable attention owing to its role in presenting rich network models to mimic real objects with self-similar structures. The generator dominates the structural and dynamic properties of edge-iteration networks. However, the general relationships between these networks' structural and dynamic properties and their generators remain unclear. We study the fractal and first-passage properties, such as the fractal dimension, walk dimension, resistance exponent, spectral dimension, and global mean first-passage time, which is the mean time for a walker, starting from a randomly selected node and reaching the fixed target node for the first time. We disclose the properties of the generators that dominate the fractal and first-passage properties of general edge-iteration networks. A clear relationship between the fractal and first-passage properties of the edge-iteration networks and the related properties of the generators are presented. The upper and lower bounds of these quantities are also discussed. Thus, networks can be customized to meet the requirements of fractal and dynamic properties by selecting an appropriate generator and tuning their structural parameters. The results obtained here shed light on the design and optimization of network structures.

The Laplacian spectrum of weighted composite networks and the applications

The topological properties of the networks can be described by the Laplacian spectra, but resolving the Laplacian spectra of networks poses difficulties. In this study, a novel approach for solving the Laplacian spectrum of weighted composite networks is presented. We first give the definitions of three weighted graph operations, namely, Cartesian product, corona, and join. Second, the Laplacian spectra of these composite networks are calculated. Finally, we use the obtained Laplacian spectrum to deduce some topological properties of the networks, such as network coherence, entire mean first-passage time, and Laplacian energy, which have several applications in physical chemistry.

Perspective study on charge time measurement of long-term stored lithium-ion batteries used in electric-powered aircraft assessed and modelled by specific growth model with diffusion process backup

Electrical energy sources and batteries are currently in great demand. The demand is growing, and their use is on the rise in a variety of applications. In contrast, there are limited both production/manufacturing options and an effort to ensure more environmentally friendly acquisition, operation and use. In this article, we study, test, and model the charging process of Li-ION batteries. We study a set of long-term stored Li-ION batteries and compare the data and results with a set of new Li-ION batteries. The charging data are obtained experimentally by long-term testing, and the volume (of the data) is very large and statistically very representative. Modelling of the Li-ION battery charging process is based on specific growth model forms. Furthermore, the charging process is simulated using special forms of the diffusion process, specifically the Geometric Ornstein-Uhlenbeck process, which seems to be the most suitable for this purpose. The research is carried out with the aim to obtain the first passage time (FPT), i.e. the charge time of the battery. The first passage time is further studied with respect to the suitability of its expression using parametric probability distributions. We also acquired the numerical statistics of the charging time FPT as well as the respective parametric distribution. The FPT results obtained for long-term stored and new Li-ION batteries were compared using well-known and lesser-known statistical tests, such as the Kullback-Leibler divergence.

Combining stochastic resetting with Metadynamics to speed-up molecular dynamics simulations

Metadynamics is a powerful method to accelerate molecular dynamics simulations, but its efficiency critically depends on the identification of collective variables that capture the slow modes of the process. Unfortunately, collective variables are usually not known a priori and finding them can be very challenging. We recently presented a collective variables-free approach to enhanced sampling using stochastic resetting. Here, we combine the two methods, showing that it can lead to greater acceleration than either of them separately. We also demonstrate that resetting Metadynamics simulations performed with suboptimal collective variables can lead to speedups comparable with those obtained with optimal collective variables. Therefore, applying stochastic resetting can be an alternative to the challenging task of improving suboptimal collective variables, at almost no additional computational cost. Finally, we propose a method to extract unbiased mean first-passage times from Metadynamics simulations with resetting, resulting in an improved tradeoff between speedup and accuracy. This work enables combining stochastic resetting with other enhanced sampling methods to accelerate a broad range of molecular simulations.

Probing the modulation in facilitated diffusion guided by DNA-protein interactions in target search processes.

Many fundamental biophysical processes involving gene regulation and gene editing rely, at the molecular level, on an intricate methodology of searching and locating the precise target base pair sequence on the genome by specific binding proteins. A unique mechanism, known as 'facilitated diffusion', which is a combination of 1D sliding along with 3D movement, is considered to be the key step for such events. This also explains the relatively much shorter timescale of the target searching process, compared to other diffusion-controlled biophysical processes. In this work, we aim to probe the modulation of target search dynamics of a protein moiety by estimating the rate of the target search process, and the statistics of the search rounds and timescales accomplished by the 1D and 3D motions, based on first passage time (FPT) calculations. This is studied with its characteristics getting influenced by various given conditions such as, when the DNA is rigid or flexible, and when the target is placed at different locations on the DNA. The current theoretical framework includes a Brownian dynamics simulation setup adopting a straightforward coarse-grained model for a diffusing protein on DNA. Moreover, this theoretical analysis provides insights into the complex target search dynamics by highlighting the significance of the chain dynamics in the mechanistic details of the facilitated diffusion process.

Overview of kinetic Monte Carlo methods used to simulate microstructural evolution of materials under irradiation

Kinetic Monte Carlo (KMC) methods are commonly used to simulate the microstructure evolution of metals under irradiation due to their ability to generate the random walks underlying defect-mediated diffusion processes at the atomic scale. However, the range of applicability of KMC methods is severely limited by the kinetic trapping of the simulated trajectories within low energy basins presenting small intra-basin barriers. This results in dramatically reducing the efficiency of the classical KMC algorithm. Kinetic trapping can be alleviated by implementing non-local jumps relying on the theory of absorbing Markov chains. A factorisation of an auxiliary absorbing transition matrix then allows to generate escaping paths and first-passage times out of trapping basins. Although, the speed-up can be of several orders of magnitudes, this is sometimes not enough for very long-term prediction. We must then turn to homogenised rate-equation formulation of the problem. Usually solved deterministically, the corresponding large ordinary differential equation system often suffers from the curse of dimensionality. Dedicated Monte Carlo schemes can simulate the coarse-grained rate equations based on a chemical master equation. Finally, we show the relevance of relaxing the rigid-lattice assumption in the calculation of the free energy barriers and attempt frequencies to capture elastic effects that are important for certain systems, such as high entropy alloys. The activation-relaxation technique can be used for this purpose in kinetic Monte Carlo studies of slow diffusion processes.

Heterogeneous binding of polymers on curved nanoparticles.

Unraveling protracted polymer binding on curved surfaces of nanoparticles (NPs) is important for the fabrication of multifunctional nanostructures in cutting-edge research disciplines such as directional self-assembly and nanomedicine. By using our newly developed Integral of First-passage Times (IFS), we demonstrate a curvature-dependent heterogeneous binding of polymers on curved NPs, not only in terms of the binding dynamics but also in terms of the final adsorption densities. The highly curved surfaces on NPs can adsorb larger density polymers with binding kinetics that are faster than those on less curved areas, which is consistent with recent experimental observations. In particular, the spherical corners on cubic NPs with a radius of R = 3.0 nm can adsorb polymers at a density 4.1 times higher than those on planar surfaces and 1.7 times higher than those on rod edge surfaces. A unified relationship between adsorption densities and surface curvatures is proposed to collapse all the data onto one master curve. The findings demonstrate a heterogeneous binding of polymers on curved NPs, providing effective guidelines for the rational design of functional nanostructures in different applications.

Stochastic resonance in Hindmarsh-Rose neural model driven by multiplicative and additive Gaussian noise

This paper investigates the occurrence of stochastic resonance in the three-dimensional Hindmarsh-Rose (HR) neural model driven by both multiplicative and additive Gaussian noise. Firstly, the three-dimensional HR neural model is transformed into the one-dimensional Langevin equation of the HR neural model using the adiabatic elimination method, and the effects of HR neural model parameters on the potential function are analyzed. Secondly the Steady-state Probability Density (SPD), the Mean First-Passage Time (MFPT), and the Signal-to-Noise Ratio (SNR) of the HR neural model are derived, based on two-state theory. Then, the effects of different parameters (a, b, c, s), noise intensity, and the signal amplitude on these metrics are analyzed through theoretical simulations, and the behavior of particles in a potential well is used to analyze how to choose the right parameters to achieve high-performance stochastic resonance. Finally, numerical simulations conducted with the fourth-order Runge–Kutta algorithm demonstrate the superiority of the HR neural model over the classical bistable stochastic resonance (CBSR) in terms of performance. The peak SNR of the HR neural model is 0.63 dB higher than that of the CBSR system. Simulation results indicate that the occurrence of stochastic resonance occur happens in HR neural model under different values of parameters. Furthermore, under certain conditions, there is a ‘suppress’ phenomenon that can be produced by changes in noise, which provides great feasibilities and practical value for engineering application.

Remaining Useful Life Prediction for Two-Phase Nonlinear Degrading Systems with Three-Source Variability.

Recently, the estimation of remaining useful life (RUL) for two-phase nonlinear degrading devices has shown rising momentum for ensuring their safe and reliable operation. The degradation processes of such systems are influenced by the temporal variability, unit-to-unit variability, and measurement variability jointly. However, current studies only consider these three sources of variability partially. To this end, this paper presents a two-phase nonlinear degradation model with three-source variability based on the nonlinear Wiener process. Then, the approximate analytical solution of the RUL with three-source variability is derived under the concept of the first passage time (FPT). For better implementation, the offline model parameter estimation is conducted by the maximum likelihood estimation (MLE), and the Bayesian rule in conjunction with the Kalman filtering (KF) algorithm are utilized for the online model updating. Finally, the effectiveness of the proposed approach is validated through a numerical example and a practical case study of the capacitor degradation data. The results show that it is necessary to incorporate three-source variability simultaneously into the RUL prediction of the two-phase nonlinear degrading systems.

Characterization of the Mean First-Passage Time Function Subject to Advection in Annular-like Domains

Cell migration in a biological medium towards a blood vessel is modeled, as a random process, sucessively inside an annulus (two-dimensional domain) and an annular cylinder (three-dimensional domain). The conditional probability function u for the cell moving inside such domains (tissue) fulfills by assumption a diffusion–advection equation that is subject to a Dirichlet boundary condition on the outer boundary and a Robin boundary condition on the inner boundary. The mean first-passage time (MFPT) function determined by u estimates the average time for the travelling cell to reach various interesting targets. The MFPT function fulfills a Poisson equation inside a domain with suitable boundary conditions, which give rise to various mathematical problems. The main novelty of this study is the characterization of such an MFPT function inside an annulus and an annular cylinder, which is subject to a Robin boundary condition on the inner boundary and a Dirichlet boundary condition on the outer one, and these are integral functions whose densities are the solution of an inhomogeneous system of linear integral equations.

On Kinetic Constraints That Catalysis Imposes on Elementary Processes.

Catalysis, the acceleration of product formation by a substance that is left unchanged, typically results from multiple elementary processes, including diffusion of the reactants toward the catalyst, chemical steps, and release of the products. While efforts to design catalysts are often focused on accelerating the chemical reaction on the catalyst, catalysis is a global property of the catalytic cycle that involves all processes. These are controlled by both intrinsic parameters such as the composition and shape of the catalyst and extrinsic parameters such as the concentration of the chemical species at play. We examine here the conditions that catalysis imposes on the different steps of a reaction cycle and the respective role of intrinsic and extrinsic parameters of the system on the emergence of catalysis by using an approach based on first-passage times. We illustrate this approach for various decompositions of a catalytic cycle into elementary steps, including non-Markovian decompositions, which are useful when the presence and nature of intermediate states are a priori unknown. Our examples cover different types of reactions and clarify the constraints on elementary steps and the impact of species concentrations on catalysis.

Periodicity-assist double delay-controlled stochastic resonance for the fault detection of bearings

This paper analyses the stochastic resonance of a double delay-controlled nonlinear system based on the high-order-degradation potential function. It provides a theoretical derivation of the stationary probability density, mean first-passage time (MFPT), and signal-to-noise ratio (SNR), while the marginal probability density is examined via Monte-Carlo simulation. In the field of fault detection for faulty bearings, the stochastic resonance of the nonlinear system encounters a challenge where the exact fault period must be known beforehand. Unfortunately, it is not available for the real engineering application. Consequently, this paper proposes the periodicity-assisted stochastic resonance method, which uses the blind deconvolution technique. The proposed method allows for adaptive determination of the fault frequency from the vibration signal and successfully enhances the power of the signal component corresponding to the determined fault frequency. Compared with the state-of-the-art methods, the proposed method shows better performance, which is validated by the simulated and experimental data. Overall, the proposed method is advantageous for enhancing the weak fault characteristic which is interfered by the strong noise and random impulses.

Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime.

We derive general bounds on the probability that the empirical first-passage time τ[over ¯]_{n}≡∑_{i=1}^{n}τ_{i}/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

Optimizing search processes with stochastic resetting on the pseudofractal scale-free web.

The pseudofractal scale-free web (PSFW) is a well-known model for a scale-free network with small-world characteristics. Understanding the dynamic properties of this network can provide valuable insights into dynamic processes occurring in general scale-free and small-world networks. In this study we investigate search processes using discrete-time random walks on the PSFW to reveal the impact of the resetting position on optimizing search efficiency, as measured by the mean first-passage time (MFPT). At each step the walker has two options: with a probability of 1-γ, it moves to one of the neighboring sites, and with a probability of γ, it resets to the predefined resetting position. We explore various choices for the resetting position, present rigorous results for the MFPT to a given node of the network, determine the optimal resetting probability γ^{*} where the MFPT reaches its minimum, and evaluate the ratio of the minimum for MFPT to the MFPT without resetting for each case. Results show that, in large PSFWs, both the degree of the resetting position and the distance between the target and the resetting position significantly affect the search efficiency. A higher degree of the resetting position leads to a slower convergence of the walker to the target, while a greater distance between the target and the resetting position also results in a slower convergence. Additionally, we observe that resetting to a vertex randomly selected from the stationary distribution can significantly expedite the process of the walker reaching the target. The findings presented in this study shed light on optimizing stochastic search processes on large networks, offering valuable insights into improving search efficiency in real-world applications, where the target node's location is unknown.

Heterogeneous Mean First-Passage Time Scaling in Fractal Media

Heterogeneous Mean First-Passage Time Scaling in Fractal Media