Mean First Passage Time Research Articles

Transition Yet Energy Consuming for Tumor Growth Regulated by the Colored Microenvironment

Transition or metastasis is a main characteristic of tumor developmental processes. However, the mechanism behind the transition and what costs involved are obscure when the tumor is exposed to a colored microscopic environment. Here, focusing on the regulatory role of noises from its strength and correlation time on phenotypic diversity of tumors, we show that (1) when the noise strength (NS) is fixed, extending the autocorrelation time (AT) of multiplicative noise can regulate bidirectionally the tumor phenotype, i.e., it can promote the diffusion also contribute to killing cancer cells simultaneously; (2) AT of additive noise can reduce the occurrence probability of cancer cells, but the NS can increase this probability; (3) the effect of the cross-correlated strength (CS) on cell phenotype is twofold, i.e., increasing CS may urge the mean first passage time (MFPT) of switching to the tumor state to have the minimum and maximum values but the cross-correlation time (CT) always makes the MFPT to have a minimum value. In addition, NS can make MFPT to have a peak. Moreover, by reconstructing the reaction network from the mesoscopic scale, we further show that AT of multiplicative noise can increase energy consumption, and there exists a trade-off between NS and AT of additive noise. We also show that the energy consumption is monotonically decreasing with increasing the CT but the CS can amplify the difference of this dependence. The overall analysis implies that tumor cells would make use of external noise to survive in fluctuating environments.

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.

Asymptotic analysis of the narrow escape problem in general shaped domain with several absorbing necks

This paper considers the two-dimensional narrow escape problem in a domain which is composed of a relatively big head and several absorbing narrow necks. The narrow escape problem is to compute the mean first passage time (MFPT) of a Brownian particle traveling from inside the head to the end of the necks. The original model for MFPT is to solve a mixed Dirichlet–Neumann boundary value problem for the Poisson equation in the composite domain, and is computationally challenging. In this paper, we compute the MFPT by solving an equivalent Neumann–Robin type boundary value problem. By solving the new model, we obtain the three-term asymptotic expansion of the MFPT. We also conduct numerical experiments to show the accuracy of the high order expansion. As far as we know, this is the first result on high order asymptotic solution for narrow escape problem (NEP) in a general shaped domain with several absorbing neck windows. This work is motivated by Li (2014 J. Phys. A: Math. Theor. 47 505202), where the Neumann–Robin model was proposed to solve the NEP in a domain with a single absorbing neck.

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

On a diffusion which stochastically restarts from moving random spatial positions: a non-renewal framework

We consider a diffusive particle that at random times, exponentially distributed with parameter β, stops its motion and restarts from a moving random position Y(t) in space. The position X(t) of the particle and the restarts do not affect the dynamics of Y(t), so our framework constitutes in a non-renewal one. We exhibit the feasibility to build a rigorous general theory in this setup from the analysis of sample paths. To prove the stochastic process X(t) has a non-equilibrium steady-state, assumptions related to the confinement of Y(t) have to be imposed. In addition we design a detailed example where the random restart positions are provided by the paradigmatic Evans and Majumdar’s diffusion with stochastic resettings (Evans M and Majumdar S 2011 Phys. Rev. Lett. 106 160601), with resetting rate β Y . We show the ergodic property for the main process and for the stochastic process of jumps performed by the particle. A striking feature emerges from the examination of the jumps, since their negative covariance can be minimized with respect to both rates β and β Y , independently. Moreover we discuss the theoretical consequences that this non-renewal model entails for the analytical study of the mean first-passage time (FPT) and mean cost up to FPT.

First-passage time statistics for non-linear diffusion

Most processes examined from a standpoint of the first-passage problem are modeled by linear diffusion equations. Here, we consider the non-linear diffusion equation in which diffusivity is power-law dependent on the concentration/probability density and explore its fundamental first-passage properties. Depending on the value of the power-law exponent, we demonstrate the exact and approximate expressions for the survival probability and the first-passage time distribution along with their asymptotic representations. These results refer to the freely and harmonically trapped diffusing particle. While in the former case the mean first-passage time is divergent, albeit the first-passage time distribution is normalized to unity, it is finite in the latter. To support this result, we derive the exact formula for the mean first-passage time to the target prescribed in the minimum of the harmonic potential.

Effective temperature for an intermittent bistable potential.

Thermodynamics of far-from-equilibrium systems often require measurement of effective parameters such as temperature. Whether such approach is valid for the general case of resetting protocols, active systems, or of confined systems under time-varying fields is still under investigation. We report on the effect of switching ON-OFF of an asymmetric bistable potential to the mean first passage time (MFPT) of a probed particle to go from one potential minima to the other. Experimental results coupled with numerical simulations shows the potential becoming more symmetric at slow switching. Moreover, the MFPT deviates from equilibrium condition with an effective temperature, Teff < T, at slow switching but approaches room temperature, T, at fast switching. For each switching rate, we quantify how far the system is from equilibrium by measuring deviation from a detailed balance like relation and the net circulation of flux present in phase-space. Both analysis suggest equilibrium condition are met at high switching.

Encounter-based reaction-subdiffusion model II: partially absorbing traps and the occupation time propagator

In this paper we develop an encounter-based model of reaction-subdiffusion in a domain Ω with a partially absorbing interior trap . We assume that the particle can freely enter and exit , but is only absorbed within . We take the probability of absorption to depend on the amount of time a particle spends within the trap, which is specified by a Brownian functional known as the occupation time A(t). The first passage time (FPT) for absorption is identified with the point at which the occupation time crosses a random threshold with probability density . Non-Markovian models of absorption can then be incorporated by taking to be non-exponential. The marginal probability density for particle position prior to absorption depends on ψ and the joint probability density for the pair , also known as the occupation time propagator. In the case of normal diffusion one can use a Feynman–Kac formula to derive an evolution equation for the propagator. However, care must be taken when combining fractional diffusion with chemical reactions in the same medium. Therefore, we derive the occupation time propagator equation from first principles by taking the continuum limit of a heavy-tailed continuous-time random walk. We then use the solution of the propagator equation to investigate conditions under which the mean FPT for absorption within a trap is finite. We show that this depends on the choice of threshold density and the subdiffusivity. Hence, as previously found for evanescent reaction-subdiffusion models, the processes of subdiffusion and absorption are intermingled.

First-passage functionals for Ornstein–Uhlenbeck process with stochastic resetting

We study the statistical properties of first-passage Brownian functionals (FPBFs) of an Ornstein–Uhlenbeck process in the presence of stochastic resetting. We consider a one dimensional set-up where the diffusing particle sets off from x 0 and resets to at a certain rate r. The particle diffuses in a harmonic potential (with strength k) which is centered around the origin. The center also serves as an absorbing boundary for the particle and we denote the first passage time (FPT) of the particle to the center as . In this set-up, we investigate the following functionals: (i) local time i.e. the time a particle spends around until the first passage, (ii) occupation or residence time i.e. the time a particle typically spends above until the first passage and (iii) the FPT to the origin. We employ the Feynman–Kac formalism for renewal process to derive the analytical expression for the first moment of all the three FPBFs mentioned above. In particular, we find that resetting can either prolong or shorten the mean residence and FPT depending on the system parameters. The transition between these two behaviors or phases can be characterized precisely in terms of optimal resetting rates, which interestingly undergo a continuous transition as we vary the trap stiffness k. We characterize this transition and identify the critical-parameter and -coefficient for both the cases. We also showcase other interesting interplay between the resetting rate and potential strength on the statistics of these observables. Our analytical results are in excellent agreement with the numerical simulations.

Splitting probabilities and mean first-passage times across multiple thresholds of jump-and-drift transition paths

Splitting probabilities and mean first-passage times across multiple thresholds of jump-and-drift transition paths

Response and reliability of suspension system under stochastic and periodic track excitations by path integral method

Considering that maglev trains are affected by factors such as track irregularities, variations in load, external wind loads, and track materials, this paper establishes a suspension system model that takes into account the combined action of periodic and stochastic excitation. It then examines the response and reliability of the suspension system from the perspective of stochastic dynamics. The path integral method is introduced to address the challenges posed by periodic excitation. Expressions for the path integral solutions of the probability density function for the responses and the reliability density function are derived directly from the suspension system. The reliability function and the mean first-passage time are then further derived. Using the path integral solutions, we investigate the effects of stochastic excitation, periodic excitation of the track, and safety domain on the suspension system. Research shows that as the intensity of stochastic excitation increases, the oscillation amplitude of the response also increases, while the reliability of the system decreases. The effect of periodic excitation on the response and reliability of the system is more complex. The most intriguing discovery is that the reliability of the suspension system weakens when the angular frequency of the periodic excitation is close to the system’s natural frequency.

Approximating First Hitting Point Distribution in Milestoning for Rare Event Kinetics.

Milestoning is an efficient method for rare event kinetics calculation using short trajectory parallelization. Mean first passage time (MFPT) is the key kinetic output of Milestoning, whose accuracy crucially depends on the initial distribution of the short trajectory ensemble. The true initial distribution, i.e., the first hitting point distribution (FHPD), has no analytic expression in the general case. Here, we introduce two algorithms, local passage time weighted Milestoning (LPT-M) and Bayesian inference Milestoning (BI-M), to accurately and efficiently approximate FHPD for systems at equilibrium condition. Starting from sampling the Boltzmann distribution on milestones, we calculate the proper weighting factor for the short trajectory ensemble. The methods are tested on two model examples for illustration purpose. Both methods improve significantly over the widely used classical Milestoning (CM) method in terms of the accuracy of MFPT. In particular, BI-M covers the directional Milestoning method as a special case in deterministic Hamiltonian dynamics. LPT-M is especially advantageous in terms of computational costs and robustness with respect to the increasing number of intermediate milestones. Furthermore, a locally iterative correction algorithm for nonequilibrium stationary FHPD is developed for exact MFPT calculation, which can be combined with LPT-M/BI-M and is much cheaper than the exact Milestoning (ExM) method.

A numerical method for solving snapping out Brownian motion in 2D bounded domains

Diffusion in heterogeneous media partitioned by semi-permeable interfaces has a wide range of applications in the physical and life sciences, including gas permeation in soils, diffusion magnetic resonance imaging (dMRI), drug delivery, thermal conduction in composite media, synaptic receptor trafficking, and intercellular gap junctions. At the single particle level, diffusion across a semi-permeable interface can be formulated in terms of so-called snapping out Brownian motion (SNOBM). The latter sews together successive rounds of reflected BM, each of which is restricted to one side of the interface. Each round of reflected BM is killed when the local time at the interface exceeds an independent, exponentially distributed random variable. (The local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of the interface.) The particle then immediately resumes reflected BM on the same side or the other side of the interface according to a stochastic switch, and the process is iterated. In this paper, we develop a Monte Carlo algorithm for simulating a two-dimensional version of SNOBM, which is used to solve a first passage time (FPT) problem for diffusion in a domain with semi-permeable partially absorbing traps. Our method combines a walk-on-spheres (WOS) method with an efficient algorithm for computing the boundary local time that uses a Skorokhod integral representation of the latter. We validate our algorithm by comparing the Monte Carlo estimates of the mean first passage time (MFPT) to the exact solution for a single circular trap, and show that our simulations are consistent with asymptotic results obtained for the 2D narrow capture problem involving multiple small circular targets. We also discuss extensions to higher dimensions.

Transient properties of grazing ecosystem driven by Lévy noise and Gaussian noise

This paper explores how Lévy noise and Gaussian noise affect the ecological grazing system by using the fourth-order Runge–Kutta method to simulate dynamic system and the Janicki-Weron algorithm to produce Lévy noise. Two deterministic quantities, the mean first passage time (MFPT) and the probability density function (PDF) of the first passage time (FPT), are utilized to explore the transient properties of grazing ecosystem. Our research results show that: (i) The transitions between two alternative stable states can be induced by the Gaussian noise intensity, Lévy noise intensity, Lévy stability index and Lévy skewness parameter. (ii) A higher Lévy noise intensity and a larger Lévy stability index or Lévy skewness parameter make the MFPT from desert state to sustainable vegetated state shorter; a higher Gaussian noise intensity makes it longer, which indicates that increased Lévy noise intensity can mitigate ecosystem degradation; increased Gaussian noise intensity will lead to desertification of vegetation. (iii) For larger Lévy noise intensity, the MFPT from sustainable vegetated state to desert state as a function of Gaussian noise intensity exhibits one maximum value when it transformation from the sustainable vegetated state to the desert state, the noise enhanced stability (NES) phenomena of grazing ecosystems are observed.

Diversity-enhanced stability.

We give compelling evidence that diversity, represented by a quenched disorder, can produce a resonant collective transition between two unsteady states in a network of coupled oscillators. The stability of a metastable state is optimized and the mean first-passage time maximized at an intermediate value of diversity. This finding shows that a system can benefit from inherent heterogeneity by allowing it to maximize the transition time from one state to another at the appropriate degree of heterogeneity.

Lifetime of locally stable states near a phase transition in the Thirring model

We study the lifetime of locally stable states in the Thirring model, which describes a system of particles whose interactions are long-range. The model exhibits first-order phase transitions in the canonical ensemble and, therefore, a free energy barrier separates two free energy minima. The energy of the system diffuses as a result of thermal fluctuations and we show that its dynamics can be described by means of a Fokker–Planck equation. Considering an initial state where the energy takes the value corresponding to one of the minima of the free energy, we can define the lifetime of the initial state as the mean first-passage time for the system to reach the top of the free energy barrier between the minima. We use an analytical formula for the mean first-passage time which is based on the knowledge of the exact free energy of the model, even at a finite number of particles. This formula shows that the lifetime of locally stable states increases exponentially in the number of particles, which is a typical feature of systems with long-range interactions. We also perform Monte Carlo simulations in the canonical ensemble in order to obtain the probability distribution of the first-passage time, which turns out to be exponential in time in a long time limit. The numerically obtained mean first-passage time agrees with the theoretical prediction. Combining theory and simulations, our work provides a new insight in the study of metastability in many-body systems with long-range interactions.

Randomness accelerates the dynamic clearing process of the COVID-19 outbreaks in China

During the implementation of strong non-pharmaceutical interventions (NPIs), more than one hundred COVID-19 outbreaks induced by different strains in China were dynamically cleared in about 40 days, which presented the characteristics of small scale clustered outbreaks with low peak levels. To address how did randomness affect the dynamic clearing process, we derived an iterative stochastic difference equation for the number of newly reported cases based on the classical stochastic SIR model and calculate the stochastic control reproduction number (SCRN). Further, by employing the Bayesian technique, the change points of SCRNs have been estimated, which is an important prerequisite for determining the lengths of the exponential growth and decline phases. To reveal the influence of randomness on the dynamic zeroing process, we calculated the explicit expression of the mean first passage time (MFPT) during the decreasing phase using the relevant theory of first passage time (FPT), and the main results indicate that random noise can accelerate the dynamic zeroing process. This demonstrates that powerful NPI measures can rapidly reduce the number of infected people during the exponential decline phase, and enhanced randomness is conducive to dynamic zeroing, i.e. the greater the random noise, the shorter the average clearing time is. To confirm this, we chose 26 COVID-19 outbreaks in various provinces in China and fitted the data by estimating the parameters and change points. We then calculated the MFPTs, which were consistent with the actual duration of dynamic zeroing interventions.

Run-and-tumble motion in a linear ratchet potential: Analytic solution, power extraction, and first-passage properties.

We explore the properties of run-and-tumble particles moving in a piecewise-linear "ratchet" potential by deriving analytic results for the system's steady-state probability density, current, entropy production rate, extractable power, and thermodynamic efficiency. The ratchet's broken spatial symmetry rectifies the particles' self-propelled motion, resulting in a positive current that peaks at finite values of the diffusion strength, ratchet height, and particle self-propulsion speed. Similar nonmonotonic behavior is also observed for the extractable power and efficiency. We find the optimal apex position for generating maximum current varies with diffusion and that entropy production can have nonmonotonic dependence on diffusion. In particular, for vanishing diffusion, entropy production remains finite when particle self-propulsion is weaker than the ratchet force. Furthermore, power extraction with near-perfect efficiency is achievable in certain parameter regimes due to the simplifications afforded by modeling "dry" active particles. In the final part, we derive mean first-passage times and splitting probabilities for different boundary and initial conditions. This work connects the study of work extraction from active matter with exactly solvable active particle models and will therefore facilitate the design of active engines through these analytic results.