The Mean First-Passage Time of the System Driven by Non-Markovian Multivalued Noises

We generalize the method for the calculation of the mean first-passage time (MFPT) developed by Weiss et al.^[1] to the situations where the processes are driven by non-Markovian multivalued noises. For simplicity, we restrict ourselves to three cases: the noises have rectangular, long-tail and combined temporal distributions. The concrete calculations of MFPT are made with an important example and explicit analytical expressions are obtained.     

First-passage behavior of periodic potential system driven by correlated noise

In this paper, the first-passage time (FPT) of periodic potential system driven by correlated noise is discussed. One-dimensional non-Markovian process in the system is stochastically equivalent to two-dimensional Markovian process according to the statistical characteristics of noise. The 5 × 10⁴ response tracks of the system is simulated by the fourth-order Runge-Kutta algorithm, and the FPT of the Brownian particle is recorded, then the mean first-passage time (MFPT) is calculated by averaging these values and the probability density function (PDF) of the FPT is simulated. Finally, the influence of the relevant parameters in the system on MFPT and the PDF of the FPT are discussed. Besides, it is found that resonance activation (RA) phenomenon appears in the system.     

Random walks in generalized delayed recursive trees

Recently a great deal of effort has been made to explicitly determine the mean first-passage time （MFPT） between two nodes averaged over all pairs of nodes on a fractal network. In this paper, we first propose a family of generalized delayed recursive trees characterized by two parameters, where the existing nodes have a time delay to produce new nodes. We then study the MFPT of random walks on this kind of recursive tree and investigate the effect of the time delay on the MFPT. By relating random walks to electrical networks, we obtain an exact formula for the MFPT and verify it by numerical calculations. Based on the obtained results, we further show that the MFPT of delayed recursive trees is much shorter, implying that the efficiency of random walks is much higher compared with the non-delayed counterpart. Our study provides a deeper understanding of random walks on delayed fractal networks.     

The Mean First-Passage Time of the System Driven by Internal and External Noise Simultaneously

We derive an equation of motion of mean firat-passage time (MFPT) for the nonlinear system driven by the external and internal noise simultaneously. An approximate formula of MFPT is obtained by the perturbation technique. We find the coupling effect of the internal and the external noise for the MFPT. Using the steepest descending method we obtain the MFPT for a specific model.     

Impact of degree heterogeneity on the behavior of trapping in Koch networks

Previous work shows that the mean first-passage time (MFPT) for random walks to a given hub node (node with maximum degree) in uncorrelated random scale-free networks is closely related to the exponent γ of power-law degree distribution P(k) ～ k(-γ), which describes the extent of heterogeneity of scale-free network structure. However, extensive empirical research indicates that real networked systems also display ubiquitous degree correlations. In this paper, we address the trapping issue on the Koch networks, which is a special random walk with one trap fixed at a hub node. The Koch networks are power-law with the characteristic exponent γ in the range between 2 and 3, they are either assortative or disassortative. We calculate exactly the MFPT that is the average of first-passage time from all other nodes to the trap. The obtained explicit solution shows that in large networks the MFPT varies lineally with node number N, which is obviously independent of γ and is sharp contrast to the scaling behavior of MFPT observed for uncorrelated random scale-free networks, where γ influences qualitatively the MFPT of trapping problem.     

Exact scaling for the mean first-passage time of random walks on a generalized Koch network with a trap

In this paper,we study the scaling for the mean first-passage time(MFPT) of the random walks on a generalized Koch network with a trap.Through the network construction,where the initial state is transformed from a triangle to a polygon,we obtain the exact scaling for the MFPT.We show that the MFPT grows linearly with the number of nodes and the dimensions of the polygon in the large limit of the network order.In addition,we determine the exponents of scaling efficiency characterizing the random walks.Our results are the generalizations of those derived for the Koch network,which shed light on the analysis of random walks over various fractal networks.     

Mean first-passage time of quantum transition processes

In this paper, we consider the problem of mean first-passage time (MFPT) in quantum mechanics; the MFPT is the average time of the transition from a given initial state, passing through some intermediate states, to a given final state for the first time. We apply the method developed in statistical mechanics for calculating the MFPT of random walks to calculate the MFPT of a transition process. As applications, we (1) calculate the MFPT for multiple-state systems, (2) discuss transition processes occurring in an environmental background, (3) consider a roundabout transition in a hydrogen atom, and (4) apply the approach to laser theory.     

Mean first passage time for a class of non-Markovian processes

We determine the probability distribution of the first passage time for a class of non-Markovian processes. This class contains, amongst others, the well-known continuous time random walk (CTRW), which is able to account for many properties of anomalous diffusion processes. In particular, we obtain the mean first passage time for CTRW processes with truncated power-law transition time distribution. Our treatment is based on the fact that the solutions of the non-Markovian master equation can be obtained via an integral transform from a Markovian Langevin process.     

Exact scaling for the mean first-passage time of random walks on a generalized Koch network with a trap

In this paper, we study the scaling for the mean first-passage time (MFPT) of the random walks on a generalized Koch network with a trap. Through the network construction, where the initial state is transformed from a triangle to a polygon, we obtain the exact scaling for the MFPT. We show that the MFPT grows linearly with the number of nodes and the dimensions of the polygon in the large limit of the network order. In addition, we determine the exponents of scaling efficiency characterizing the random walks. Our results are the generalizations of those derived for the Koch network, which shed light on the analysis of random walks over various fractal networks.     

Escape over fluctuating potential barrier for system only driven by dichotomous noise

The escape for the mean first passage time (MFPT) over the fluctuating potential barrier for system only driven by a dichotomous noise is investigated. It is found that, in some circumstances, the dichotomous noise can induce the resonant activation for the MFPT over the fluctuating potential barrier, but in other circumstances, it cannot. There are two resonant activations for the MFPT. One is the MFPT as a function of the flipping rate of the fluctuating potential barrier, the other is the MFPT as a function of the transition rate of the dichotomous noise.     

Mean First-Passage Time for Non-Markovian Processes Driven by Continuous Noise

In this article, we discuss the mean first-passage time for non-Markovian processes driven by continuous noise. Applying the iterative method to the Volterra integral equation, we obtain for the first time the series solution to the mean first-passage time problem and present the exact expression for the mean first-passage time in both the cases of rectangular distribution and long-tail distribution.     

Active biopolymers confer fast reorganization kinetics

Many cytoskeletal biopolymers are "active," consuming energy in large quantities. In this Letter, we identify a fundamental difference between active polymers and passive, equilibrium polymers: for equal mean lengths, active polymers can reorganize faster than equilibrium polymers. We show that equilibrium polymers are intrinsically limited to linear scaling between mean lifetime (or mean first-passage time, or MFPT) and mean length, MFPT～, by analogy to 1D Potts models. By contrast, we present a simple active-polymer model that improves upon this scaling, such that MFPT～(1/2). Since, to be biologically useful, structural biopolymers must typically be many monomers long yet respond dynamically to the needs of the cell, the difference in reorganization kinetics may help to justify the active polymers' greater energy cost.     

Mean first-passage time of an asymmetric bistable system driven by colour-correlated noise

In this paper, the effect of every parameter (including p, q, r, \la, \tau) on the mean first-passage time (MFPT) is investigated in an asymmetric bistable system driven by colour-correlated noise. The expression of MFPT has been obtained by applying the steepest-descent approximation. Numerical results show that (1) the intensity of multiplicative noise p and the intensity of additive noise q play different roles in the MFPT of the system, (2) suppression appears on the curve of the MFPT with small \la (e.g. \la<0.5) but there is a peak on the curve of the MFPT when \la is big (e.g. \la >0.5), and (3) with different values of r (e.g. r=0.1, 0.5, 1.5), the effort of \tau on the MFPT is diverse.     

Mean First Passage Time for System with Fluctuating Potential Barrier and Coupled Noise

In this paper we study the mean first passage time (MFPT) over a fluctuation potential barrier driven by a coupled noise. It is shown that the MFPT over the fluctuation potential barrier displays resonant activations as the function of the flipping rate of the fluctuation potential barrier, and as the function of the dichotomous noise transition rate.     

High spin metastable state relaxation of spin-crossover solids driven by white noise

The mean first passage time (MFPT) as dwell time in high spin metastable state of spin-crossover solids is investigated both analytically and numerically. Calculations showed that the MFPT decreases with the increases of a distance from metastable state. The probability density function has also been examined.     

Enhancement of stability in randomly switching potential with metastable state

The overdamped motion of a Brownian particle in randomly switching piece-wise metastable linear potential shows noise enhanced stability (NES): the noise stabilizes the metastable system and the system remains in this state for a longer time than in the absence of white noise. The mean first passage time (MFPT) has a maximum at a finite value of white noise intensity. The analytical expression of MFPT in terms of the white noise intensity, the parameters of the potential barrier, and of the dichotomous noise is derived. The conditions for the NES phenomenon and the parameter region where the effect can be observed are obtained. The mean first passage time behaviors as a function of the mean flipping rate of the potential for unstable and metastable initial configurations are also analyzed. We observe the resonant activation phenomenon for initial metastable configuration of the potential profile.Received: 16 June 2004, Published online: 31 August 2004PACS: 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 02.50.-r Probability theory, stochastic processes, and statistics - 05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)     

Mean First-Passage Time of an Asymmetric Kinetic Model Driven by Two Different Kinds of Colored Noises

The mean first-passage time （MFPT） of an asymmetric bistable system between multiplicative non-Gaussian noise and additive Gaussian white noise with nonzero cross-correlation time is investigated. Firstly, the non-Markov process is reduced to the Markov process through a path-integral approach; Secondly, the approximate Fokker Planck equation is obtained by applying the unified colored noise approximation and the.Novikov Theorem. The steady-state probability distribution （SPD） is also obtained. The basal functional analysis and simplification are employed to obtain the approximate expressions of MFPT T^±. The effects of the asymmetry parameter β, the non-Gaussian parameter （measures deviation from Gaussian character） r, the noise correlation times τ and τ2, the coupling coefficient A, the intensities D and a of noise on the MFPT are discussed. It is found that the asymmetry parameter β, the non-Gaussian parameter r and the coupling coefficient A can induce phase transition. Moreover, the main findings are that the effect of self-existent parameters （D, α, and τ） of noise and cross-correlation parameters （A, 7-2） between noises on MFPT T^± is different.     

Effect of time delay in FitzHugh-Nagumo neural model with correlations between multiplicative and additive noises

We study the effect of time delay in the FitzHugh-Nagumo neural model with correlations between multiplicative and additive noise terms. Based on the corresponding Fokker-Planck equation, the explicit expressions of the stationary probability distribution function (SPDF), the mean first passage time (MFPT) and the signal-to-noise ratio (SNR) are obtained, respectively. Research results show that: (i) the system undergoes a succession of two phase transitions (i.e., the reentrance phenomenon) as the noise correlation parameter is increased and a (single) phase transition as the time delay is increased. (ii) The MFPT as a function of the multiplicative noise intensity exhibits a maximum. This maximum for MFPT identifies the noise enhanced stability (NES) effect, the noise correlation parameter intensifies the NES effect while the time delay, and the additive noise intensity weakens it. (iii) The existence of a maximum in the SNR as a function of the multiplicative noise intensity is the identifying characteristic of the stochastic resonance (SR) phenomenon, the noise correlation parameter enhances the SR while the time delay, and the additive noise intensity weaken it.     

色关联的色噪声驱动的分段非线性模型的平均首次穿越时间

研究了色关联的乘性高斯色噪声和加性高斯色噪声驱动的分段非线性系统中, 噪声强度和相关时间对平均首次穿越时间的影响. 利用一致有色噪声近似方法和最速下降方法, 推导出系统平均首次穿越时间的表达式. 研究结果表明: 系统的平均首次穿越时间随着乘性噪声的增加会出现单峰结构, 即“共振”现象, 峰值会随着加性噪声强度和噪声之间关联强度的增加而减小. 而平均首次穿越时间作为加性噪声的函数呈单调曲线, 说明乘性噪声和加性噪声对平均首次穿越时间的影响不同. 此外, 乘性和加性噪声关联时间以及互关联时间在正关联时和负关联时 对系统平均首次穿越时间的影响是不同的. 关键词： 色噪声 分段非线性系统 平均首次穿越时间     

Random walks on dual Sierpinski gaskets

We study an unbiased random walk on dual Sierpinski gaskets embedded in d-dimensional Euclidean spaces. We first determine the mean first-passage time (MFPT) between a particular pair of nodes based on the connection between the MFPTs and the effective resistance. Then, by using the Laplacian spectra, we evaluate analytically the global MFPT (GMFPT), i.e., MFPT between two nodes averaged over all node pairs. Concerning these two quantities, we obtain explicit solutions and show how they vary with the number of network nodes. Finally, we relate our results for the case of d = 2 to the well-known Hanoi Towers problem.