First Passage Time Statistics Research Articles

Sample-dependent first-passage-time distribution in a disordered medium.

Above two dimensions, diffusion of a particle in a medium with quenched random traps is believed to be well described by the annealed continuous-time random walk. We propose an approximate expression for the first-passage-time (FPT) distribution in a given sample that enables detailed comparison of the two problems. For a system of finite size, the number and spatial arrangement of deep traps yield significant sample-to-sample variations in the FPT statistics. Numerical simulations of a quenched trap model with power-law sojourn times confirm the existence of two characteristic time scales and a non-self-averaging FPT distribution, as predicted by our theory.

First-passage time statistics in a bistable system subject to Poisson white noise by the generalized cell mapping method

The first-passage time statistics in a bistable system subject to Poisson white noise is studied by using the generalized cell mapping method. Specifically, an approximate solution for the first-passage time statistics in a second-order bistable system is developed by analyzing the motions in double-well potential and the global dynamics in phase space. Both symmetric and asymmetric cases have been investigated, and the effects of noise intensity and mean arrival rate of impulse on the first-passage time statistics are discussed respectively. It shows that the effect of Poisson white noise excitation on the first-passage time is quite different from that of the Gaussian one. With the same noise intensity, Poisson white noise can make for a faster first-passage.

Time Delay Induces Resonant Activation in Intracellular Calcium Oscillations

The role of time delay of the active transmission processes with colored noises of the intracellular Ca^(2+) on the intracellular calcium oscillation (ICO) is investigated by means of a first-order stochastic simulation algorithm. By simulating the time series and stationary probability distribution (SPD) of the ICO system, as the time delay and colored noises increase periodicity is respectively strengthened and destroyed; bistable and monostable states appear as the time delay varies. Then, from the statistics of the mean first-passage time (MFPT) of the system from the secondary peak to the main peak of the SPD, the results indicate: as colored noises strengthen, time delay induces a resonant activation (RA) in the ICO system.

First-passage time statistics of Markov gamma processes

The analysis of the First-Passage Time (FPT) statistics has a relevant importance either in theoretical or practical sense for the signal processing design in communications. This paper introduces a simple approach that allows a rather accurate calculation of an arbitrary number of cumulants of the Probability Density Function (PDF) of the FPT for the relevant case of Markov gamma processes.

Effects of confinement on the statistics of encounter times: exact analytical results for random walks in a partitioned lattice

We study the effects of temporarily and permanently confining domains on the statistics of first-passage times in finite lattices in one and two dimensions. We present exact results for the mean and variance of the first-passage time between arbitrary sites in the following: (1) a finite one-dimensional lattice partitioned into temporarily confining domains and (2) a finite two-dimensional lattice with reflecting boundaries for a single random walker and an immobile target. In the one-dimensional case, we also present the full first-passage time distribution via numerical inversion of Laplace transforms.

The geodynamo as a random walker: A view on reversal statistics

[1] The sequence of durations of the geomagnetic polarity intervals is often described in terms of a nonhomogenous Poisson process with time-dependent reversal rate, reflecting the nonstationarity of the underlying geodynamo process. This view has recently been challenged, and here we show that the first-passage time statistics of random walks taking place on a flat potential relief yields a much more consistent interpretation of the distribution of geomagnetic polarity intervals. A possible physical explanation suggests that the stability of a polarity chron of the Earth's magnetic field is controlled by a sum of statistically independent, randomly behaving, dynamo processes in the outer core. The random-walk hypothesis naturally includes the observed occurrences of very long superchrons, and it provides a new, considerably longer statistical estimate for the total duration of the present Brunhes chron predicting that the probability for a geomagnetic field reversal within the next 30 Ka is ≈5% and the probability that we live in a chron longer than 2 Ma is about 30%.

DNA electrophoresis in confined, periodic geometries: A new lakes-straits model

We present a method to study the dynamics of long DNA molecules inside a cubic array of confining spheres, connected through narrow openings. Our method is based on the coarse-grained, lakes-straits model of Zimm and is therefore much faster than Brownian dynamics simulations. In contrast to Zimm's approach, our method uses a standard stochastic kinetic simulation to account for the mass transfer through the narrow straits and the formation of new lakes. The different rates, or propensities, of the reactions are obtained using first-passage time statistics and a Monte Carlo sampling to compute the total free energy of the chain. The total free energy takes into account the self-avoiding nature of the chain as well as confinement effects from the impenetrable spheres. The mobilities of various chains agree with biased reptation theory at low and high fields. At moderate fields, confinement effects lead to a new regime of reptation where the mobility is a linear function of molecular weight and the dispersion is minimal.

Random walk in a two-dimensional self-affine random potential: Properties of the anomalous diffusion phase at small external force

We study the dynamical response to an external force F for a particle performing a random walk in a two-dimensional quenched random potential of Hurst exponent H=1/2 . We present numerical results on the statistics of first-passage times that satisfy closed backward master equations. We find that there exists a zero-velocity phase in a finite region of the external force 0<F<F{c} , where the dynamics follows the anomalous diffusion law x(t)∼ξ(F)t^{μ(F)} . The anomalous exponent 0<μ(F)<1 and the correlation length ξ(F) vary continuously with F . In the limit of vanishing force F→0 , we measure the following power laws: the anomalous exponent vanishes as μ(F)∝F^{a} with a≃0.6 (instead of a=1 in dimension d=1 ), and the correlation length diverges as ξ(F)∝F^{-ν} with ν≃1.29 (instead of ν=2 in dimension d=1 ). Our main conclusion is thus that the dynamics renormalizes onto an effective directed trap model, where the traps are characterized by a typical length ξ(F) along the direction of the force, and by a typical barrier 1/μ(F) . The fact that these traps are "smaller" in linear size and in depth than in dimension d=1 , means that the particle uses the transverse direction to find lower barriers.

Statistics of first-passage times in disordered systems using backward master equations and their exact renormalization rules

We consider the non-equilibrium dynamics of disordered systems as defined by a master equation involving transition rates between configurations (detailed balance is not assumed). To compute the important dynamical time scales in finite-size systems without simulating the actual time evolution which can be extremely slow, we propose to focus on first-passage times that satisfy ‘backward master equations’. Upon the iterative elimination of configurations, we obtain the exact renormalization rules that can be followed numerically. To test this approach, we study the statistics of some first-passage times for two disordered models: (i) for the random walk in a two-dimensional self-affine random potential of Hurst exponent H, we focus on the first exit time from a square of size L × L if one starts at the square center and (ii) for the dynamics of the ferromagnetic Sherrington–Kirkpatrick model of N spins, we consider the first passage time tf to zero-magnetization when starting from a fully magnetized configuration. Besides the expected linear growth of the averaged barrier , we find that the rescaled distribution of the barrier (ln tf) decays as for large u with a tail exponent of order η ≃ 1.72. This value can be simply interpreted in terms of rare events if the sample-to-sample fluctuation exponent for the barrier is ψwidth = 1/3.

The first-passage problem for diffusion through a cylindrical pore with sticky walls

We calculate the first-passage time distribution for diffusion through a cylindrical pore with sticky walls. A particle diffusively explores the interior of the pore through a series of binding and unbinding events with the cylinder wall. Through a diagrammatic expansion we obtain first-passage time statistics for the particle's exit from the pore. Connections between the model and nucleocytoplasmic transport in cells are discussed.

Sudden Stratospheric Warmings as Noise-Induced Transitions

Sudden stratospheric warmings (SSWs) are usually considered to be initiated by planetary wave activity. Here it is asked whether small-scale variability (e.g., related to gravity waves) can lead to SSWs given a certain amount of planetary wave activity that is by itself not sufficient to cause a SSW. A highly vertically truncated version of the Holton–Mass model of stratospheric wave–mean flow interaction, recently proposed by Ruzmaikin et al., is extended to include stochastic forcing. In the deterministic setting, this low-order model exhibits multiple stable equilibria corresponding to the undisturbed vortex and SSW state, respectively. Momentum forcing due to quasi-random gravity wave activity is introduced as an additive noise term in the zonal momentum equation. Two distinct approaches are pursued to study the stochastic system. First, the system, initialized at the undisturbed state, is numerically integrated many times to derive statistics of first passage times of the system undergoing a transition to the SSW state. Second, the Fokker–Planck equation corresponding to the stochastic system is solved numerically to derive the stationary probability density function of the system. Both approaches show that even small to moderate strengths of the stochastic gravity wave forcing can be sufficient to cause a SSW for cases for which the deterministic system would not have predicted a SSW.

Firing time statistics for driven neuron models: analytic expressions versus numerics.

Analytical expressions are put forward to investigate the forced spiking activity of abstract neuron models such as the driven leaky integrate-and-fire model. The method is valid in a wide parameter regime beyond the restraining limits of weak driving (linear response) and/or weak noise. The novel approximation is based on a discrete state Markovian modeling of the full long-time dynamics with time-dependent rates. The scheme yields excellent agreement with numerical Langevin and Fokker-Planck simulations of the full nonstationary dynamics, not only for the first-passage time statistics, but also for the important interspike interval (residence time) distribution.

Hysteresis loss and stochastic resonance: A numerical study of a double-well potential.

A Langevin dynamic simulation is carried out in order to understand the phenomena of hysteresis in a double-well system represented by a Landau (${\mathit{m}}^{4}$) potential, where m is the order parameter, with a symmetrical sawtooth-type periodic external field. The calculation of a hysteresis loop is based on the statistics of first-passage time to make a transition from one well to the other across the potential barrier as the external field (of symmetrical sawtooth type) is swept in time. The basic construction of our model used to understand hysteresis rules out any dynamical (symmetry breaking) phase transition predicted by other workers in Ising- and O(N\ensuremath{\rightarrow}\ensuremath{\infty})-model systems. Also, we do not find any universal scaling relationship between the hysteresis loss and the field-sweep ``frequency.'' Our treatment, however, makes close contact with a recently observed phenomenon of stochastic resonance: the hysteresis loss shows a stochastic resonant behavior with respect to the noise strength. We discuss the recent experiment on the observation of the Kramers rate and stochastic resonance by Simon and Libchaber [Phys. Rev. Lett. 68, 3375 (1992)] in light of our results.

Integral-equation approach to delayed bifurcation in a noisy dynamical system.

The Siegert integral-equation approach is used to study the statistics of first passage time in a noisy linear dynamical system with a swept control parameter. Small-noise-theory results are recovered from the source terms in the integral equations. Numerical results obtained using a simple algorithm are found to agree well with simulation results. An interesting case in which the first-passage-time probability density has the two most probable values is pointed out.

Relaxation from a marginal state in optical bistability. Anomalous fluctuations and first-passage time statistics

We show that it is possible to characterize anomalous fluctuations in transients from marginal states using first-passage time statistics. The maximum of the fluctuations is found to occur in the median of the first-passage time distribution. The value of this maximum is given in terms of the initial and final state. This characterization of anomalous fluctuations is related to the phenomenon of transient bimodality. We analyze the existence of such a bimodality in terms of an adimensional parameter. Our results are substantiated by numerical simulations. Applications to the Bonifacio-Lugiato model of absorptive bistability are presented.

First-passage time statistics: Processes driven by Poisson noise.

First-passage time statistics: Processes driven by Poisson noise.

Comment on first-passage times for processes driven by dichotomous fluctuations.

The recent derivation of the mean first-passage time for one-dimensional processes driven by additive dichotomous random processes [J. Masoliver, K. Lindenberg, and B. J. West, Phys. Rev. A 34, 2351 (1986)] can be extended to situations where the noise occurs multiplicatively and nonlinearly in the stochastic differential equation. For equations with Markovian dichotomous fluctuations in particular, this result allows for a complete and general probabilistic description of the statics and dynamics of the non-Markovian solution process in terms of stationary probability distributions and first-passage time statistics.

First-passage times for non-Markovian processes: Correlated impacts on bound processes.

Our previously developed stochastic trajectory analysis technique has been applied to the calculation of first-passage time statistics of bound processes. Explicit results are obtained for linearly bound processes driven by dichotomous fluctuations having exponential and rectangular temporal distributions.

First-passage times for non-Markovian processes: Correlated impacts on a free process.

Our previously developed stochastic trajectory analysis technique has been applied to the calculation of first-passage time statistics of bound processes. Explicit results are obtained for linearly bound processes driven by dichotomous fluctuations having exponential and rectangular temporal distributions.

First-passage times for non-Markovian processes.

First-passage time statistics for non-Markovian processes have heretofore only been developed for processes driven by dichotomous fluctuations that are themselves Markov. Herein we develop a new method applicable to Markov and non-Markovian dichotomous fluctuations and calculate analytic mean first-passage times for particular examples.