Mean escape time over a fluctuating barrier

Dichotomous flow with thermal diffusion and stochastic resetting

We consider properties of one-dimensional diffusive dichotomous flow and discuss effects of stochastic resonant activation (SRA) in the presence of a statistically independent random resetting mechanism. Resonant activation and stochastic resetting are two similar effects, as both of them can optimize the noise-induced escape. Our studies show completely different origins of optimization in adapted setups. Efficiency of stochastic resetting relies on elimination of suboptimal trajectories, while SRA is associated with matching of time scales in the dynamic environment. Consequently, both effects can be easily tracked by studying their asymptotic properties. Finally, we show that stochastic resetting cannot be easily used to further optimize the SRA in symmetric setups.

Diffusion crossing over a barrier in a random rough metastable potential

We carry out a detailed study of escape dynamics of a particle driven by a white noise over a metastable potential corrugated by spatial disorder in the form of zero-mean random correlated potential. The approach of double-averaging over test particles and statistic ensemble is proposed to calculate the escape rate in a finite-size random rough metastable potential, moreover, the interference mechanism of test particles multi-passing over the saddle point is considered. Through analyzing the dependence of the steady escape rate on various modelled potentials and parameters, we demonstrate that the obstruction induced by roughness leads to a decrease in the steady escape rate with the increase of rough intensity. We also add the random correlated potential into the vicinity of the saddle-point of metastable potentials of three kinds and show an enhancement phenomenon of escape rate similar to the previous study of a surmounting fluctuating sharp barrier.

Resonant activation of population extinctions

Understanding the mechanisms governing population extinctions is of key importance to many problems in ecology and evolution. Stochastic factors are known to play a central role in extinction, but the interactions between a population's demographic stochasticity and environmental noise remain poorly understood. Here we model environmental forcing as a stochastic fluctuation between two states, one with a higher death rate than the other. We find that, in general, there exists a rate of fluctuations that minimizes the mean time to extinction, a phenomenon previously dubbed "resonant activation." We develop a heuristic description of the phenomenon, together with a criterion for the existence of resonant activation. Specifically, the minimum extinction time arises as a result of the system approaching a scenario wherein the severity of rare events is balanced by the time interval between them. We discuss our findings within the context of more general forms of environmental noise and suggest potential applications to evolutionary models.

Quantifying a resonant-activation-like phenomenon in non-Markovian systems

Resonant activation is an effect of a noise-induced escape over a modulated potential barrier. The modulation of an energy landscape facilitates the escape kinetics and makes it optimal as measured by the mean first-passage time. A canonical example of resonant activation is a Brownian particle moving in a time-dependent potential under action of Gaussian white noise. Resonant activation is observed not only in typical Markovian-Gaussian systems but also in far-from-equilibrium and far-from-Markovianity regimes. We demonstrate that using an alternative to the mean first-passage time, robust measures of resonant activation, the signature of this effect can be observed in general continuous-time random walks in modulated potentials, even in situations when the mean first-passage time diverges.

Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response

In this paper spatial dynamics of the Beddington-DeAngelis predator-prey model is investigated. We analyze the linear stability and obtain the condition of Turing instability of this model. Moreover, we deduce the amplitude equations and determine the stability of different patterns. In Turing space, we found that this model has coexistence of H(0) hexagon patterns and stripe patterns, H(π) hexagon patterns, and H(0) hexagon patterns. To better describe the real ecosystem, we consider the ecosystem as an open system and take the environmental noise into account. It is found that noise can decrease the number of the patterns and make the patterns more regular. What is more, noise can induce two kinds of typical pattern transitions. One is from the H(π) hexagon patterns to the regular stripe patterns, and the other is from the coexistence of H(0) hexagon patterns and stripe patterns to the regular stripe patterns. The obtained results enrich the finding in the Beddington-DeAngelis predator-prey model well.

Kinetic models for stochastically modified ionic channels

Ionic channels form pores in biomembranes. These pores are large macromolecular structures. Due to thermal fluctuations of countless degrees-of-freedom of the biomembrane material, the actual form of the pores is permanently subject to modification. Furthermore, the arrival of an ion at the binding site can change this form by repolarizing the surrounding aminoacids. In any case the variations of the pore structure are stochastic. In this paper, we discuss the effect of such modifications on the channel conductivity. Applying a simple kinetic description, we show that stochastic variations in channel properties can significantly alter the ionic current, even leading to its substantial increase or decrease for the specific matching of some time-scales of the system.

Analytical solution to transport in Brownian ratchets via the gambler's ruin model

We present an analogy between the classic gambler's ruin problem and the thermally activated dynamics in periodic Brownian ratchets. By considering each periodic unit of the ratchet as a site chain, we calculated the transition probabilities and mean first passage time for transitions between energy minima of adjacent units. We consider the specific case of Brownian ratchets driven by Markov dichotomous noise. The explicit solution for the current is derived for any arbitrary temperature, and is verified numerically by Langevin simulations. The conditions for current reversal in the ratchet are obtained and discussed.

Stochastic resonant signaling in enzyme cascades

We observe the phenomenon of stochastic resonant signaling in signal amplification enzyme cascades, where certain optimal reaction rates minimize the average threshold-crossing time. We develop a new analytical technique to obtain the mean first passage time, based on a novel decomposition of the master equation. Our analytical results are in good agreement with the exact numerical simulations. We demonstrate that resonant behavior may be a ubiquitous phenomenon in stochastic threshold crossing in cell signaling. The physical principles behind this phenomenon are elucidated.

Scaling properties for a classical particle in a time-dependent potential well

Some scaling properties for a classical particle interacting with a time-dependent square-well potential are studied. The corresponding dynamics is obtained by use of a two-dimensional nonlinear area-preserving map. We describe dynamics within the chaotic sea by use of a scaling function for the variance of the average energy, thereby demonstrating that the critical exponents are connected by an analytic relationship.

Chaos-induced escape over a potential barrier

We investigate the statistical parity of a class of chaos-generated noises on the escape of strongly damped particles out of a potential well. We show that statistical asymmetry in the chaotic fluctuations can lead to a skewed Maxwell-Boltzmann distribution in the well. Depending on the direction of skew, the Kramers escape rate is enhanced or suppressed accordingly. Based on the Perron-Frobenious equation, we determine an analytical expression for the escape rate's prefactor that accounts for this effect. Furthermore, our perturbative analysis proves that in the zeroth-order limit, the rate of particle escape converges to the Kramers rate.

Path integral approach to Brownian motion driven with an ac force

Brownian motion in a periodic potential driven by an ac (oscillatory) force is investigated for the full range of damping constant from the overdamped limit to the underdamped limit. The path (functional) integral approach is advanced to produce formulas for the probability distribution function and for the current of the Brownian particle in response to an ac driving force. The negative friction Langevin dynamics technique is employed to evaluate the dc current for various parameters without invoking the overdamped or the underdamped approximation. The dc current is found to have nonlinear dependence upon the damping constant, the potential parameter, and the ac force magnitude and frequency.

Chaotic properties of a time-modulated barrier

Some chaotic properties of a classical particle interacting with a time-modulated barrier are studied. The dynamics of this problem is obtained by use of a two-dimensional nonlinear area-preserving map. The chaotic low energy region is characterized in terms of Lyapunov exponents. The time that the particle stays trapped in the well is such that the distributions of successive reflections, and of the corresponding successive reflection times, obey power laws with the same exponent. Using time series analysis, we show that the chaotic sea exhibits an interesting scaling property over a large range of control parameters. Our results indicate that the particle experiences unlimited energy growth when the barrier behaves randomly.

Transient response of a Brownian particle with general damping

We study the transient response of a Brownian particle with general damping in a system of metastable potential well. The escape rate is evaluated as a function of time after an infinite wall is removed from the potential barrier. It takes a relaxation time for the rate to reach its limit value and this rate relaxation time differs from the relaxation time of the majority of the probability around the bottom of the potential well. The rate relaxation time is found to depend on the temperature as well as the damping constant. It involves the diffusion time and the instanton time, in general agreement with recent studies of the overdamped case by Bier et al. [Phys. Rev. E 59, 6422 (1999)].