Diffusion in flashing periodic potentials

The one-dimensional overdamped Brownian motion in a symmetric periodic potential modulated by external time-reversible noise is analyzed. The calculation of the effective diffusion coefficient is reduced to the mean first passage time problem. We derive general equations to calculate the effective diffusion coefficient of Brownian particles moving in arbitrary supersymmetric potential modulated by: (i) an external white Gaussian noise and (ii) a Markovian dichotomous noise. For both cases the exact expressions for the effective diffusion coefficient are derived. We obtain acceleration of diffusion in comparison with the free diffusion case for fast fluctuating potentials with arbitrary profile and for sawtooth potential in case (ii). In this case the parameter region where this effect can be observed is given. We obtain also a finite net diffusion in the absence of thermal noise. For rectangular potential the diffusion slows down, for all parameters of noise and of potential, in comparison with the case when particles diffuse freely.     

Scaling behavior of a nonlinear oscillator with additive noise,white and colored

We study analytically and numerically the problem of a nonlinear mechanical oscillator with additive noise in the absence of damping. We show that the amplitude, the velocity and the energy of the oscillator grow algebraically with time. For Gaussian white noise, an analytical expression for the probability distribution function of the energy is obtained in the long-time limit. In the case of colored, Ornstein-Uhlenbeck noise, a self-consistent calculation leads to (different) anomalous diffusion exponents. Dimensional analysis yields the qualitative behavior of the prefactors (generalized diffusion constants) as a function of the correlation time. Received 10 October 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: mallick@spht.saclay.cea.fr     

Role of noise in a market model with stochastic volatility

We study a generalization of the Heston model, which consists of two coupled stochastic differential equations, one for the stock price and the other one for the volatility. We consider a cubic nonlinearity in the first equation and a correlation between the two Wiener processes, which model the two white noise sources. This model can be useful to describe the market dynamics characterized by different regimes corresponding to normal and extreme days. We analyze the effect of the noise on the statistical properties of the escape time with reference to the noise enhanced stability (NES) phenomenon, that is the noise induced enhancement of the lifetime of a metastable state. We observe NES effect in our model with stochastic volatility. We investigate the role of the correlation between the two noise sources on the NES effect.     

Hypersensitivity to small signals in a stochastic system with multiplicative colored noise

Recently we discovered the phenomenon of hypersensitivity to small time-dependent signals in a simple stochastic system, the Kramers oscillator with multiplicative white noise. In the present work we study, theoretically and experimentally with analog simulations, an influence of noise correlation time on hypersensitivity in a nonlinear oscillator with piecewise-linear current-voltage characteristic and multiplicative colored dichotomous noise. We found that the region of hypersensitive behavior is defined by universal scaling index, whereas the specifics of a particular system reveals itself only in the dependence of the above index on system parameters. The dependence of gain factor on noise correlation time is of bell-shaped (resonant) type. Received 27 April 2000 and Received in final form 2 November 2000     

Associated relaxation time and intensity correlation function of a bistable system driven by cross-correlation additive and multiplicative coloured noise sources

The associated relaxation time and the intensity correlation function of a bistable system driven by an additive and a multiplicative coloured noise with coloured cross-correlation are investigated. Using the Novikov theorem and the projection operator method, the analytic expressions of the stationary probability distribution P_st(x), the relaxation time T_c, and the normalized correlation function C(s) of the system are obtained. The effects of the noise intensity, the cross-correlation strength λ and the cross-correlation time τ are discussed. By numerical computation, it is found that the cross-correlation strength |λ| and the quantum noise intensity D decrease the relaxation of the system from unstable points. The cross-correlation time τ delays relaxation of the system from unstable points. The cross-correlation strength λ and the cross-correlation time τ can alter the effects of the pump noise intensity Q. Thus, the relaxation time T_c is a stochastic resonant phenomenon, and distribution curves exhibit a single-maximum structure.     

Escape of Brownian particles in an asymmetric bistable sawtooth potential driven by cross-correlated noises

The relative escape rate (RER) for Brownian particles in an asymmetric bistable sawtooth potential driven by cross correlations between multiplicative white noise and additive white noise is studied. A new expression of the mean first-passage time is derived under the condition of piecewise linear potentials and discontinuous diffusion function. Based on the results of RER numerically calculated, it is found that (i) under positively correlated noises action (i.e. λ > 0, and λ is the correlation strength), the escape rate exhibits the suppression platform as the intensity of multiplicative noise varies. The effect of suppression becomes more pronounced with the growth of height of the deterministic potential barrier for transition, and with the increase of λ. However, for the case of uncorrelated noises (λ = 0) and of negatively correlated noises (λ < 0), the suppression platform disappears. (ii) The positive correlation between noises amplifies the change of the escape rate with the height of barrier for transition, while the negative correlation between noises suppresses this change. Received 20 November 2002 / Received in final form 19 October 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: kmdcmei@public.km.yn.cn     

The nonlinear Fokker-Planck equation with state-dependent diffusion - a nonextensive maximum entropy approach

Nonlinear Fokker-Planck equations (e.g., the diffusion equation for porous medium) are important candidates for describing anomalous diffusion in a variety of systems. In this paper we introduce such nonlinear Fokker-Planck equations with general state-dependent diffusion, thus significantly generalizing the case of constant diffusion which has been discussed previously. An approximate maximum entropy (MaxEnt) approach based on the Tsallis nonextensive entropy is developed for the study of these equations. The MaxEnt solutions are shown to preserve the functional relation between the time derivative of the entropy and the time dependent solution. In some particular important cases of diffusion with power-law multiplicative noise, our MaxEnt scheme provides exact time dependent solutions. We also prove that the stationary solutions of the nonlinear Fokker-Planck equation with diffusion of the (generalized) Stratonovich type exhibit the Tsallis MaxEnt form. Received 26 February 1999     

Dynamics of a FitzHugh-Nagumo system subjected to autocorrelated noise

We analyze the dynamics of the FitzHugh-Nagumo (FHN) model in the presence of colored noise and a periodic signal. Two cases are considered: (i) the dynamics of the membrane potential is affected by the noise, (ii) the slow dynamics of the recovery variable is subject to noise. We investigate the role of the colored noise on the neuron dynamics by the mean response time (MRT) of the neuron. We find meaningful modifications of the resonant activation (RA) and noise enhanced stability (NES) phenomena due to the correlation time of the noise. For strongly correlated noise we observe suppression of NES effect and persistence of RA phenomenon, with an efficiency enhancement of the neuronal response. Finally we show that the self-correlation of the colored noise causes a reduction of the effective noise intensity, which appears as a rescaling of the fluctuations affecting the FHN system.     

Energy diffusion controlled reaction rate of reacting particle driven by broad-band noise

The energy diffusion controlled reaction rate of a reacting particle with linear weak damping and broad-band noise excitation is studied by using the stochastic averaging method. First, the stochastic averaging method for strongly nonlinear oscillators under broad-band noise excitation using generalized harmonic functions is briefly introduced. Then, the reaction rate of the classical Kramers' reacting model with linear weak damping and broad-band noise excitation is investigated by using the stochastic averaging method. The averaged It? stochastic differential equation describing the energy diffusion and the Pontryagin equation governing the mean first-passage time (MFPT) are established. The energy diffusion controlled reaction rate is obtained as the inverse of the MFPT by solving the Pontryagin equation. The results of two special cases of broad-band noises, i.e. the harmonic noise and the exponentially corrected noise, are discussed in details. It is demonstrated that the general expression of reaction rate derived by the authors can be reduced to the classical ones via linear approximation and high potential barrier approximation. The good agreement with the results of the Monte Carlo simulation verifies that the reaction rate can be well predicted using the stochastic averaging method.     

非各态历经噪声的产生及其应用

用n阶时间导数噪声作为内部热噪声驱动自由粒子运动时,若该噪声所对应的非各态历经强度b?=0,且系统初始处于热平衡,则此时系统的长时稳态速度可以作为非各态历经噪声使用.非各态历经噪声具有谱密度在零频处发散的特点,一维半无界耦合简谐振子链对与之相连的气体分子的作用具有非各态历经噪声的特点,是非各态历经噪声的实例.最后将非各态历经噪声作为外噪声驱动倾斜周期势中的粒子运动,系统出现扩散指数α2的超弹道扩散现象.同时发现系统的速度分布将呈现出迁移态和锁定态两种不同状态,并且处于迁移态的粒子的速度分布为双峰.迁移态中双峰的出现是一种比较新奇的现象.     

Stochastic resonance on paced genetic regulatory small-world networks: effects of asymmetric potentials

We study the phenomenon of stochastic resonance on small-world networks consisting of bistable genetic regulatory units, whereby the external subthreshold periodic forcing is introduced as a pacemaker trying to impose its rhythm on the whole network through the single unit to which it is introduced. Without the addition of additive spatiotemporal noise, however, the whole network remains forever trapped in one of the two stable steady states of the local dynamics. We show that the correlation between the frequency of subthreshold pacemaker activity and the response of the network is resonantly dependent on the intensity of additive noise. The reported pacemaker driven stochastic resonance depends significantly on the asymmetry of the two potential wells characterizing the bistable dynamics, which can be tuned via a single system parameter. In particular, we show that the ratio between the clustering coefficient and the characteristic path length is a suitable quantity defining the ability of a small-world network to facilitate the outreach of the pacemaker-emitted subthreshold rhythm, but only if the asymmetry between the potentials is practically negligible. In case of substantially asymmetric potentials the impact of the small-world topology is less profound and cannot warrant an enhancement of stochastic resonance by units that are located far from the pacemaker.     

Resonant-Like Activation in a Bistable System with Noise and Time Delay

A bistable system with noise and time delay is investigated. Theoretical analysis and stochastic simulation show that： （i） In the ease of a system driven only by multiplicative Gaussian white noise, the mean first-passage time for a particle to reach the other stable state from one stable state exhibits a minimum with respect to delay time, i.e., a resonant-like activation （RA） phenomenon. （ii） In the action of additive and multiplicative noise, as the additive noise intensity increases, no matter whether a correlation between the two types of noise exists or not, the RA gradually disappears. （iii） The correlation strength between the two types of noise does not influence the existence of the RA.     

Role of the driving frequency in a randomly perturbed Hodgkin-Huxley neuron with suprathreshold forcing

The present paper examines the influence of the forcing frequency on the response of a randomly perturbed Hodgkin-Huxley system in the realm of suprathreshold amplitudes. Our results show that, in the presence of noise, the choice of driving frequency can seriously affect the precision of the external information transmission. At the same level of noise the precision can either decrease or increase depending on the driving frequency. We demonstrate that the destructive influence of noise on the interspike interval can be effectively reduced. That is, with driving signals in certain frequency ranges, the system can show stable periodic spiking even for relatively large noise intensities. Here, the most accurate transmission of an external signal occurs. Outside these frequency ranges, noise of the same intensity destroys the regularity of the spike trains by suppressing the generation of some spikes. On the other hand, we show that noise can have a reconstructive role for certain driving frequencies. Here, increasing noise intensity enhances the coherence of the neuronal response.     

Stochastic resonance in an ion channel following the non-Arrhenius gating rate

Stochastic resonance (SR) is a novel cooperative phenomenon occurring in nonlinear systems due to coupling of an ambient noise and an external signal. Biological systems may use SR mechanism to detect the signal efficiently from an external environment. A number of studies have addressed the SR in artificial ion channels considering external voltages as noises. More important than these external noises is the internal, thermal noise which changes the channel conformations essential for biological functions. In this work, we consider that the channel gating rates follow a non-Arrhenius temperature dependence derived from experimental data of a real biological channel. Using the Monte-Carlo simulations, we find that in this channel SR occurs near a physiological temperature in a very distinctive manner compared with that for the Arrhenius gating model.     

Anharmonic Oscillator Driven by Additive Ornstein–Uhlenbeck Noise

We present an analytical study of a nonlinear oscillator subject to an additive Ornstein–Uhlenbeck noise. Known results are mainly perturbative and are restricted to the large dissipation limit (obtained by neglecting the inertial term) or to a quasi-white noise (i.e., a noise with vanishingly small correlation time). Here, in contrast, we study the small dissipation case (we retain the inertial term) and consider a noise with finite correlation time. Our analysis is non perturbative and based on a recursive adiabatic elimination scheme a reduced effective Langevin dynamics for the slow action variable is obtained after averaging out the fast angular variable. In the conservative case, we show that the physical observables grow algebraically with time and calculate the associated anomalous scaling exponents and generalized diffusion constants. In the case of small dissipation, we derive an analytic expression of the stationary probability distribution function (PDF) which differs from the canonical Boltzmann–Gibbs distribution. Our results are in excellent agreement with numerical simulations.     

Phase transitions in a bistable system driven by two colored noises

The colored noise problem is studied from the point of view of consistent Markovian approximations through extending unified colored-noise approximation to the case of two-colored-noise driving systems. A bistable system simultaneously driven by multiplicative and additive colored noise is investigated by means of the extended unified colored-noise approximation. It is found that, for weak strength and color of the additive noise, the form of the stationary probability distribution changes from a unimodal to a bimodal structure via a three modal one as the correlation time of the multiplicative colored noise increases, showing the system undergoes a first order phase transition from a monostable to a bistable state. Numerical simulations support our results. Received 10 August 1998 and Received in final form 23 April 1999     

乘性噪音和加性噪音效应对脉孢菌生物钟体系内信号随机共振的增强

利用有和无外信号作用的脉孢菌生物钟体系,研究了与加性噪音相关或不相关的乘性噪音对加性噪音诱导出的内信号随机共振的影响作用.结果表明：无外信号的情况下,不论加性和乘性噪音相关与否,当乘性噪音强度小于临界值时,乘性噪音的加入使加性噪音诱导产生的内随机共振强度得到增强;当大于其临界值时,加性噪音的随机共振强度却得不到进一步增强,这说明脉孢菌生物钟体系能抵抗外噪音的干扰而维持自身的生理节奏.当加入外信号时,对于乘性和加性噪音不相关的情况,发现存在最佳频率（0.003 Hz）的外信号能使加性噪音诱导出的内信号随机共 关键词： 噪音 脉孢菌生物钟体系 内信号随机共振     

Generalized Fokker-Planck equation: Derivation and exact solutions

We derive the generalized Fokker-Planck equation associated with the Langevin equation (in the Ito sense) for an overdamped particle in an external potential driven by multiplicative noise with an arbitrary distribution of the increments of the noise generating process. We explicitly consider this equation for various specific types of noises, including Poisson white noise and Lévy stable noise, and show that it reproduces all Fokker-Planck equations that are known for these noises. Exact analytical, time-dependent and stationary solutions of the generalized Fokker-Planck equation are derived and analyzed in detail for the cases of a linear, a quadratic, and a tailored potential.