Approximate Fokker-Planck Equation Research Articles

Transient Properties of a Bistable System with Delay Time Driven by Non-Gaussian and Gaussian Noises: Mean First-Passage Time

The mean first-passage time of a bistable system with time-delayed feedback driven by multiplicative non-Gaussian noise and additive Gaussian white noise 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 coloured noise approximation, the small time delay approximation and the Novikov Theorem. The functional analysis and simplification are employed to obtain the approximate expressions of MFPT. The effects of non-Gaussian parameter (measures deviation from Gaussian character) r, the delay time τ, the noise correlation time τ0, the intensities D and α of noise on the MFPT are discussed. It is found that the escape time could be reduced by increasing the delay time τ, the noise correlation time τ0, or by reducing the intensities D and α. As far as we know, this is the first time to consider the effect of delay time on the mean first-passage time in the stochastic dynamical system.

Time Dependence of Entropy Flux and Entropy Production of a Dissipative Dynamical System Driven by Non-Gaussian Noise

A stochastic dissipative dynamical system driven by non-Gaussian noise is investigated. A general approximate Fokker–Planck equation of the system is derived through a path-integral approach. Based on the definition of Shannon's information entropy, the exact time dependence of entropy flux and entropy production of the system is calculated both in the absence and in the presence of non-equilibrium constraint. The present calculation can be used to interpret the interplay of the dissipative constant and non-Gaussian noise on the entropy flux and entropy production.

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 λ, the intensities D and α of noise on the MFPT are discussed. It is found that the asymmetry parameter β, the non-Gaussian parameter r and the coupling coefficient λ 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 (λ,τ2) between noises on MFPT T± is different.

Effect of colored cross-correlated noise on the gene transcriptional regulatory system

The transient properties (e.g., the mean first passage time T) of the gene transcriptional regulatory system driven by colored cross-correlated noise are investigated. The approximate Fokker-Planck equation is obtained based on the Novikov theorem and the Fox approach and the explicit expression of the mean first passage time is derived. The numerical computation results show that for the case of strong correlation intensity and short correlation time，a successive switch process (i.e.，“on”→“off”→“on”) occurs; for tha case of weak correlation intensity and long correlation time，only one switch process occurs (i.e.，“on”→“off”). The reentrance phenomena appear in the gene transcriptional regulatory system.

Transient properties of a tumor cell growth system driven by color Gaussian noises: Mean first-passage time

The transient properties (mean first passage time T)of the tumor cell growth system driven by color Gaussian noise are investigated. The approximate Fokker-Planck Equation is obtained based on the Novikov theorem and the Fox approach and the explicit expression of the mean first passage time is derived. The numerical computation results show that T behaves with different characters in the cases of 0λλλλT exhibits complex behavior, namely T has one maximum with the multiplicative noise intensity D and its self-correlation time τ1 increasing.

Influence of Coloured Correlated Noises on ProbabilityDistribution and Mean of Tumour Cell Number in the Logistic Growth Model

An approximate Fokker–Planck equation for the logistic growthmodel which is driven by coloured correlated noises is derived byapplying the Novikov theorem and the Fox approximation. The steady-stateprobability distribution (SPD) and the mean of the tumour cell number areanalysed. It is found that the SPD is the single extremum configurationwhen the degree of correlation between the multiplicative and additivenoises, λ, is in −1<λ⩽0 and can be the double extrema in0<λ<1. A configuration transition occurs because of the variation ofnoise parameters. A minimum appears in the curve of the mean ofthe steady-state tumour cell number, ⟨x⟩, versus λ.The position and the value of the minimum are controlled by thenoise-correlated times.

Stochastic resonance in a bistable system with time-delayed feedback and non-Gaussian noise

The phenomenon of stochastic resonance in a bistable system with time-delayed feedback driven by non-Gaussian noise is investigated. Combining the small time delay approximation, the path-integral approach and the unified colored noise approximation, a general approximate Fokker–Planck equation of a stochastic system is obtained. The effects of the parameter q indicating the departure from the Gaussian noise, the delay time τ, and the correlation time τ0 of the non-Gaussian noise on the quasi-steady-state probability distribution function (SPD) and the signal-to-noise ratio (SNR) are discussed. It is found that the number of peaks in SPD and the reentrant transition between one peak and two peaks and then to one peak again in the curve of SNR depends on the parameter q, the delay time τ, and the noise correlation time τ0.

Statistical fluctuations in a saturation laser model with correlated noises

We study the effects of correlations between quantum and pump noises on fluctuations of the laser intensity in a saturation laser model. An approximative Fokker–Planck equation and analytic expressions of the steady-state probability distribution function (SPD) of the laser system are derived. Based on the SPD, the normalized mean, the normalized variance, and the normalized skewness of the steady-state laser intensity are calculated numerically. The results indicate that (i) the correlation strength λ of correlated noises always enhances the fluctuation of laser intensity; (ii) the correlation time τ of correlated noises strengthens the fluctuation of laser intensity for the below-threshold case but τ weakens it for the above-threshold case.

EFFECTS OF CORRELATED NOISES IN A SATURATION LASER MODEL

The effects of correlations between additive and multiplicative noises in a saturation laser model are investigated. The approximative Fokker–Planck equation and analytic expressions of the steady-state probability distribution function (SPD) of the laser system are derived. Based on the SPD, the normalized mean, the normalized variance, and the normalized skewness of the steady-state laser intensity are calculated numerically. Our results indicate that: (i) For the laser being operated above threshold, the correlation strength λ reduces the intensity fluctuation; (ii) For the laser being operated near threshold and below threshold, the correlation strength λ enhances the intensity fluctuation.

Stochastic system with coupling between non-Gaussian and Gaussian noise terms

A stochastic system with coupling between non-Gaussian and Gaussian noise terms is investigated. A general approximate Fokker–Planck equation of the system is derived through a path-integral approach. For a bistable system, the coupling λ between noise terms can induce the reentrance-like phase transition while the parameter q of the departure from the Gaussian noise can induce the first-order-like phase transition. Both the coupling λ and the parameter q can change the curve of the mean first passage time (MFPT) from monotonically decreasing function to a peak in the MFPT. Numerical simulations are carried out to check the approximate theoretical results. Reasonably good agreement is obtained.

Approximate Fokker-Planck equation for a single-mode laser driven by quadratic pump noise and quantum noise with cross-correlation between real and imaginary parts of noise

Approximate Fokker-Planck equation for a single-mode laser driven by quadratic pump noise and quantum noise with cross-correlation between real and imaginary parts of noise

Effects of Cross-Correlation Colour Noises on Tumour GrowthProcess

We present a tumour cell growth process model including a multiplicativecoloured noise and an additive coloured noise correlated. How the noisecross-correlation intensity λ and correlation time τ canaffect the steady state properties of tumour cell growth model arediscussed by solving an approximative Fokker–Planck equation. It isfound that the increase of noise correlation time τ can cause thetumour cell number increasing, but the increase of multiplicative noiseintensity can cause the tumour cell number extinction. We also find thatthe increase of cross-correlation intensity λ in the case of 0<λ<1 can cause the tumour cell number extinction, whereasincrease of cross-correlation intensity λ in the case ofλ<0 can cause the tumour cell number increasing.

The stationary properties and the state transition of the tumor cell growth mode

We study the stationary properties and the state transition of the tumor cell growth model (the logistic model) in presence of correlated noises for the case of nonzero correlation time. We derived an approximative Fokker-Planck equation and the stationary probability distribution (SPD) of the model. Based the SPD, we investigated the effects of both correlation strength ( $\lambda$ ) and correlation time ( $\tau$ ) of cross-correlated noises on the SPD, the mean of the tumor cell population and the normalized variance ( $\lambda_2$ ) of the system, and calculated the state transition rate of the system between two stable states. Our results indicate that: (i) $\lambda$ and $\tau$ play opposite roles in the stationary properties and the state transition of the system, i.e. increase of $\lambda$ can produce a smaller mean value of the cell population and slow down the state transition, but increase of $\tau$ can produce a larger mean value of the cell population and enhance state transition; (ii) For large $\lambda$ , there a peak structure on both $\lambda_2$ - $\lambda$ plot and $\lambda_2$ - $\tau$ plot. For the small $\lambda$ , $\lambda_2$ increases with increasing $\lambda$ , but $\lambda_2$ increases with decreasing $\tau$ .

The state variable correlation function of the bistable system subject to the cross-correlated noises

The effects of correlations between additive and multiplicative noises on the state variable correlation function of a bistable system driven by cross-correlated noises were studied. Applying an approximate Fokker–Planck equation to the system, the analytic expression for the state variable correlation function of the system is derived by projection operator method. From numerical computations we find that the normalized correlation function C( s) decreases exponentially as the decay time s increases and decreases with increasing cross-correlated strength λ, however C( s) increases with increasing cross-correlated time τ. λ and τ play opposing roles in the C, i.e., λ enhances the rate of fluctuation decay of dynamical variable and the τ slows down the rate of fluctuation decay of dynamical variable.

Approximate Fokker–Planck equation of system driven by multiplicative colored noises with colored cross-correlation

Using the Hanggi ansatz to truncate the evolution equation for probability density, an approximate Fokker–Planck equation (AFPE) is derived. This AFPE is valid for one-dimensional general systems driven by two multiplicative colored noises ( τ 1≠0 and τ 2≠0) that are correlated in color ( τ 3≠0) under the condition for τ 1, τ 2, and τ 3 to satisfy some inequalities. We apply this AFPE to a symmetrical bistable potential system driven by a colored multiplicative noise and a white additive noise with white cross-correlation. To verify the validity of our analytical approximation the numerical simulations for this system is performed. We discovered a new phenomenon that the symmetry of stationary probability density broken by the correlation between the noises can be gradually recovered as the noise self-correlation time is increased.

Stationary Intensity Distribution of Single-Mode Laser Driven byAdditive and Multiplicative Colored Noises with ColoredCross-Correlation**The project supported by National Natural Science Foundation of China under Grant No. 10275025

Applying the approximate Fokker-Planck equation we derived, we obtain the analytic expression of the stationary laser intensity distribution by studying the single-mode laser cubic model subject to colored cross-correlation additive and multiplicative noise, each of which is colored. Based on it, we discuss the effects on the stationary laser intensity distribution by cross-correlation between noises and “color” of noises (non-Markovian effect) when the laser system is above the threshold. In detail, we analyze two cases: One is that the three correlation-times (i.e. the self-correlation and cross-correlation times of the additive and multiplicative noise) are chosen to be the same value . For this case, the effect of noise cross-correlation is investigated emphatically, and we detect that only when can the noise-induced transition occur in the curve, and only when and , can the “reentrant noise-induced transition” occur. The other case is that the three correlation times are not the same value, . For this case, we find that the noise-induced transition occurring in the curve is entirely different when the values of and are changed respectively. In particular, when (self-correlation time of additive noise) is changing, the ratio of the two maximums of the curve R exhibits an interesting phenomenon, “reentrant noise-induced transition”, which demonstrates the effect of noise “color” (non-Markovian effect).

Moments of intensity of single-mode laser driven by additive and multiplicative colored noises with colored cross-correlation

Applying an approximative Fokker–Planck equation we recently derived for a system driven by a multiplicative colored noise and an additive colored noise that are cross-correlated, we study, in this Letter, the stationary intensity distribution of the single-mode laser cubic model with correlated pump noise and quantum noise which each is colored. We present a thorough discussion of how the noise cross-correlation intensity λ and correlation time τ determine the behaviors of the mean intensity 〈 I〉 and variance λ 2(0). A nonzero value of noise cross-correlation ( λ≠0) leads to the entire λ 2(0)– P′ curve being suppressed (here P′ is the pump noise intensity). This indicates increased precision of laser beam due to the existence of the cross-correlation λ. A combined consequence of τ≠0 and λ≠0 is the changing number of extrema in the λ 2(0)– P′ curve, which is similar to the different phases of stationary intensity distribution. When the net gain coefficient is increased monotonously, the λ 2(0)– P′ curves exhibit reentrant phase transition, in similarity to what occurs in the stationary intensity distribution.

Delayed stochastic systems

Noise and time delay are two elements that are associated with many natural systems, and often they are sources of complex behaviors. Understanding of this complexity is yet to be explored, particularly when both elements are present. As a step to gain insight into such complexity for a system with both noise and delay, we investigate such delayed stochastic systems both in dynamical and probabilistic perspectives. A Langevin equation with delay and a random-walk model whose transition probability depends on a fixed time-interval past (delayed random walk model) are the subjects of in depth focus. As well as considering relations between these two types of models, we derive an approximate Fokker-Planck equation for delayed stochastic systems and compare its solution with numerical results.

Steady-state analysis of a bistable system with additive and multiplicative noises.

An approximate Fokker-Planck equation for a general one-dimensional system driven by correlated noises is derived; the correlation times of the correlations between the noises are nonzero. The steady-state properties of the bistable kinetic model are analyzed. We find the following. (1) In the $\ensuremath{\alpha}\ensuremath{-}D$ parameter plane ($\ensuremath{\alpha}$ and $D$ are the additive noise and multiplicative noise intensities, respectively), the area of the bimodal region of the stationary probability distribution (SPD) is contracted as $\ensuremath{\lambda}$ is increased ($\ensuremath{\lambda}$ is the strength of the correlations between noises), but the area of the bimodal region of the SPD is enlarged as $\ensuremath{\tau}$ is increased ($\ensuremath{\tau}$ is the correlation time of the correlations between noises). (2) $\ensuremath{\lambda}$ and $\ensuremath{\tau}$ play opposing roles in the transition of the SPD of the system. (3) For the case of perfectly correlated noises ($\ensuremath{\lambda}=1$), there is not the phenomenon of the critical ratio ($\frac{\ensuremath{\alpha}}{D}=1$) which was shown by Wu, Cao, and Ke [Phys. Rev. E 50, 2496 (1994)]. (4) The change of the mean of the state variable is very remarkable in the small $\ensuremath{\tau}$ and large $\ensuremath{\lambda}$ regimes. (5) The normalized variance of the state variable increases with increasing $\ensuremath{\tau}$, but decreases with increasing $\ensuremath{\lambda}$.

Stochastic Hopf bifurcation: The effect of colored noise on the bifurcation interval.

We consider a general nonlinear dynamical system undergoing a supercritical Hopf bifurcation, and driven by a Gaussian colored noise. Using normal form theory and an approximate effective Fokker-Planck equation, valid for weak intensity and small correlation time of the noise, we obtain an explicit expression for the stationary probability distribution of the full general system close to the bifurcation, and analyze the changes in the shape of the distribution to discuss the effect of the noise on the bifurcation interval. \textcopyright{} 1996 The American Physical Society.