Nano-pattern stabilization by multiplicative noise

Within the reaction-diffusion framework, a one-component system was confined by means of a multiplicative noise into the attraction basin of a patterning attractor in its (infinite-dimensional) configuration space. In this way, inhomogeneities that otherwise would have been “expelled” from this basin, and subsequently eliminated through the diffusive process, were stabilized. For the present study, a model describing the physics of adsorbed particles on a metallic surface has been used. In particular, an underlying deterministic inhomogeneity-building mechanism was exploited, that acts driven by lateral interactions among the adsorbed particles. This process cannot by itself sustain and stabilize the inhomogeneities, but together with the contribution of a particular form of multiplicative noise, it is able to confine the system into the region of configuration space where this mechanism is enabled, hence stabilizing the pattern. Although the proposal could be applied to more general situations, for the particular model studied here we have found that nanopatterns that without the indicated noise source would be eliminated by diffusion, under its effect can grow and be stabilized.     

Power-law tails from multiplicative noise

We show that the well-known linear Langevin equation, modeling the Brownian motion and leading to a Gaussian stationary distribution of the corresponding Fokker-Planck equation, is changed by the smallest multiplicative noise. This leads to a power-law tail of the distribution for sufficiently large momenta. At finite ratio of the correlation strength for the multiplicative and the additive noises the stationary energy distribution becomes exactly the Tsallis distribution.     

Effect of multiplicative noise on parametric instabilities

We report a study on the effect of external multiplicative noise on parametric instabilities using two different experimental systems: an electronic RLC circuit, parametrically pumped with a voltage-variable capacitor, and surface waves generated by vertically vibrating a layer of fluid (the Faraday instability). Both systems are forced by the superposition of a sinusoidal and a noisy component. We study the statistical properties of the response of both systems to noisy parametric forcing and compare them with theoretical predictions. When the detuning from parametric resonance is such that the bifurcation in the absence of noise is supercritical, both systems behave in the same way under the influence of noise. We find that the effect of noise is twofold: on one hand, it triggers the instability before its deterministic onset under the form of oscillatory bursts; on the other hand, it inhibits the nonlinearly saturated oscillatory response above the deterministic onset. When the detuning is such that the bifurcation is subcritical, we find that the two systems behave differently. In the case of the electronic oscillator, noise mostly triggers random transitions between the two states of the bistable region that exists in the absence of noise, whereas in the surface wave experiment new states are created by noise and the bistable region is strongly enlarged.     

General transformations from multiplicative noise to additive noise

General transformations from multiplicative noise to additive noise are obtained for systems whose dynamical evolution is given by multidimensional Langevin equations.     

Kink propagation induced by multiplicative noise

The influence of spatio-temporal external multiplicative fluctuations on a single kink in a bistable distributed system is studied. For this purpose we derive a stochastic dynamic equation for the position of the shifted kink. An analytical estimate for spatio-temporally uncorrelated fluctuations is represented and discussed. We draw the conclusion that multiplicative noise induces a propagation of the most probable kink into the region of larger noise. This effect is demonstrated in numerical simulations.     

On the interpretation of multiplicative white noise

The interpretation of a stochastic differential equation as an approximation to a stochastic difference equation is discussed. A criterion is presented for determining which stochastic calculus should be used.     

Renormalized perturbation theory for a multiplicative noise system

We study an anharmonic overdamped oscillator driven by both additive and multiplicative Gaussian white noise. Within the Martin-Siggia-Rose formalism we employ renormalized perturbation theory to calculate the stationary correlation function and the response function of the oscillator variable. Due to the absence of a simple fluctuation-dissipation theorem renormalization cannot make use of exact stationary moments, but is perormed with the help of renormalization conditions.     

Kinetics of phase transitions with singular multiplicative noise

The behavior of the most probable values of the order parameter x and the amplitude p of conjugate force fluctuations is studied for a stochastic system with a noise amplitude depending on x as |x|^a. It is shown that the phase half-plane x>0 for the canonical pair x, p is divided into isolated regions of large, intermediate, and small values of x. In the first region, the trajectories converge to values of x, p → ∞ as the time t → ∞, and the probability of their realization is negligibly small. In the intermediate region, the configuration point tends to the attraction center corresponding to a stationary ordered state. In the region , the trajectories converge to the point x=p=0 for 0<a<1/2 and to x=0, p → ∞ for 1/2<a≤1. In the former case, the probability of realization of trajectories is finite, while, in the latter case, it is negligibly small, and an absorbing state can be formed.     

Stochastic resonance in linear system driven by multiplicative noise and additive quadratic noise

In this paper the stochastic resonance (SR) is studied in an overdamped linear system driven by multiplicative noise and additive quadratic noise. The exact expressions are obtained for the first two moments and the correlation function by using linear response and the properties of the dichotomous noise. SR phenomenon exhibits in the linear system. There are three different forms of SR: the bona fide SR, the conventional SR and SR in the broad sense. Moreover, the effect of the asymmetry of the multiplicative noise on the signal-to-noise ratio (SNR) is different from that of the additive noise and the effect of multiplicative noise and additive noise on SNR is different.     

Phase separation in binary systems with internal multiplicative noise

A study of a phase separation process in stochastic systems with a field dependent kinetic coefficient and an internal multiplicative noise is presented. Dynamics of spinodal decomposition at early and late stages is investigated by computer simulations where the domain growth law is generalized. A mean field approach was carried out in order to obtain the stationary probability, bifurcation and phase diagrams displaying reentrant phase transitions. We relate our approach to entropy driven phase transitions theory.     

The asymptotic dynamics of processes with multiplicative quadratic noise

The mean value of a non-Markovian stochastic process modelled by the rate equation with a multiplicative Ornstein-Uhlenbeck noise is investigated. The stability properties are discussed and the temporal evolution of the mean value and fluctuations of the process are presented. The effective relaxation time of the system is analysed.The research reported in this paper is supported in part by the Polish Academy of Sciences under the contract CPBP 1.1.4. The authors acknowledge the referee for valuable remarks.     

染料激光中的饱和效应和倍增噪声

研究了单模激光中的倍增噪声和饱和效应,导出了定态激光强度的平均值和归一化方差.同实验值相比较,理论与实验在阈值附近符合得很好.而在远高于阈值时,含有饱和效应的激光模型能更好地描述实际应用中的激光.     

Emergent bistability: Effects of additive and multiplicative noise

Positive feedback and cooperativity in the regulation of gene expression are generally considered to be necessary for obtaining bistable expression states. Recently, a novel mechanism of bistability termed emergent bistability has been proposed which involves only positive feedback and no cooperativity in the regulation. An additional positive feedback loop is effectively generated due to the inhibition of cellular growth by the synthesized proteins. The mechanism, demonstrated for a synthetic circuit, may be prevalent in natural systems also as some recent experimental results appear to suggest. In this paper, we study the effects of additive and multiplicative noise on the dynamics governing emergent bistability. The calculational scheme employed is based on the Langevin and Fokker-Planck formalisms. The steady state probability distributions of protein levels and the mean first passage times are computed for different noise strengths and system parameters. In the region of bistability, the bimodal probability distribution is shown to be a linear combination of a lognormal and a Gaussian distribution. The variances of the individual distributions and the relative weights of the distributions are further calculated for varying noise strengths and system parameters. The experimental relevance of the model results is also pointed out.     

Influence of multiplicative noise on properties of first order dynamical phase transition

Critical phenomena in distributed dynamical two-dimensional nonlinear system near the point of the Turing instability are discussed. The system is considered in the presence of thermal fluctuations and multiplicative noise (MN) representing fluctuations of the bifurcation parameter. Since such a noise of the control parameter can have macroscopic (not thermal) nature, the intensity is considered as sufficiently large in comparison with the amplitude of thermal fluctuations, and it is shown that in the system the first order phase transition occurs with the characteristics which are independent on the thermal noise. Hence the discontinuous transtion could be observable in experimental situations where this would not be possible in the absence of MN (like the Rayleigh-Benard problem). When the correlation length of MN is small, the transition results in the formation of a complex state possessing only short-range order, and when MN is spatially uniform, a quasi-one-dimensional structure will be formed.     

Optimum spatial filter for image restoration degraded by multiplicative noise

We discuss the problem of image restoration when the image is degraded by multiplicative noise. In order to solve this problem, we use as a criterion the minimization of the mean-square error between the restored and the original image. The transfer function of the optimum filter derived in this paper is described by the product of the conjugate of the transfer function of the imaging system and the power spectrum density of the degraded image divided by the convolution of the power spectrum density of the image and that of the noise. Some special cases of the filter are discussed. A numerical experiment has been performed and satisfactory results have been obtained.     

Correlation function of an optical bistable system with cross-correlated additive white noise and multiplicative colored noise

Considering an optical bistable system with cross-correlated additive white noise and multiplicative colored noise, we study the effects of correlation between the noises on the correlation function C（s） using the unified colored noise approximation and the Stratonovich deeoupling ansatz formalism. The effects of the self-correlation time τ of the multiplicative colored noise and the correlation intensity λ between the two noises are studied with numerical calculation. It is found that C（s） increases with the increase of the self-correlation timeτ, but decreases with the increase of the correlation intensity λ. At large value of τ, there is almost no change for C（s） when τ changes.