Colored Noise in Dynamical Systems

Natural systems are undeniably subject to random fluctuations, arising from either environmental variability or thermal effects. The consideration of those fluctuations supposes to deal with noisy quantities whose variance might at times be a sizable fraction of their mean levels. It is known that, under these conditions, noisy fluctuations can interact with the system's nonlinearities to render counterintuitive behavior, in which an increase in the noise level produces a more regular behavior. In systems with spatial degrees of freedom, this regularity takes the form of spatiotemporal order. An overview is presented of the mechanisms through which noise induces, enhances, and sustains ordered behavior in passive and active nonlinear media, and different spatiotemporal phenomena are described resulting from these effects. The general theoretical framework used in the analysis of these effects is reviewed, encompassing the theory of stochastic partial differential equations and coupled sets of ordinary stochastic differential equations. Experimental observations of self-organized behavior arising out of noise are also described, and details on the numerical algorithms needed in the computer simulation of these problems are given.

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Spatiotemporal order out of noise

Hydrodynamic and hydromagnetic stability

Stochastic motion of a point – known as Brownian motion – has many successful applications in science, thanks to its scale invariance and consequent universal features such as Gaussian fluctuations. In contrast, the stochastic motion of a line, though it is also scale-invariant and arises in nature as various types of interface growth, is far less understood. The two major missing ingredients are: an experiment that allows a quantitative comparison with theory and an analytic solution of the Kardar-Parisi-Zhang (KPZ) equation, a prototypical equation for describing growing interfaces. Here we solve both problems, showing unprecedented universality beyond the scaling laws. We investigate growing interfaces of liquid-crystal turbulence and find not only universal scaling, but universal distributions of interface positions. They obey the largest-eigenvalue distributions of random matrices and depend on whether the interface is curved or flat, albeit universal in each case. Our exact solution of the KPZ equation provides theoretical explanations.

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Growing interfaces uncover universal fluctuations behind scale invariance

We experimentally investigate the critical behavior of a phase transition between two topologically different turbulent states of electrohydrodynamic convection in nematic liquid crystals The statistical properties of the observed spatiotemporal intermittency regimes are carefully determined, yielding a complete set of static critical exponents in full agreement with those defining the directed percolation class in 2+1 dimensions This constitutes the first clear and comprehensive experimental evidence of an absorbing phase transition in this prominent nonequilibrium universality class

http://daisy.phys.s.u-tokyo.ac.jp/student/kuroda/prl_2007.pdf

Directed percolation criticality in turbulent liquid crystals.

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Pattern Dynamics in the Electrohydrodynamics of Nematic Liquid Crystals

We show experimentally that for the electrohydrodynamic instability in nematic liquid crystals the threshold for the onset of spatial turbulence can be increased by at least a factor of 2 by superimposing noise on the applied voltage. We find that for sufficiently high noise intensities a direct transition from the spatially homogeneous state to turbulence via intermittency can be induced.

External noise can suppress the onset of spatial turbulence

Various types of macroscopic nonlinear-waves were observed on the interface between two liquid phases, water and nitrobenzen, which contained a surfactant and an iodine respectively. These two solutes react with each other and produce precipitate. Some of macroscopic structures have been caused by a kind of the Marrangoni effect with the second order reaction. Main motions were classified into four kinds of waves, i.e. solitary, oscillatory, rotational and chaotic ones. Other submotions were also observed. The typical motions, the characteristies and the concentration phase diagram are given.

Experimental Study of Nonlinear Waves on Interface between Two Liquid Phases with Chemical Reaction

A chemical trigger wave is initiated which propagates with accelerating speed in a thin liquid layer of the oscillatory Belousov-Zhabotinsky reaction. The localized phase gradient of the oscillation and induction of a hydrodynamic motion traveling with the chemical wave are possible causes for the acceleration. Evidence for pronounced hydrodynamic flow is supplied by digital processing of images visualizing flow patterns under the microscope. Surface deformations due to the hydrodynamic motion are analyzed by Mach-Zehnder interferometry.

Accelerating chemical waves accompanied by traveling hydrodynamic motion and surface deformation

Pattern-formation processes and the associated defect motions are experimentally studied for various rectangular cells with aspect ratios \ensuremath{\Gamma} ranging from 8 to 15 in the electrohydrodynamic instability (EHD). Wave-number selection arises from the competition between the most rapidly growing mode ${k}_{c}$, which dominates initially, and the final stable mode ${k}_{m}$ in the Williams domain (WD) state, which is influenced by the cell boundaries as well as nonlinear effects. Defect motion is associated with such competitive growth into the final stage. The fluctuating Williams domains (FWD's) are strongly related to the oscillatory gliding motion of defects. A temporally nonperiodic change of the number of defects is observed in the FWD state with a power spectrum of 1/f type (defect chaos). For a step change in the external voltage, a clear threshold for the onset of FWD is determined. On the other hand, when the voltage is increased continuously no defect is formed at the threshold for WD, and nondecaying defect motion starts at a certain threshold value. A large hysteresis is observed near the onset of FWD.The step change of the external field leads to defect formation more easily than the continuous change. A detailed phase diagram of stable convective patterns in the plane spanned by the threshold voltage and the applied frequency is presented. The characteristic behavior of defect motions, for example, its oscillatory activity of gliding and climbing motions, is strongly related to the pattern following the second pattern instability. In the case of the strong gliding oscillation, a pattern with temporal order appears above the second threshold. Contrary to this, in the case of the weakly oscillatory motion of defects, a stationary pattern is usually observed above its threshold. The magnetic field suppresses the defect chaos and stabilizes the system, that is, the threshold of the applied electric field for the onset of EHD is raised when the magnetic field is increased. The magnetic field can also change the direction of the roll axis so that it is perpendicular to the external field, which leads more easily to the zigzag and oblique patterns. A detailed phase diagram for the magnetic field effect is also given.

Shoichi Kai

Papers

Pattern Dynamics in the Electrohydrodynamics of Nematic Liquid Crystals

External noise can suppress the onset of spatial turbulence

Experimental Study of Nonlinear Waves on Interface between Two Liquid Phases with Chemical Reaction

Accelerating chemical waves accompanied by traveling hydrodynamic motion and surface deformation

Spatial and temporal behavior of pattern formations and defect motions in the electrohydrodynamic instability of nematic liquid crystals