Scaling of 1/<Emphasis Type="Italic">f</Emphasis> noise in nonequilibrium phase transitions

Investigation of the dynamics of fluctuations of heat and mass transfer reveals that its crisis and transient modes exhibit high-energy pulsations with a power spectrum that is inversely proportional to frequency (flicker or 1/f fluctuations). Such a spectrum suggests energy transfer from high-to low-frequency modes and the possibility of large-scale catastrophic outbursts in the system being considered. The theory shows that such fluctuations arise in the system owing to the simultaneous occurrence of interacting phase transitions in the presence of white noise having a sufficiently high intensity. The distribution of fluctuations for scale transformations of the set of stochastic equations that describe the generation of 1/f noise is investigated. It is shown that, under a scale transformation, the Gaussian distribution of a random process having a 1/f spectrum passes to an exponential distribution, which is characteristic of the statistics of extreme outbursts. The probability of such outbursts must be taken into account in predicting the stability of various heat-transfer modes.     

Low frequency fluctuations of laser beam intensity run through the system of cavitation water clusters

Experiments on water cavitation in ultrasonic field have been carried out. Low frequency fluctuations of the intensity of laser beam run through the cavitation area have been studied. Experiments have proved presence of low-frequency random fluctuations with frequency dependence of power spectra, S ∼ 1/f ^α where the exponent α ranged within 0.8 ≤ α ≤ 1.2. From experimental realizations, large-scale low frequency pulsations characterized by scale invariance, which duration is distributed according to the power law, have been distinguished. The results are explained on the basis of mathematical model for the rise of scale invariant fluctuations with 1/f ^α power spectrum in the system of two nonlinear stochastic differential equations describing interaction of heterogeneous phase transitions. Distribution of extreme low frequency emissions obtained from numerical solutions to stochastic equations takes the power series form. Correlations of dynamic scaling between critical indicators determining frequency dependence of pulsations power spectra α and distribution functions of extreme low frequency pulsation amplitudes β have been determined. It is shown that both in the experiments on acoustic cavitation of water and in the theoretical model of interacting phase transitions critical indicators are bound with the correlation α + β = 2. Spectra of fluctuation power are determined in the experiments simpler and more accurately than the function of extreme amplitudes distribution. In case when only one frequency dependence of fluctuation capacity spectra is known correlations between indicators serve to obtain information on the distribution of large scale emissions and to estimate critical amplitudes.     

Relaxation in establishing a steady-state stochastic process with 1/<Emphasis Type="Italic">f</Emphasis> spectrum and low-frequency spike statistics

According to theory, fluctuations with a power spectrum inversely proportional to frequency (1/f processes) may arise when dissimilar phase transitions simultaneously take place in physical systems with intense white noise. In this work, relaxation effects in establishing a steady-state stochastic process with non-equilibrium phase transitions are described in terms of two nonlinear stochastic differential equations. The results thus obtained carry information on the statistics of large-scale low-frequency spikes. Step “forgetting” of initial conditions is noted. It is numerically shown that the distributions of the durations and maximal values of extreme low-frequency spikes have a power-type form.     

Boiling up of jets of superheated ethanol-water solutions

The stability and evolution of the shapes of jets of strongly superheated aqueous solutions of ethanol flowing out of a high-pressure chamber are experimentally investigated at various initial parameters. It has been found that a low-boiling liquid causes shifting to lower temperatures of the boundaries between various kinds of boiling-up jets, such as cylindrical, conical, and open. Large-scale pulsations of the jet shapes are observed in the temperature range from 454 to 484 K, where the jet completely breaks down. The conditions for the formation of large-scale low-frequency pulsations with a power spectrum diverging inversely proportional to the frequency f (1/f spectra) have been revealed. Such fluctuations prove that the flow is unstable and large-scale low-frequency ejections are possible.     

1/f Noise in a nonequilibrium phase transition: Experiment and mathematical model

The results of an experimental investigation of a high-power source of broad-band 1/f noise, which can be generated in a system of two interacting nonequilibrium phase transitions, are presented. This process takes place when a normal conductor-superconductor phase transition is superposed on the critical liquid-vapor transition in a boiling coolant. A mathematical model describing a nonequilibrium phase transition in a complicated nonlinear system with two interacting order parameters, which involves the conversion of white noise into stochastic fluctuations of the order parameters with 1/f and 1/f ² spectra, is proposed. The properties of the model fluctuations with a 1/f spectrum agree qualitatively with the experimentally observed properties. A characteristic difference between the model fluctuations with a 1/f ² spectrum and random walks is also noted. Zh. éksp. Teor. Fiz. 113, 1748–1757 (May 1998)     

Low-frequency fluctuations in stochastic processes with a 1/<Emphasis Type="Italic">f</Emphasis><Superscript>α</Superscript> spectrum

The results of numerical analysis of the Brownian movement of a particle in the force field of the potential corresponding to interacting subcritical and supercritical phase transitions are considered. If the white noise intensity corresponds to the critical intensity of the noise-induced transition, the system of stochastic differential equations describes random steady-state processes with fluctuation power spectra inversely proportional to frequency f, S(f) ∼ 1/f ^α, where exponent α varies in the interval 0.8 ≤ α ≤ 1.8. Exponent β of distribution function P(τ) ∼ τ^−β for the duration of low-frequency extremal fluctuations, which are analogous to avalanches considered in the models of self-organized criticality in many respects, varies between the same limits. It is shown that exponents α and β are connected through the relation α + β = 2.     

Entropy maximum in a nonlinear system with the 1/<Emphasis Type="Italic">f</Emphasis> fluctuation spectrum

Analysis of the control and subordination is carried out for the system of nonlinear stochastic equations describing fluctuations with the 1/f spectrum and with the interaction of nonequilibrium phase transitions. It is shown that the control equation of the system has a distribution function that decreases upon an increase in the argument in the same way as the Gaussian distribution function. Therefore, this function can be used for determining the Gibbs-Shannon informational entropy. The local maximum of this entropy is determined, which corresponds to tuning of the stochastic equations to criticality and indicates the stability of fluctuations with the 1/f spectrum. The values of parameter q appearing in the definition of these entropies are determined from the condition that the coordinates of the Gibbs-Shannon entropy maximum coincide with the coordinates of the Tsallis entropy maximum and the Renyi entropy maximum for distribution functions with a power dependence.     

Stochastic resonance in a nonlinear system with a 1/f spectrum

The system of two nonlinear stochastic equations simulating 1/f fluctuations during the interaction of nonequilibrium phase transitions in the presence of an external harmonic force is analyzed using numerical methods. It is shown that the stochastic resonance occurring in the system enhances the output periodic signal under the action of noise. A random process with a 1/f power spectrum corresponds to the Gibbs-Shannon information entropy peak. In stochastic resonance, the information entropy is minimal.     

Inversion of the relative probabilities of the f-f and f-d transitions in the Ln3+ lanthanide ions at their radio- and sonoexcitation as compared to photoexcitation

The results of measuring the efficiencies of the formation of electronically excited states of the Ln³⁺ lanthanide ions in aqueous solutions in the processes of radioluminescence and multibubble sonoluminescence are analyzed. In both cases, electronic excitation occurs due to inelastic collisions of Ln³⁺ ions with (for radioluminescence) charged ionizing particles in liquid and (for multibubble sonoluminescence) high-energy particles, primarily electrons, in the gas phase of cavitation bubbles. In both processes, the efficiencies of exciting ions whose luminescence states appear in the 4f-5d transitions (Ce³⁺ and Pr³⁺) are significantly lower (by an order of magnitude or larger) than the efficiencies of exciting ions whose luminescence states appear in the 4f-4f transitions (Gd³⁺ and Tb³⁺). Therefore, the probability of the f-d transitions is lower than the probability of the f-f transitions in lanthanide ions excited by collisions with the charged particles and the relative probabilities of these transitions are inverted in these processes as compared to photoexcitation. Original Russian Text ? G.L. Sharipov, 2007, published in Pis’ma v Zhurnal éksperimental’noĭ i Teoreticheskoĭ Fiziki, 2007, Vol. 85, No. 9, pp. 559–562.     

Oscillations and Waves in a Nonlinear System with the 1/<Emphasis Type="Italic">f</Emphasis> Spectrum

Numerical methods are used to study a spatially distributed system of two nonlinear stochastic equations that simulate interacting phase transitions. Conditions for self-oscillations and waves are determined. The 1/f and 1/k spectra of extreme fluctuations are formed when waves emerge and move under the action of white noise. The distribution of the extreme fluctuations corresponds to the maximum entropy, which is proven by the stability of the 1/f and 1/k spectra. The formation and motion of waves under external periodic perturbation are accompanied by spatiotemporal chaotic resonance in which the domain of periodic pulsations is extended under the action of white noise.     

Dynamics of transition processes and structure formation in critical heat-mass transfer regimes during liquid boiling and cavitation

The results of experimental studies concerning the development of critical phenomena and structure formation in the process of boiling in falling films and during liquid cavitation are given. In conditions of stepwise and periodic pulsed surges of a thermal load, the parameters of formed metastable regular structures and critical parameters of heat-releasing surface drying are shown to be determined by the dynamics of moving wetting boundaries in the process of system self-organization. In the case of high-intensity heat fluxes, decomposition of a falling film is determined by propagation regimes of self-maintaining boiling fronts with a complex shape of intermediate structures. The study of ultrasonic cavitation of water, glycerin, and vacuum oil shows that structures of interacting gas-vapor bubbles (having the form of fractal clusters) are formed near the emitter surface. Spatial structures are characterized by a low-frequency divergence of the power spectra and a scale-invariant function of the fluctuation distributions. The experimental results are in good qualitative agreement with the numerical simulations performed within the theory of 1/f fluctuations in the case of nonequilibrium phase transitions in a spatially distributed system.     

Critical behavior and 1/f noise in lumped systems undergoing two phase transitions

It is shown that the interaction of order parameters when subcritical and supercritical phase transitions take place simultaneously may result in a self-organized critical state and cause a 1/f ^α fluctuation spectrum, where 1≤α≤2. Such behavior is inherent in potential and nonpotential systems of nonlinear Langevin equations. A numerical analysis of the solutions to the proposed systems of stochastic differential equations showed that the solutions correlate with fractional integration and differentiation of white noise. The general behavior of such a system has features in common with self-organized criticality.     

Coupled phase transitions under periodic perturbation

The influence of periodic perturbation on the system of two nonlinear stochastic equations, which model low-frequency pulsations in crisis and transient modes of heat-and-mass transfer with phase transitions, has been investigated by numerical methods. When studying the influence of the periodic perturbation on the system, a researcher should largely take into account the phase diagram. It is shown that nonequilibrium phase transitions from asymmetric cycles of phase trajectories to centrally symmetric ones occur in the absence of noise. These transitions are accompanied by the stochastic resonance response, which enhances as the frequency of the external periodic force decreases.     

Self-organized criticality and 1/f fluctuation under conditions of nonequilibrium phase transitions

The results are given of an experimental investigation of fluctuation phenomena under conditions of electric arc discharge. Fluctuations are observed whose spectral density is inversely proportional to frequency (1/f noise). Power dependences are revealed of the fluctuation distribution functions. The behavior of spectral density and of distribution functions is associated with the simultaneous occurrence of various nonequilibrium phase transitions. Within the framework of the mean field theory, a mathematical model is suggested of interacting nonequilibrium phase transitions in a distributed system, which predicts the self-organization of the critical state and the generation of fluctuations with diverging spectral characteristics. An adequate agreement is observed between the suggested model and experimental data.     

A finite fractal cluster theory of 1/f noise in percolation systems near metal-insulator transition

The problem of 1/f noise in thin metal films and metal-insulator composites in the scaling fractal regime near percolation threshold is considered. The correspondence between a percolation transition and a second order phase transition is extended from the point of view of electronic polarization and electrical fluctuations. The charge fluctuations on finite fractal clusters are argued to be analogous to spontaneous order parameter fluctuations in phase transitions, being correlated upto percolation correlation length. The charge relaxation times are shown to be related to the cluster sizes having distribution function of the formg()^–b , whereb is connected to Euclidean and fractal dimensionalities and critical exponents. This produces the 1/f noise spectrum. Below percolation threshold, the nodes-links-blobs picture is invoked such that the blobs represent metallic conductances of the finite clusters and the links are tunnelling conductances between them through narrowest barrier regions. Above threshold, the finite cluster network is visualized as connected to the infinite cluster through narrowest tunnelling regions. The correlated spontaneous charge fluctuation on finite fractal clusters is held responsible for conductance fluctuation on either side of the metal-insulator transition via tunnelling processes. Finally, the scaling behaviour of noise magnitude near percolation threshold is explained.     

Autowaves in a near-threshold medium

Approximate and numerical methods are used to study the behavior of autowaves for parameters close to the propagation threshold. Under these conditions, the variations in wave velocity and amplitude are slow. A quasi-steady-state equation is derived for the velocity. This equation describes the relaxation to a steady state (uniform motion) in the above-threshold region and the initial damping stage that determines the time scale of this process in the below-threshold region. As the threshold is approached, the time scales indefinitely increase in the above-and below-threshold regions of parameters. Small random inhomogeneities of the active medium and other “ noise” sources produce intense velocity pulsations. These pulsations are comparable in scale to the mean velocity (as in the case of strong turbulence) and resemble the critical fluctuations in order parameter near the point of a continuous phase transition in their statistical properties. The pulsation spectrum exhibits a sharp peak at zero frequency. In contrast to flicker noise, this peak disappears as one recedes from the threshold. The solutions to the quasi-steady-state equation and the results of numerical simulations agree as long as the fluctuations are small— as in the theory of continuous transitions, beyond the fluctuation region.     

Universality in the length spectrum of integrable systems

The length spectrum of periodic orbits in integrable hamiltonian systems can be expressed in terms of the set of winding numbers {M ₁,…,M _f} on thef-tori. Using the Poisson summation formula, one can thus express the density, Σδ(T−T _M), as a sum of a smooth average part and fluctuations about it. Working with homogeneous separable potentials, we explicitly show that the fluctuations are due to quantal energies. Further, their statistical properties are universal and typical of a Poisson process as in the corresponding quantal energy eigenvalues. It is interesting to note however that even though long periodic orbits in chaotic billiards have similar statistical properties, the form of the fluctuations are indeed very different.     

Phase separation in binary systems with competing internal and external multiplicative noises

We have considered phase separation processes in binary stochastic systems with thermal diffusion and ballistic mixing representing irradiation influence. Introducing fluctuations of thermal flux and an external source of atom relocations due to ballistic diffusion into dynamics of a globally conserved field, we have shown that there are two competing mechanisms of phase transitions type of “order-disorder”: thermally assisted diffusion and irradiation induced atomic exchange. We have studied dynamics of the structure function at early stages of decomposition. In the framework of the mean field theory we have derived the effective Fokker-Planck equation to describe phase separation processes. It was shown that the ordering processes can be controlled by both regular and stochastic parts of external source influence. A reentrant behavior of a mean field order parameter versus the external noise intensity and fluctuations correlation radius is found.     

Stability of a random process with a 1/f spectrum under deterministic action

The stability of the resultant process, which appears upon the interaction of a random process having a 1/f spectrum with a deterministic action, is analyzed using the entropy maximum principle. Under the action of a harmonic force, stable resultant processes are divided into two branches depending on the amplitude of the harmonic force. Due to an exponential relaxation upon an increase in the damping coefficient, the resultant process acquires the Lorentz spectrum without high-energy low-frequency spikes.     

Fluctuations and waves in a spatially distributed system with the 1/<Emphasis Type="Italic">f</Emphasis> spectrum

A spatially distributed system with the 1/ _f spectrum of fluctuation power is modeled by two nonlinear stochastic equations. The numerical methods show the formation of 1/ _f and 1/ _k spectra of extreme fluctuations against the background of the formation and motion of waves under the effect of white noise. The distribution of extreme fluctuations corresponds to the maximum of entropy, which testifies to their stability. Under an external periodic perturbation in the system, it is possible to observe space?time stochastic resonance.