Noise-induced synchronization in a spatially distributed system with a 1/f spectrum

Abstract

193Stable physical systems and processes are charac terized by the Lorentzian spectrum of fluctuation power, at which low frequency pulsations are inde pendent of frequency, while high frequency ones descend inversely proportional to the frequency quad rate. Fluctuations are small as a rule and follow the Gaussian distribution. The relaxation of perturbation in such processes occurs following the descend expo nent. The entropy of the stable state is maximal. Large fluctuations spontaneously appear in complex statistic systems that are far from equilibrium. A considerable part of the energy of such pulsations is associated with slow processes and is accumulated at low frequencies; therefore, large scale high energy emissions are possi ble [1]. Such fluctuations have a power spectrum inversely proportional to the frequency: .Another characteristic feature of such fluctuations is the scale invariant distribution function, which has power "tails." In this case, the perturbation relaxation descends not exponentially but following the power law. Such a situation occurs in the thermodynamic critical point of the liquid-vapor phase transition. The scale invariance of fluctuations of thermodynamic quantities near the critical point [2] is determined by the approaching conditions of properties of various phases and requires the exact tuning and large relax ation times. In contrast with the thermodynamic crit ical point, nonequilibrium processes with large fluctu ations point to their stable spatial and temporal scale invariance without fine tuning the parameters. There fore, the appearance of 1/f fluctuations is often associ 1 ( )S f f ated with the concept of self organized criticality [3], which describes the avalanche dynamics and is applied to demonstrate the criticality of the behavior in numerous computer models. Fractional integration of white noise is applied for the mathematical description of random processes with the 1/f spectrum of fluctuations [4], but it is diffi cult to associate it with the physical properties of the system. Fluctuations in a broad range of scales are also developed in turbulent flows of liquid [5]. The appear ance of extreme fluctuations is possible under the effect of white noise in interacting phase transitions. Extreme fluctuations manifest themselves in thermalphysical systems with phase transitions as pulsations of flows in critical and transient modes of heat and mass transfer [6]. When describing the processes with extreme fluctuations, a set of nonlinear stochastic equations that describe the interacting nonequilib rium phase transitions is used [7]:(1)where and are dynamic variables, and is the Gaussian white noise, = , and is the amplitude of the white noise. The criticality of set (1) corresponds to a level of white noise that corresponds to the transition induced by the noise [7]. A random process, which is described by set (1), has a 1/f spectrum of fluctuation power. This is accompanied by the accumulation of fluctuation energy at low frequencies, and the process is character...