Statistics of avalanches in stochastic processes with a 1/fα spectrum

The results of numerical investigation of the Brownian motion in a two-dimensional potential field formed under the coupling of phase transitions at the conditions of the criticality induced by white noise are presented. The suggested system of stochastic equations at the white noise intensity that corresponds to the criticality of a noise-induced transition describes stationary random processes with power spectra S(f) f−α, where the exponent α varies in the range 0.8≤α≤1.8. The exponent α was found by the direct FFT method from numerical realizations of the Brownian motion processes. The exponent β of the distribution function P(τ) τ−β of the duration of low frequency extreme fluctuations was determined by numerical methods from the distributions of time of the first passage across a potential barrier with different realizations of white noise. The low frequency extreme fluctuations in many properties are similar to avalanches considered in models of self-organized criticality. The exponents α and β were determined directly from numerical realizations of random processes independently of each other. It is shown that the exponents α and β are related by the relation α+β=2.