Journal of Physical Studies 7(1), 1–26 (2003) DOI: https://doi.org/10.30970/jps.07.1 THEORY OF SELF-SIMILAR STOCHASTIC SYSTEMS. PART II A. I. Olemskoi, D. O. Kharchenko Sumy State University 2 Rimskiy-Korsakov Str., Sumy, UA-40007, Ukraine

We show that the description of the evolution of statistical moments, which are reduced to the order parameter, autocorrelator and response function yield to calculation of the average of the power-law function with a fractional exponent. Such a kind of calculation can be performed due to using of the self-similar properties of the system. In such a case the reversible transition ‟disorder-order-disorder” is observed when temperature is changed. Using the fractional Lorenz system with additive noises above mentioned scheme is applied to the problem of self-organized criticality. Conditions of the avalanche formation show that its behaviour depends on intensities of control parameter fluctuations in a critical manner. According to the phase diagram, a discontinues regime of self-organized criticality is realized if intensities of fluctuations of energy of avalanches and complexity of their ensemble pass to the critical magnitudes. The power-law distribution over the avalanche size is reproduced. Relations between the exponent of the size distribution, fractal dimension of phase space, characteristic exponent of multiplicative noise, dynamical exponents, and non-extensivity parameter are found.

PACS number(s): 02.50.-r, 47.53.+n, 64.60.Cn, 64.60.Lx, 72.70.+m.

ps pdf