Maximum entropy in a nonlinear system with a power spectrum

An analysis of master–slave hierarchy has been made in a system of nonlinear stochastic equations describing fluctuations with a 1 / f spectrum at coupled nonequilibrium phase transitions. It is shown that for a system of stochastic equations there exist different probability distribution functions with power-law (non-Gaussian) and Gaussian tails. The governing equation of a system has a probability distribution function with Gaussian tails. Therefore, distribution functions for governing equations may be used for finding the Gibbs–Shannon entropy. The local maximum of this entropy has been found. It corresponds to the tuning of the parameters of the equations to criticality and points to the stability of fluctuations with a 1 / f spectrum. The Tsallis entropy and the Renyi entropy for the probability distribution functions with power-law tails have been calculated. The parameter q , which is included in the determination of these entropies has been found from the condition that the coordinates of the maximum Gibbs–Shannon entropy coincide with the maxima of the Tsallis and Renyi entropies.