Information entropy and power-law distributions for chaotic systems

The power law is found for density distributions for the chaotic systems of most different nature (physical, geophysical, biological, economical, social, etc.) on the basis of the maximum entropy principle for the Renyi entropy. Its exponent q is expressed as a function q(β) of the Renyi parameter β. The difference between the Renyi and Boltzmann–Shannon entropies (a modified Lyapunov functional ΛR) for the same power-law distribution is negative and as a function of β has a well-defined minimum at β∗ which remains within the narrow range from 1.5 to 3 when varying other characteristic parameters of any concrete systems. Relevant variations of the exponent q(β∗) are found within the range 1–3.5. The same range of observable values of q is typical for the various applications where the power-law distribution takes place. It is known under the following names: “triangular or trapezoidal” (in physics and technics), “Gutenberg–Richter law” (in geophysics), “Zipf–Pareto law” (in economics and the humanities), “Lotka low” (in science of science), etc. As the negative ΛR indicates self-organisation of the system, the negative minimum of ΛR corresponds to the most self-organised state. Thus, the comparison between the calculated range of variations of q(β∗) and observable values of the exponent q testifies that the most self-organised states are as a rule realised regardless of the nature of a chaotic system.