Unifying Laws in Multidisciplinary Power-Law Phenomena: Fixed-Point Universality and Nonextensive Entropy

Critical, power-law behavior in space and/or time manifests in a large variety of complex systems [12] within physics and, nowadays, more conspicuously in other fields, such as biology, ecology, geophysics, and economics. Universality, the same power law holding for completely different systems, is a consequence of the characteristic self-similar, scale-invariant property of criticality, and can be understood in terms of basins of attraction of the renormalization-group (RG) fixed points. However, the guiding quality of a variatkmal approach has been seemingly lacking in the theoretical studies of critical phenomena. Here we give an account of entropy extrema associated with fixed points of RG transformations. As illustrations, we consider simple one-dimensional models of random walks and nonlinear dynamical systems. In describing these systems we consider distribution and/or time relaxation functions with power-law decay that may have infinite first- or second- and higher-order moments. When these moments diverge, we observe the emergence of nonexponential or non-Gaussian fractal properties that can be measured by the nonextensive Tsallis entropy index q. We note that the presence of nonextensive properties may signal situations of hindered movement among the system's possible configurations. Some representative applications within physics, but with suggested or recognized connections to other fields, are critical behavior in fluids and magnets, anomalous diffusion processes, transitions to chaos in nonlinear systems, and relaxation properties of supercooled liquids near the glass formation. Two prototypical model systems serve to illustrate the development of critical states characterized by power laws from generic states described by exponential behavior. These are random walks and nonlinear iterated maps that we discuss below in some detail. Random walks [18] are suitable, for example, for representing Brownian motion (molecular thermal motion under the microscope), but also for many types of data originating from diverse disciplines. One type is that which comes in the form of a "time series," a temporal sequence of measured values, for instance, stock market prices in economics or electroencephalographic potentials in medicine.