Comments on the renormalization group, scaling and measures of complexity

One definition of complexity is implicit in phenomena that are neither completely ordered nor completely random. This lack of certitude can be traced to the fact that the phenomenon has multiple scales, all of which are coupled to one another. This multiplicity of scales is the harbinger of fractals and scaling. Herein we discuss some of the methods, such as renormalization group theory, scaling and fractal geometry, that have been applied to the understanding of such complex phenomena in physics, economics and medicine. The simplest and probably the most familiar model of statistical processes in the physical sciences is the random walk. Herein we extend the random walk model to include long-time memory in the dynamics and find that this gives rise to a fractional-difference stochastic equation of motion for complex phenomena. The continuum limit of this latter random walk is a fractional differential equation driven by random fluctuations, which in physics is called a Langevin equation for integer dimensions. The index of the inverse power-law spectrum in many biophysical processes can be related to the non-integer order of the fractional derivative in a fractional Langevin equation. Such a fractional stochastic model suggests that a scale-free process guides the dynamics of many complex biophysical phenomena, which are typically multifractal.