Title:  A Primer on Complexity, Part 1:  Inverse Power Laws – October 23, 2006, Part 2:  Fractional Calculus – October 24, 2006

Speaker: Bruce J. West, Chief Scientist, Mathematics and Information Science Directorate Army
Research Office

Date/Time: Monday & Tuesday, October 23 & October 24, 2006, 1:00 – 2:00 pm

Location: CSRI Building/Room 90 (Sandia-NM)

Brief Abstract:   Part 1:  Complex adaptive systems have become popular in the last few years, particularly in the field of medicine, but the modeling and understanding of complex phenomena has always been a major concern of the physical scientist. In my presentation I will adopt a historical perspective and review ways in which complexity has been described by mathematical models over the past few hundred years, but in this first lecture I will avoid the explicit introduction of the formalism. There have been two main strategies for modeling the time development of complex physical phenomena over this period; dynamic equations and phase space equations. These approaches are intended to capture the chaos of nonlinear dynamics, the uncertainty of statistics and the limited predictability of probabilities.
However these approaches are no longer adequate when phenomena contain long-range memory in time and/or nonlocal interactions in space. Such mechanisms are manifest in inverse power-law correlation functions and/or inverse power-law probability densities, which introduce the notion of fractal stochastic processes. We will focus on the exotic nature of inverse power law statistics and discuss how this replacement of the statistics of Gauss has revolutionized medicine.
Brief Abstract:  Part 2:  Physiology is found to be replete with fractal time series, for example, heart beats, respiration rate, walking and cerebral blood flow are all fractal processes. The evolution of such fractal stochastic phenomena is found to be well described by fractional partial differential equations for the evolution of the probability density and fractional Langevin equations for the evolution of particle trajectories. These ‘modern’ descriptions of complexity require the application of the fractional calculus, which we interpret in the context of a number of biomedical phenomena.
We review how the traditional mathematics of random walks, Langevin equations and Fokker-Planck equations are replaced by fractional random walks, fractional Langevin equations and fractional phase space equations. These fractional operators are necessary for a proper description of a calculus of medicine, since they are required to determine the time evolution of fractal phenomena. We apply the methods of the fractional calculus to heart beat, stride interval, blood flow to the brain and breathe interval time series.