Some Thoughts on Modeling Complexity
Bruce J. West, ST / Mathematical and Informational Science Directorate, US Army Research Office
Complex adaptive systems have become popular in the last few years, but the modeling and understanding complex phenomena has always been a major concern of the physical/life scientist. In my presentation I will adopt a historical perspective and review ways in which complexity has been described by physical/biological/engineering models over the past hundred years or so. There have been two main strategies for modeling the time development of complex phenomena over this period; dynamic equations and phase space equations. The dynamic equations start with Newton’s law, or their equivalent, and incorporate complexity through random forces, for example, Langevin’s development of ordinary stochastic differential equations for particle trajectories (eg. Brownian motion). The phase space equations are obtained by averaging over the single particle trajectories to obtain equations of motion for the probability density, usually partial differential equations that are first order in time and second order in space, such as the Fokker-Planck equation (eg. Einstein diffusion).
However these approaches are no longer adequate when phenomena contain long-term memory in time 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. (I will give many examples of these.) 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 physiological phenomena.
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