> Jim Fehlinger wrote:
> The analogy here, of course, is a view of the nested processes
> of selection, with evolution as the outermost level and the
> fleeting activation patterns of consciousness as the innermost,
> as all parts of a unified fractal pattern spread out in time,
> spanning timescales ranging from milliseconds to billions of
> years, and broadly self-similar across all those scales.
Beyond the (qualitative) unified fractal pattern there are also fractal
functions. One of the properties of a fractal function is that it does not
possess a characteristic scale length, and consequently its derivatives
diverge.
Fortunately, the dynamics of *complex* phenomena - described by fractal
functions - can be expressed, often, in terms of fractional differential
equations of motion.
Fractional diffusion equations, in example, have been used to model the
evolution of stochastic phenomena with *long-time* memory, that is,
phenomena with correlations that decay as inverse power laws (rather than
exponentially in time).
Actually, fractal functions have, often, fractional derivatives. Therefore
*complex* phenomena - having a fractal dimension - are now reasonably
modeled using fractional equations of motion.
- P. Meakin
"Fractals, Scaling and Growth far from Equilibrium",
Cambridge Nonlinear Science, Series 5,
Cambridge University Press, Cambridge (1998).
- M. Schroeder
"Fractals, Chaos, Power Laws",
W.H. Freeman and Comp., New York (1991).
- B.B. Mandelbrot
"The Fractal Geometry of Nature",
W.H. Freeman and Co., San Francisco (1977).
- B.J. West
"Physiology, Promiscuity and Prophecy at the Millennium: A Tale of Tails",
Studies of Nonlinear Phenomena in the Life Sciences, Vol. 9,
World Scientific, Singapore (1998).
__________________
Entropy can also be regarded as the fractal dimension of an appropriate
compact set.
- Chris Hillman
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