Complexus MUNDI: Emergent Patterns in Nature

Complexus Mundi Emergent Patterns in Nature

Miroslav M. Novak Editor

~f

^kJ.

> >

• -

„ _

* %

N/

J*
a

(1)

where the factorial has been replaced by the gamma function, which has an analytic continuation into the complex domain for non-integer and non-positive values of its argument. The fractional derivative corresponding to the fractional integral in (1) is given by

D,W=— D , M °

'

dtn"

(2)

'

where [a] + 1 > n > [or] and the bracket denotes the integer value n closest to OC. Consequently for GC 0.

2.1 Derivative of a fractal function Richardson, in his 1926 investigation of turbulence, observed that the velocity field of the atmospheric wind is so erratic that it probably cannot be described by an analytic function16. He suggested a Weierstrass function as a candidate to represent the velocity field, since the function is continuous everywhere, but nowhere differentiable, properties he observed in the wind-field data. Here we investigate a generalization of the Weierstrass function in order to simplify some of the discussion

W(t)=f4\[^Mb"t)]

(3)

under the conditions that b>a>l. Note that the time derivative of (3) yields a series in which the successive terms increase by a factor of b/a>l and therefore diverges. The generalized Weierstrass function (GWF) satisfies the scaling relation

13

W{bt) = aW{t),

(4)

resulting from a simple shift of the index n in the summation. Equation (4) has the form of a renormalization group scaling relation5, which can be solved by assuming a solution of the form

W(t)=±Ant»»

(5)

n=-co

Inserting (5) into (3) yields the equation for the scaling index

H= "

ln\A

(12)

where the phase is a random quantity uniformly distributed on the interval(0,2^"J. Equation (12) would be one way to introduce a random function that has the desired scaling properties and we could discuss the dynamics of this process in time using the fractional operators. An alternative approach is to start with continuous functions and that is what we do now. Consider a function defined by the Fourier transform

t{t) = ~)e-i""i{co)da). In J

(13)

Operating on this function with the 9L£-fractional operator, with a lower limit of negative infinity, defines a new function

Fa-.lt>^)]

= j - )e^'{-ico)-ai{co)dco

(14)

-00

and the weighting in the iniegraiid is obtained by operating on the exponential in the Fourier transform _XD\^a'\e"° ' ] = e'°" (ico)a . If we now interpret the function (13) as a stochastic quantity, then we can evaluate the correlation of the function (14) at two time points separated by a time interval X

15

(^('K('+4 =7ry H J ^ e ^ ^ V ^ ^ r ^ ) ! ' ^ ) )

(15)

where the brackets denote an average over an ensemble of realizations of random fluctuations. If the random fluctuations correspond to a Wiener process the average in the integrand reduces to

£(a>$\a>2^=C8(