9780521861724htl 1..2

The probability of finding theith species and particle at the scalel is defined

aspi¼Ni/N, whereNis the total number of species or particles andNiis theith

species and particle at scalel. This probability,pi, is used to compute the

partition function (Evertsz & Mandelbrot, 1992 ) as:

qðlÞ¼

XNðlÞ

i¼ 1

pqilðqÞ ( 8 : 7 )

where(q) is the Re ́nyi or mass exponent ofq-moments, which is given as:

ðqÞ¼limlogqðlÞ

logl

( 8 : 8 )

Moments of order q act as a scanning tool scrutinizing the denser and

sparser regions of the measure(Chhabra & Jensen, 1989 ). Whenq>0,empha-

sizes regions in the distribution with higher proportions (concentrations) of the

measure, whereas the opposite is true forq<0. More generally, theq-moments

are linked to the dynamics of the scale-related distributions and measure the

amount of non-additivity in the system (Keylock, 2005 ).

It follows that Re ́nyi’s generalized dimension,Dq, which is based on the

concept of generalized entropies (Re ́nyi, 1955 ), is given as:

DqðlÞ¼

1

1 qllim!1

logqðlÞ

logl ¼

ðqÞ

ð 1 qÞ forq6¼^1 (^8 :^9 )

forq¼1,

D 1 ðlÞ¼llim!1



NPðlÞ

i¼ 1

pilogpi

logl ¼

dðqÞ

dq (^8 :^10 )

and forq¼ 0

D 0 ðlÞ¼lim

l!1

logNðlÞ

logl

( 8 : 11 )

whereD 0 is the capacity,D 1 the entropy andD 2 the correlation dimension.D 0 is

equivalent to the single fractal dimension of the measure,D 1 is the entropy

dimension and quantifies the degree of disorder present in the distribution, and

D 2 is mathematically linked to the correlation function, estimating the correla-

tion of measures contained in intervals of sizel.Asqvaries from1toþ1,

the functionDqdescribes the spectrum of generalized fractal dimensions. One

can theoretically encounter difficulties in establishing reliable estimates ofDq

forq>0, because positiveq-moments diminish the terms with smallerpiand the

limit converges very slowly (Schroeder, 1991 ). To overcome this potential draw-

back, the numerator in Eq. (8.9) was calculated for bothl and 2l following the

procedure outlined in Halseyet al.(1986).

154 P.E. SCHMID AND J. M. SCHMID-ARAYA