Non-extensive diffusion entropy analysis: non-stationarity in teen birth phenomena Page: 3
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where the coefficient b is the scaling exponent.
The Shannon entropy is defined by
S(t)= - dx p(x, t) ln[p(x, t)] . (3)
Using the stationary condition of Eq.(2) we obtain
S(t) = A + b In(t), (4)
with
A - dyF(y) ln[F(y)] , (5)
where y = x/t". Eq. (4) indicates that in the case of stationarity, the entropy
S(t) increases linearly with ln(t). Numerically, the scaling exponent 6 can be
evaluated by using fitting curves with function of the form fs(t) = K + Iln(t)
that, when graphed on linear-log graph paper yields straight lines.
The breakdown of the stationary condition may be simulated by assuming
that the scaling exponent 6 of Eq. (2) changes with time. This can be imple-
mented by assuming Eq. (2) has the non-stationary general form
p(x,t) = ) F . (6)
If we assume that
6(t) = So + 9 In(t), (7)
where do and r are two constants, we notice that, in the new non-stationary
condition, the traditional entropy (3) yields:
S(t) = A + So ln(t) + r [ln(t)]2. (8)
The quadratic form of Eq. (8) suggests that the choice of 6(t) given by Eq. (7)
has the mathematical meaning of the quadratic term in the Taylor expansion of
the diffusion entropy (3). As a consequence, we should expect that, in general,
6(t) always assumes the form of Eq. (7), at least for small values of ln(t).
Let us see how all this may be related to the non-extensive Tsallis q-indicator
[8]. The Tsallis non-extensive entropy reads
1 - f+ dx p(x, t)q
Sq(t)= " (9)
q-1
It is straightforward to prove that this entropic indicator coincides with that of
Eq.(3) in the limit where the entropic index q - 1. Let us make the assumption
that in the diffusion regime the departure from this traditional value is weak
and assume c q - 1 << 1. This allows us to use the following approximate
expression for the non-extensive entropy
/+Sq() dx p(x, t)00ln[p(x, ) dx p(x, t)n p(x, )]. (10)
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Scafetta, Nicola; Grigolini, Paolo; Hamilton, P. & West, Bruce J. Non-extensive diffusion entropy analysis: non-stationarity in teen birth phenomena, paper, February 6, 2008; (https://digital.library.unt.edu/ark:/67531/metadc174685/m1/3/: accessed March 29, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.