130
11
Randomness and Complexity
and
Sw(f ) ∝ 1 ;
(11.23)
the power spectrum is convergent at low frequencies, but if one integrates up from
some finite frequency towards infinity, one finds a divergence: there is an infinite
amount of power at the highest frequencies; that is, a plot ofw left parenthesis t right parenthesisw(t) is infinitely choppy
and the instantaneous value of w left parenthesis t right parenthesisw(t) is undefined!
White noise is also called Johnson (who first measured it experimentally, in 1928)
or Nyquist (who first derived its power spectrum theoretically) noise. It is character-
istic of the voltage across a resistor measured at open circuit and is due to the random
motions of the electrons. The integral of white noise,
B(t) =
{
w(t) dt ,
(11.24)
corresponds to a random walk or Brownian motion (hence, “brown noise”). Its power
spectrum is
SB(f ) ∝ 1/f 2 ;
(11.25)
that is, it is convergent when integrating to infinity, but divergent when integrating
down to zero frequency. In other words, the function has a well-defined value at each
point, but wanders ever further from its initial value at longer and longer times; that
is, it does not have a well-defined mean value.
If current is flowing across a resistor, then the power spectrum of the voltage fluc-
tuations upper S Subscript upper F Baseline left parenthesis f right parenthesis proportional to 1 divided by fSF(f ) ∝1/f [“1 divided by f1/f noise”, sometimes called “fractional Gaussian noise”
(FGN), as a special case of fractionally integrated white noise]. FGNs are char-
acterized by a parameter upper FF: the mean distance travelled in the process described
by its integral upper G Subscript upper F Baseline left parenthesis t right parenthesis equals integral x Subscript upper F Baseline left parenthesis t right parenthesis d tGF(t) =
{
xF(t) dt is proportional to t Superscript upper FtF, and the power spectrum
upper S Subscript upper G Baseline left parenthesis f right parenthesis proportional to 1 divided by f Superscript 2 upper F minus 1SG(f ) ∝1/f 2F−1. White noise has upper F equals one halfF = 1
2, and 1 divided by f1/f noise has upper F equals 1F = 1. It is divergent
when integrated to infinite frequency and when integrated to zero frequency, but the
divergences are only logarithmic. 1 divided by f1/f noise exhibits very long-range correlations,
the physical reason for which is still a mystery. Many natural processes exhibit 1 divided by f1/f
noise.
11.5
Complexity
The notion of complexity occurs rather frequently in biology, where one often refers
to the complexity of this or that organism (cf. biological complexity, Sect. 11.6).
Several procedures for ascribing a numerical value to it have been devised, but for
all that it remains somewhat elusive. When we assert that a mouse is more complex
than a bacterium (or than a fly), what do we actually mean? Intuitively, the assertion
is unexceptionable—most people would presumably readily agree that man is the
most complex organism of all. Is our genome the biggest (as may once have been