". . . ecological systems are
organized such that the different species 'support'
each other in a way which cannot be understood by studying the
individual constituents in isolation. The same interdependence of
species also makes the ecosystem very susceptible to small
changes or 'noise'. However, the system cannot be too
sensitive since then it could not have evolved into its present
state in the first place. Owing to this balance we may say that
such a system is 'critical'. We shall see that this
qualitative concept of criticality can be put on a firm
quantitative basis.

Such critical systems are abundant in
nature. We shall see that the dynamics of a critical state has a
specific temporal fingerprint, namely 'flicker noise',
in which the power spectrum S(*f*) scales as 1/*f* at
low frequencies. Flicker noise is characterized by**
correlations extended over a wide range of timescales, a clear
indication of some sort of cooperative effect.** Flicker noise
has been observed, for example, in the light from quasars, the
intensity of sunspots, the current through resistors, the sand
flow in an hourglass, the flow of rivers such as the Nile, and
even stock exchange price indices. Despite the ubiquity of
flicker noise, its origin is not well understood. Indeed, one may
say that *because* of its ubiquity, no proposed mechanism to
data can lay claim as the single general underlying root of 1/*f*
noise. **We shall argue that flicker noise is in fact not noise
but reflects the intrinsic dynamics of self-organized critical
systems**. Another signature of criticality is spatial
self-similarity. It has been pointed out that nature is full of
self-similar 'fractal' structures, though the physical
reason for this is not understood."

Bak,
Tang and
Wiesenfeld,
"Self-organized criticality" (*Physical Review* A **38**
(1988) p. 364)

notes on the controversy
surrounding the above article

Chao Tang's homepage

A few reviews of Per
Bak's 1997 book
*How Nature Works: The Science of Self-Organized Criticality*

*Sadly, Per Bak died in October 2002 at the age of 54 (1948-2002).*

Wentian Li's extensive
1/f Noise Bibliography

Sergei Maslov's homepage

Maslov on 1/f noise

Maslov on self-organized criticality

Maslov's 1/f noise 'sandpile' applet

M. Planat,
"1/f frequency noise in a communication receiver and the Riemann
Hypothesis", from M. Planat (editor),
Noise, Oscillators and Algebraic Randomness:
From Noise in Communication Systems to Number Theory,
(conference proceedings - Chapelle des Bois, France, April 5-10, 1999),
Lecture Notes in Physics **550** (Springer-Verlag,
2000)

M. Planat,
"1/*f* noise, the measurement of time and number theory",
*Fluctuation and Noise Letters* **0** (2001).

The following is a brief review paper:

M. Planat, "The impact of prime number theory
on frequency metrology", to appear in Proceedings of 6th Symposium on
Frequency Standards and Metrology, St. Andrews, Scotland, 9-14 September
2001 (World Scientific)

This extends the work in some very interesting directions:

M. Planat, H. Rosu and S. Perrine, "Ramanujan sums for signal processing
of low frequency noise" (submitted to *Physical Review E*

[Abstract:] "An aperiodic (low frequency) spectrum may originate from
the error term in the mean value of an arithmetical function such as
Möbius function or Mangoldt function, which are coding sequences for
prime numbers. In the discrete Fourier transform the analyzing wave is
periodic and not well suited to represent the low frequency regime. In
place we introduce a new signal processing tool based on the Ramanujan
sum *c*_{q}(*n*), well adapted to the analysis of
arithmetical sequences with many resonances *p*/*q*. The sums
are quasi-periodic versus the time *n* of the resonance and aperiodic
versus the order *q* of the resonance. New results arise from the
use of this Ramanujan-Fourier transform (RFT) in the context of arithmetical
and experimental signals."

The final paragraph:

"The other challenge behind Ramanujan sums relates to prime number
theory. We just focused our interest to the relation between 1/*f*
noise in communication circuits and the still unproved Riemann hypothesis
[13]. The mean value of the modified Mangoldt function *b*(*n*),
introduced in (29), links Riemann zeros to the 1/*f*^{2\alpha}
noise and to the Mobius function. This should follow from generic
properties of the modular group *SL*(2,*Z*), the group of 2
by 2 matrices of determinant 1 with integer coefficients [15], and to
the statistical physics of Farey spin chains [16]. See also the link to
the theory of Cantorian fractal spacetime [17]."

[13] M. Planat, "Noise, oscillators and algebraic randomness", *Lecture
Notes in Physics* **550** (Springer, Berlin, 2000)

[15] S. Perrine, *La theorie de Markoff et ses developpements*,
ed. T. Ashpool (Chantilly, 2000)

[16] A. Knauf, *Communications in Mathematical Physics* **3**
(1998) 703

[17] C. Castro and J. Mahecha, arXiv: hep-th/0009014v3 (8 Apr. 2001)

M. Pitkanen,
"Quantum criticality and 1/*f* noise"
(submitted for publication in *Fluctuation and Noise Letters* **0**)

R.L. Bagula, "The
information in the prime sequence and new chaos" (03/10/01)

"Abstract: The variable dimension of the Cantor set created by the
sequence of the primes contains a new kind of 1/*f* noise. In the
investigation of this type of pattern I have created a new kind of dimensional self-similarity
which I have named new chaos."

B. Mandelbrot,
*Multifractals and 1/f Noise (Wild Self-Affinity
in Physics)* (Springer-Verlag, New York, 1998)

B. Luque, O. Miramontes and L. Lacasa, "Number theoretic example of scale-free topology inducing self-organized criticality", *Phys. Rev. Lett.* **101** (2008) 158702

[abstract:] "In this Letter we present a general mechanism by which simple dynamics running on networks become self-organized critical for scale-free topologies. We illustrate this mechanism with a simple arithmetic model of division between integers, the division model. This is the simplest self-organized critical model advanced so far, and in this sense it may help to elucidate the mechanism of self-organization to criticality. Its simplicity allows analytical tractability, characterizing several scaling relations. Furthermore, its mathematical nature brings about interesting connections between statistical physics and number theoretical concepts. We show how this model can be understood as a self-organized stochastic process embedded on a network, where the onset of criticality is induced by the topology."

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