1/f noise and self-organised critical systems

". . . 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 cq(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/f2\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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