Editing Pink noise (section)

== Origin ==
{{See also|Supersymmetry#Supersymmetry in dynamical systems}}
There are many theories about the origin of pink noise. Some theories attempt to be universal, while others apply to only a certain type of material, such as [[semiconductor]]s. Universal theories of pink noise remain a matter of current research interest.

A hypothesis (referred to as the Tweedie hypothesis) has been proposed to explain the genesis of pink noise on the basis of a mathematical convergence theorem related to the [[central limit theorem]] of statistics.<ref name="Kendal2011b">{{cite journal |vauthors=Kendal WS, Jørgensen BR | year = 2011 | title = Tweedie convergence: a mathematical basis for Taylor's power law, 1/''f'' noise and multifractality. | journal = Phys. Rev. E | volume = 84 | issue = 6| page = 066120 | doi = 10.1103/physreve.84.066120 | pmid = 22304168 | bibcode = 2011PhRvE..84f6120K | url = https://findresearcher.sdu.dk:8443/ws/files/55639035/e066120.pdf }}</ref> The [[Tweedie convergence theorem]]<ref>{{cite journal |author1=Jørgensen B |author2=Martinez JR |author3=Tsao M |name-list-style=vanc |title=Asymptotic behaviour of the variance function|journal=Scandinavian Journal of Statistics |year=1994 |volume=21 |pages=223–243}}</ref> describes the convergence of certain statistical processes towards a family of statistical models known as the [[Tweedie distribution]]s. These distributions are characterized by a variance to mean [[power law]], that have been variously identified in the ecological literature as [[Taylor's law]]<ref name="Taylor1961">{{cite journal |vauthors=Taylor LR | year = 1961 | title = Aggregation, variance and the mean | journal = Nature | volume = 189 | issue = 4766| pages = 732–735 | doi = 10.1038/189732a0 | bibcode = 1961Natur.189..732T | s2cid = 4263093 }}</ref> and in the physics literature as ''fluctuation scaling''.<ref name="Eisler2008">{{cite journal |vauthors=Eisler Z, Bartos I, Kertesz J | year = 2008 | title = Fluctuation scaling in complex systems: Taylor's law and beyond | journal = Advances in Physics | volume = 57 | issue = 1| pages = 89–142 | doi = 10.1080/00018730801893043 | bibcode = 2008AdPhy..57...89E | arxiv = 0708.2053 | s2cid = 119608542 }}</ref> When this variance to mean power law is demonstrated by the method of expanding enumerative bins this implies the presence of pink noise, and vice versa.<ref name="Kendal2011b" /> Both of these effects can be shown to be the consequence of [[convergence in distribution|mathematical convergence]] such as how certain kinds of data will converge towards the [[normal distribution]] under the central limit theorem. This hypothesis also provides for an alternative paradigm to explain [[power law]] manifestations that have been attributed to [[self-organized criticality]].<ref name=Kendal2015>
{{cite journal
 | author = Kendal WS
 | year = 2015
 | title = Self-organized criticality attributed to a central limit-like convergence effect
 | journal =Physica A 
 | volume = 421
 | pages =141–150
 | doi = 10.1016/j.physa.2014.11.035
|bibcode = 2015PhyA..421..141K
 }}</ref>

There are various mathematical models to create pink noise.  The superposition of exponentially decaying pulses is able to generate a signal with the <math>1/f</math>-spectrum at moderate frequencies, transitioning to a constant at low frequencies and <math>1/f^2</math> at high frequencies.<ref>{{Cite arXiv |title=1/f noise: a pedagogical review |eprint=physics/0204033 |date=2002-04-12 |first=Edoardo |last=Milotti}}</ref>  In contrast, the sandpile model of self-organized criticality, which exhibits quasi-cycles of gradual stress accumulation between fast rare stress-releases, reproduces the flicker noise that corresponds to the intra-cycle dynamics.<ref>{{cite journal | first1 = Alexander | last1 = Shapoval | first2 = Mikhail | last2 = Shnirman | title = Explanation of flicker noise with the Bak-Tang-Wiesenfeld model of self-organized criticality | journal = Physical Review E | volume = 110 | page = 014106 | year = 2024 | doi = 10.1103/PhysRevE.110.014106 | url = https://journals.aps.org/pre/abstract/10.1103/PhysRevE.110.014106 | arxiv = 2212.14726 }}</ref> The statistical signature of self-organization is justified in <ref>{{Cite journal|title = Statistical signatures of self-organization|journal = Physical Review A|date = 1992-10-01|pages = R4475–R4478|volume = 46|issue = 8|doi = 10.1103/PhysRevA.46.R4475|first1 = Kevin P.|last1 = O'Brien|first2 = M. B.|last2 = Weissman|pmid = 9908765|bibcode = 1992PhRvA..46.4475O }}</ref>  It can be generated on computer, for example, by filtering white noise,<ref>{{Cite web|title = Noise in Man-generated Images and Sound|url = http://mlab.uiah.fi/~eye/mediaculture/noise.html|website = mlab.uiah.fi|access-date = 2015-11-14}}</ref><ref>{{Cite web|title = DSP Generation of Pink Noise|url = http://www.firstpr.com.au/dsp/pink-noise/|website = www.firstpr.com.au|access-date = 2015-11-14}}</ref><ref>{{Cite journal|url = http://linkage.rockefeller.edu/wli/moved.8.04/1fnoise/mcclain01.pdf|title = Numerical Simulation of Pink Noise|last = McClain|first = D.|date = May 1, 2001|journal = Preprint|archive-url = https://web.archive.org/web/20111004100713/http://linkage.rockefeller.edu/wli/moved.8.04/1fnoise/mcclain01.pdf|archive-date = 2011-10-04}}</ref> [[inverse Fourier transform]],<ref>{{Cite journal|title = On Generating Power Law Noise |journal = Astronomy and Astrophysics|date = 1995-01-01|pages = 707–710|volume = 300|first1 = J.|last1 = Timmer|first2 = M.|last2 = König |bibcode= 1995A&A...300..707T }}</ref> or by multirate variants on standard white noise generation.<ref name="Voss-1978" /><ref name="Gardner-1978" />

In [[Supersymmetric theory of stochastic dynamics|supersymmetric theory of stochastics]],<ref>{{cite journal|year=2016|title=Introduction to supersymmetric theory of stochastics|journal=Entropy|volume=18|issue=4|pages=108|doi=10.3390/e18040108|author=Ovchinnikov, I.V.|arxiv=1511.03393|bibcode=2016Entrp..18..108O|s2cid=2388285|doi-access=free}}</ref> an approximation-free theory of [[stochastic differential equation]]s, 1/''f'' noise is one of the manifestations of the spontaneous breakdown of topological [[supersymmetry]]. This supersymmetry is an intrinsic property of all stochastic differential equations and its meaning is the preservation of the continuity of the [[phase space]] by continuous time dynamics. Spontaneous breakdown of this supersymmetry is the stochastic generalization of the concept of [[chaos theory|deterministic chaos]],<ref>{{ cite journal | author1 = Ovchinnikov, I.V.| author2 = Schwartz, R. N. | author3 = Wang, K. L. | title = Topological supersymmetry breaking: Definition and stochastic generalization of chaos and the limit of applicability of statistics | journal = Modern Physics Letters B | volume = 30 | issue = 8 | year = 2016 | pages = 1650086 | doi = 10.1142/S021798491650086X | arxiv = 1404.4076 | bibcode = 2016MPLB...3050086O | s2cid = 118174242 }}</ref> whereas the associated emergence of the long-term dynamical memory or order, i.e., 1/''f'' and [[Crackling noise|crackling]] noises, the [[Butterfly effect]] etc., is the consequence of the [[Goldstone boson|Goldstone theorem]] in the application to the spontaneously broken topological supersymmetry.