Supersymmetric theory of stochastic dynamics: Difference between revisions

Revision as of 14:39, 18 July 2025

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| volume = 2

| issue = 4

| pagesarticle-number = 100226

| year = 2021

| doi = 10.1016/j.patter.2021.100226

| pmid = 33982021

| pmc = 8085613

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| url = https://www.sciencedirect.com/science/article/pii/S2666389921000404

| access-date = 2025-06-05

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:''... chaos is counter-intuitively the "ordered" phase of dynamical systems. Moreover, a pioneer of complexity, [[Ilya Prigogine|Prigogine]], would define chaos as a spatiotemporally complex form of order...''

The [[Goldstone theorem]] necessitates the long-range response, which may account for [[pink noise|1/f noise]]. The [[Edge of Chaos]] is interpreted as noise-induced chaos -- a distinct phase where TS is broken in a specific manner and dynamics is dominated by noise-induced instantons. In the deterministic limit, this phase collapses onto the critical boundary of conventional chaos.

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| publisher = Springer

| year = 1997

| doi = 10.1007/978-1-4612-2256-9

| isbn = 978-0-387-94785-3

| url = https://link.springer.com/book/10.1007/978-1-4612-2256-9

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Its phenomenological understanding is largely based on the concepts of [[Adaptive system|self-adaptation]] and [[self-organized criticality|self-organization]].<ref>{{Cite journal|last1=Watkins|first1=N. W.|last2=Pruessner|first2=G.|last3=Chapman|first3=S. C.|last4=Crosby|first4=N. B.|last5=Jensen|first5=H. J.|date=2016-01-01|title=25 Years of Self-organized Criticality: Concepts and Controversies|journal=Space Science Reviews|language=en|volume=198|issue=1–4|pages=3–44|doi=10.1007/s11214-015-0155-x|issn=0038-6308|bibcode=2016SSRv..198....3W|arxiv=1504.04991|s2cid=34782655}}</ref><ref>{{Cite journal|last1=Bak|first1=P.|last2=Tang|first2=C.|last3=Wiesenfeld|first3=K.|date=1987|title=Self-organized criticality: An explanation of the 1/f noise|journal=Physical Review Letters|volume=59|issue=4|pages=381–384|doi=10.1103/PhysRevLett.59.381|pmid=10035754|bibcode=1987PhRvL..59..381B|s2cid=7674321 }}</ref>

STS offers the following explanation for the [[Edge of chaos]] (see figure on the right).,<ref name=":10"/> <ref>{{Cite journal |last=Ovchinnikov |first=I.V. |title=Ubiquitous order known as chaos |date=2024-02-15 |journal=Chaos, Solitons & Fractals |language=en |volume=181 |issue=5 |pagesarticle-number=114611 |doi=10.1016/j.chaos.2024.114611 |arxiv=2503.17157 |bibcode=2024CSF...18114611O |url=https://www.sciencedirect.com/science/article/abs/pii/S0960077924001620 |issn = 0960-0779|url-access=subscription }}</ref> In the presence of noise, the TS can be spontaneously broken not only by the [[Integrable system|non-integrability]] of the flow vector field, as in deterministic chaos, but also by noise-induced instantons.

<ref> {{cite journal|last1=Witten|first1=Edward|title=Dynamical breaking of supersymmetry|journal=Nuclear Physics B|date=1988|volume=188|issue=3|pages=513–554|doi=10.1016/0550-3213(81)90006-7}} </ref>

Under this condition, the dynamics must be dominated by instantons with power-law distributions, as dictated by the Goldstone theorem. In the deterministic limit, the noise-induced instantons vanish, causing the phase hosting this type of noise-induced dynamics to collapse onto the boundary of the deterministic chaos (see figure on the right).