Supersymmetric theory of stochastic dynamics: Difference between revisions

Revision as of 19:30, 7 August 2025

Revision as of 19:48, 11 August 2025

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== History and relation to other theories ==

The first relation between supersymmetry and stochastic dynamics was established in two papers in 1979 and 1982 by [[Giorgio Parisi]] and Nicolas Sourlas,<ref name=":9">{{Cite journal|last1=Parisi|first1=G.|last2=Sourlas|first2=N.|date=1979|title=Random Magnetic Fields, Supersymmetry, and Negative Dimensions|journal=Physical Review Letters|volume=43|issue=11|pages=744–745|doi=10.1103/PhysRevLett.43.744|bibcode=1979PhRvL..43..744P}}</ref><ref name=":15">{{Cite journal|last=Parisi|first=G.|title=Supersymmetric field theories and stochastic differential equations|journal=Nuclear Physics B|language=en|volume=206|issue=2|pages=321–332|doi=10.1016/0550-3213(82)90538-7|year=1982|bibcode=1982NuPhB.206..321P}}</ref> where [[Langevin equation|Langevin SDEs]] -- SDEs with linear phase spaces, gradient flow vector fields, and additive noises -- were given supersymmetric representation with the help of the [[BRST quantization|BRST]] gauge fixing procedure. While the original goal of their work was [[dimensional reduction]], <ref name=Aharony1976>{{cite journal|author=Aharony, A.|author2=Imry, Y.|author3=Ma, S.K.|year=1976|title=Lowering of dimensionality in phase transitions with random fields|journal=Physical Review Letters|volume=37|issue=20|pages=1364–1367|doi=10.1103/PhysRevLett.37.1364|bibcode=1976PhRvL..37.1364A }}</ref> the so-emerged supersymmetry of Langevin SDEs has since been addressed from a few different angles <ref>{{Cite journal|last1=Cecotti|first1=S|last2=Girardello|first2=L|date=1983-01-01|title=Stochastic and parastochastic aspects of supersymmetric functional measures: A new non-perturbative approach to supersymmetry|journal=Annals of Physics|volume=145|issue=1|pages=81–99|doi=10.1016/0003-4916(83)90172-0|bibcode=1983AnPhy.145...81C|doi-access=free}}</ref><ref>{{Cite journal|last=Zinn-Justin|first=J.|date=1986-09-29|title=Renormalization and stochastic quantization|journal=Nuclear Physics B|volume=275|issue=1|pages=135–159|doi=10.1016/0550-3213(86)90592-4|bibcode=1986NuPhB.275..135Z}}</ref><ref>{{Cite journal|last1=Dijkgraaf|first1=R.|last2=Orlando|first2=D.|last3=Reffert|first3=S.|date=2010-01-11|title=Relating field theories via stochastic quantization|journal=Nuclear Physics B|volume=824|issue=3|pages=365–386|doi=10.1016/j.nuclphysb.2009.07.018|bibcode=2010NuPhB.824..365D|arxiv=0903.0732|s2cid=2033425}}</ref><ref name=":12">{{Cite journal|last=Kurchan|first=J.|date=1992-07-01|title=Supersymmetry in spin glass dynamics|journal=Journal de Physique I|language=en|volume=2|issue=7|pages=1333–1352|doi=10.1051/jp1:1992214|issn=1155-4304|bibcode=1992JPhy1...2.1333K|s2cid=124073976|url=https://hal.science/jpa-00246625/document }}</ref><ref name=":14" /> including the [[fluctuation-dissipation theorem]]s,<ref name=":12" /> [[Jarzynski equality]],<ref name=":20">{{cite arXiv|last1=Mallick|first1=K.|last2=Moshe|first2=M.|last3=Orland|first3=H.|date=2007-11-13|title=Supersymmetry and Nonequilibrium Work Relations|eprint=0711.2059|class=cond-mat.stat-mech}}</ref> [[Onsager reciprocal relations|Onsager principle of microscopic reversibility]],<ref name=":11">{{Cite journal|last=Gozzi|first=E.|date=1984|title=Onsager principle of microscopic reversibility and supersymmetry|journal=Physical Review D|volume=30|issue=6|pages=1218–1227|doi=10.1103/physrevd.30.1218|bibcode=1984PhRvD..30.1218G}}</ref> solutions of [[Fokker–Planck equation]]s,<ref>{{Cite journal|last=Bernstein|first=M.|date=1984|title=Supersymmetry and the Bistable Fokker-Planck Equation|journal=Physical Review Letters|volume=52|issue=22|pages=1933–1935|doi=10.1103/physrevlett.52.1933|bibcode=1984PhRvL..52.1933B}}</ref> [[self-organization]],<ref>{{Cite journal|last1=Olemskoi|first1=A. I|last2=Khomenko|first2=A. V|last3=Olemskoi|first3=D. A|date=2004-02-01|title=Field theory of self-organization|journal=Physica A: Statistical Mechanics and Its Applications|volume=332|pages=185–206|doi=10.1016/j.physa.2003.10.035|bibcode=2004PhyA..332..185O|url=http://essuir.sumdu.edu.ua/handle/123456789/16485}}</ref> etc.

The Parisi-Sourlas method has been extended to several other classes of dynamical systems, including [[classical mechanics]],<ref name=":1">{{Cite journal|last1=Gozzi|first1=E.|last2=Reuter|first2=M.|title=Classical mechanics as a topological field theory|journal=Physics Letters B|language=en|volume=240|issue=1–2|pages=137–144|doi=10.1016/0370-2693(90)90422-3|year=1990|bibcode=1990PhLB..240..137G|url=https://cds.cern.ch/record/204132|url-access=subscription}}</ref><ref name=":16">{{Cite journal|last=Niemi|first=A. J.|title=A lower bound for the number of periodic classical trajectories|journal=Physics Letters B|language=en|volume=355|issue=3–4|pages=501–506|doi=10.1016/0370-2693(95)00780-o|year=1995|bibcode=1995PhLB..355..501N}}</ref> its stochastic generalization,<ref name=":13">{{Cite journal|last1=Tailleur|first1=J.|last2=Tănase-Nicola|first2=S.|last3=Kurchan|first3=J.|date=2006-02-01|title=Kramers Equation and Supersymmetry|journal=Journal of Statistical Physics|language=en|volume=122|issue=4|pages=557–595|doi=10.1007/s10955-005-8059-x|issn=0022-4715|bibcode=2006JSP...122..557T|arxiv=cond-mat/0503545|s2cid=119716999}}</ref> and higher-order Langevin SDEs.<ref name=":14">{{Cite journal|last1=Kleinert|first1=H.|last2=Shabanov|first2=S. V.|date=1997-10-27|title=Supersymmetry in stochastic processes with higher-order time derivatives|journal=Physics Letters A|volume=235|issue=2|pages=105–112|doi=10.1016/s0375-9601(97)00660-9|bibcode=1997PhLA..235..105K|arxiv=quant-ph/9705042|s2cid=119459346}}</ref>

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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 |article-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).