View source for Noise-induced order

'''Noise-induced order''' is a mathematical phenomenon appearing in the Matsumoto-Tsuda<ref name="MT1983">{{cite journal |last1=Matsumoto |first1= K. |last2= Tsuda |first2= I. |s2cid= 189855973 |date= 1983 |title=Noise-induced order
 |journal= J Stat Phys |volume=31 |issue=1 |pages=87–106  |doi=10.1007/BF01010923 |bibcode= 1983JSP....31...87M }}
</ref> model of the Belosov-Zhabotinski reaction.

In this model, adding noise to the system causes a transition from a "chaotic" behaviour to a more "ordered" behaviour; this article was a seminal paper in the area and generated a big number of citations<ref>{{cite journal |title= Citation Details for "Noise-induced order"  |publisher=Springer |doi= 10.1007/BF01010923 |s2cid=189855973 }}</ref> and gave birth to a line of research in [[Applied mathematics|applied mathematics]] and [[Physics:Physics|physics]].<ref>
{{cite journal |last1=Doi |first1= S. |s2cid= 122930351 |date= 1989 |title=A chaotic map with a flat segment can produce a noise-induced order
 |journal= J Stat Phys |volume=55 |issue=5–6 |pages=941–964 |doi=10.1007/BF01041073 |bibcode= 1989JSP....55..941D }}</ref><ref>
{{cite journal |last1=Zhou |first1= C.S. |last2=Khurts |first2= J. |last3=Allaria |first3= E. |last4=Boccalletti |first4= S. |last5=Meucci |first5= R. |last6=Arecchi |first6= F.T. |date= 2003 |title=Constructive effects of noise in homoclinic chaotic systems
 |journal= Phys. Rev. E |volume=67 |issue= 6 |page= 066220 |doi=10.1103/PhysRevE.67.066220 |pmid= 16241339 |bibcode= 2003PhRvE..67f6220Z }}
</ref>
This phenomenon was later observed in the Belosov-Zhabotinsky reaction.<ref>{{cite journal |doi=10.1063/1.2946710|pmid=18624484|title=Noise-induced order in the chaos of the Belousov–Zhabotinsky reaction|journal=The Journal of Chemical Physics|volume=129|pages=014508|year=2008|last1=Yoshimoto|first1=Minoru|last2=Shirahama|first2=Hiroyuki|last3=Kurosawa|first3=Shigeru|issue=1|bibcode=2008JChPh.129a4508Y}}</ref>

==Mathematical background==
Interpolating experimental data from the Belosouv-Zabotinsky reaction,<ref>{{cite journal |last1=Hudson |first1= J.L. |last2=Mankin |first2= J.C. |date= 1981 |title=Chaos in the Belousov–Zhabotinskii reaction
 |journal= J. Chem. Phys. |volume=74 |issue= 11 |pages=6171–6177 |doi=10.1063/1.441007 |bibcode= 1981JChPh..74.6171H }} </ref> Matsumoto and Tsuda introduced a one dimensional model, a [[Random dynamical system|random dynamical system]] with uniform additive noise, driven by the map:

<math>
T(x)=\begin{cases} 
(a+(x-\frac{1}{8})^{\frac{1}{3}})e^{-x}+b, & 0\leq x\leq 0.3 \\ 
c(10xe^{\frac{-10x}{3}})^{19}+b & 0.3\leq x\leq 1
\end{cases}
</math>

where
* <math>a=\frac{19}{42}\cdot\bigg(\frac{7}{5}\bigg)^{1/3}</math> (defined so that <math>T'(0.3^-)=0</math>),
* <math>b=0.02328852830307032054478158044023918735669943648088852646123182739831022528_{158}^{213}</math>, such that <math>T^5 (0.3)</math> lands on a repelling fixed point (in some way this is analogous to a [[Misiurewicz point]])
* <math>c=\frac{20}{3^{20}\cdot 7}\cdot\bigg(\frac{7}{5}\bigg)^{1/3}\cdot e^{187/10}</math> (defined so that <math>T(0.3^-)=T(0.3^+)</math>).

This random dynamical system is simulated with different noise amplitudes using [[Floating-point arithmetic|floating-point arithmetic]] and the [[Lyapunov exponent]] along the simulated orbits is computed; the Lyapunov exponent of this simulated system was found to transition from positive to negative as the noise amplitude grows.<ref name="MT1983" /> 

The behavior of the floating point system and of the original system may differ;<ref>{{cite journal |last1=Guihéneuf |first1= P. |date= 2018 |title=Physical measures of discretizations of generic diffeomorphisms
 |journal= Erg. Theo. And Dyn. Sys. |volume=38 |issue=4 |pages= 1422–1458 |doi=10.1017/etds.2016.70 |arxiv=1510.00720 |s2cid= 54986954 }}
 </ref> therefore, this is not a rigorous mathematical proof of the phenomenon.

A computer assisted proof of noise-induced order for the Matsumoto-Tsuda map with the parameters above was given in 2017.<ref>{{cite journal | doi=10.1088/1361-6544/ab86cd| title=Existence of noise induced order, a computer aided proof| year=2020| last1=Galatolo| first1=Stefano| last2=Monge| first2=Maurizio| last3=Nisoli| first3=Isaia| s2cid=119141740| journal=Nonlinearity| volume=33| issue=9| pages=4237–4276| arxiv=1702.07024| bibcode=2020Nonli..33.4237G}}</ref>
In 2020 a sufficient condition for noise-induced order was given for one dimensional maps:<ref>{{cite arXiv
 | eprint     = 2003.08422
 | title      =How does noise induce order?
 | class      =math.DS
 | last1 =Nisoli
 | first1 =Isaia
 | year =2020
 }}</ref> the Lyapunov exponent for small noise sizes is positive, while the average of the logarithm of the derivative with respect to Lebesgue is negative.

==See also==
* [[Physics:Self-organization|Self-organization]]
* Stochastic Resonance

==References==
{{Reflist}}

[[Category:Non-equilibrium thermodynamics]]
[[Category:Name reactions]]
[[Category:Pattern formation]]

{{Sourceattribution|Noise-induced order}}