Noise-induced order
{{Short description|Mathematical phenomenon}}
Noise-induced order is a mathematical phenomenon appearing in the Matsumoto-Tsuda{{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 }} 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 and gave birth to a line of research in applied mathematics and physics. {{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 }}
{{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 }}
This phenomenon was later observed in the Belosov-Zhabotinsky reaction.{{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}}
Mathematical background
Interpolating experimental data from the Belosouv-Zabotinsky reaction,{{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 }} Matsumoto and Tsuda introduced a one-dimensional model, a random dynamical system with uniform additive noise, driven by the map:
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}
where
- (defined so that ),
- , such that lands on a repelling fixed point (in some way this is analogous to a Misiurewicz point)
- (defined so that ).
This random dynamical system is simulated with different noise amplitudes using 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.
The behavior of the floating point system and of the original system may differ;{{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 }}
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.{{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}}
In 2020 a sufficient condition for noise-induced order was given for one dimensional maps:{{cite journal | last=Nisoli | first=Isaia | title=How Does Noise Induce Order? | journal=Journal of Statistical Physics | volume=190 | issue=1 | date=2023 | arxiv= 2003.08422 | issn=0022-4715 | doi=10.1007/s10955-022-03041-y | page=}} the Lyapunov exponent for small noise sizes is positive, while the average of the logarithm of the derivative with respect to Lebesgue is negative.