Noise-induced order

Mathematical phenomenon

Noise-induced order is a mathematical phenomenon appearing in the Matsumoto-Tsuda[1] 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 [1] and gave birth to a line of research in applied mathematics and physics. [2] [3] This phenomenon was later observed in the Belosov-Zhabotinsky reaction.[4]

Mathematical background

Interpolating experimental data from the Belosouv-Zabotinsky reaction,[5] Matsumoto and Tsuda introduced a one-dimensional model, a random dynamical system with uniform additive noise, driven by the map:

T ( x ) = { ( a + ( x − 1 8 ) 1 3 ) e − x + b , 0 ≤ x ≤ 0.3 c ( 10 x e − 10 x 3 ) 19 + b 0.3 ≤ x ≤ 1 {\displaystyle 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

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

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.[1]

The behavior of the floating point system and of the original system may differ;[6] 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.[7] In 2020 a sufficient condition for noise-induced order was given for one dimensional maps:[8] 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

  • Self-organization
  • Stochastic Resonance

References

  1. ^ a b c Matsumoto, K.; Tsuda, I. (1983). "Noise-induced order". J Stat Phys. 31 (1): 87–106. Bibcode:1983JSP....31...87M. doi:10.1007/BF01010923. S2CID 189855973.
  2. ^ Doi, S. (1989). "A chaotic map with a flat segment can produce a noise-induced order". J Stat Phys. 55 (5–6): 941–964. Bibcode:1989JSP....55..941D. doi:10.1007/BF01041073. S2CID 122930351.
  3. ^ Zhou, C.S.; Khurts, J.; Allaria, E.; Boccalletti, S.; Meucci, R.; Arecchi, F.T. (2003). "Constructive effects of noise in homoclinic chaotic systems". Phys. Rev. E. 67 (6): 066220. Bibcode:2003PhRvE..67f6220Z. doi:10.1103/PhysRevE.67.066220. PMID 16241339.
  4. ^ Yoshimoto, Minoru; Shirahama, Hiroyuki; Kurosawa, Shigeru (2008). "Noise-induced order in the chaos of the Belousov–Zhabotinsky reaction". The Journal of Chemical Physics. 129 (1): 014508. Bibcode:2008JChPh.129a4508Y. doi:10.1063/1.2946710. PMID 18624484.
  5. ^ Hudson, J.L.; Mankin, J.C. (1981). "Chaos in the Belousov–Zhabotinskii reaction". J. Chem. Phys. 74 (11): 6171–6177. Bibcode:1981JChPh..74.6171H. doi:10.1063/1.441007.
  6. ^ Guihéneuf, P. (2018). "Physical measures of discretizations of generic diffeomorphisms". Erg. Theo. And Dyn. Sys. 38 (4): 1422–1458. arXiv:1510.00720. doi:10.1017/etds.2016.70. S2CID 54986954.
  7. ^ Galatolo, Stefano; Monge, Maurizio; Nisoli, Isaia (2020). "Existence of noise induced order, a computer aided proof". Nonlinearity. 33 (9): 4237–4276. arXiv:1702.07024. Bibcode:2020Nonli..33.4237G. doi:10.1088/1361-6544/ab86cd. S2CID 119141740.
  8. ^ Nisoli, Isaia (2023). "How Does Noise Induce Order?". Journal of Statistical Physics. 190 (1). arXiv:2003.08422. doi:10.1007/s10955-022-03041-y. ISSN 0022-4715.