Journal of Statistical Physics

, Volume 174, Issue 2, pp 365–403 | Cite as

Turing Instability in a Model with Two Interacting Ising Lines: Non-equilibrium Fluctuations

  • Monia Capanna
  • Nahuel Soprano-LotoEmail author


This is the second of two articles on the study of a particle system model that exhibits a Turing instability type effect. About the hydrodynamic equations obtained in Capanna and Soprano (Markov Proc Relat Fields 23(3):401–420, 2017), we find conditions under which Turing instability occurs around the zero equilibrium solution. In this instability regime: for long times at which the process is of infinitesimal order, we prove that the non-equilibrium fluctuations around the hydrodynamic limit are Gaussian; for times converging to the critical time defined as the one at which the process starts to be of finite order, we prove that the \(\pm \,1\)-Fourier modes are uniformly away from zero.


Non-equilibrium fluctuations Turing instability Ising Kac potential 



It is a great pleasure to thank Errico Presutti for suggesting us the problem and for his continuous advising. We also acknowledge (in alphabetical order) fruitful discussions with Inés Armendáriz, Anna De Masi, Pablo Ferrari, Ellen Saada, Livio Triolo, and Maria Eulália Vares. The authors also acknowledge the hospitality of Laboratoire MAP5 at Université Paris Descartes. We finally thank the anonymous referees for several comments that helped us improve the presentation of the article.

Compliance with Ethical Standards

Conflict of interest

We declare our research does not involve potential conflicts of interest nor participation with Humans and/or Animals.


  1. 1.
    Asslani, M., Di Patti, F., Fanelli, D.: Stochastic turing patterns on a network. Phys. Rev. E 86, 046105 (2012)ADSCrossRefGoogle Scholar
  2. 2.
    Andreu-Vaillo, F., Mazón, J.M., Rossi, J.D., Toledo-Melero, J.J.: Nonlocal Diffusion Problems. Mathematical Surveys and Monographs, vol. 165. American Mathematical Society, Providence (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Biancalani, T., Fanelli, D., Di Patti, F.: Stochastic turing patterns in the Brusselator model. Phys. Rev. E 81, 046215 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    Brémaud, P.: Markov Chains: Gibbs Fields, Monte Carlo Simulation and Queues, Texts in Applied Mathematics. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cao, Y., Erban, R.: Stochastic Turing patterns: analysis of compartment-based approaches (2010). arXiv:1310.7634
  6. 6.
    Capanna, M., Soprano-Loto, N.: Turing instability in a model with two interacting Ising lines: hydrodynamic limit. Markov Proc. Relat. Fields 23(3), 401–420 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    De Masi, A., Ferrari, P.A., Lebowitz, J.L.: Reaction–diffusion equations for interacting particle systems. J. Stat. Phys. 44(3), 589–644 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Han, Q.: A Basic Course in Partial Differential Equations, Graduate Studies in Mathematics, vol. 120. American Mathematical Society, Providence (2011)Google Scholar
  9. 9.
    Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 320. Springer, Berlin (1999)zbMATHGoogle Scholar
  10. 10.
    Meester, R., Roy, R.: Continuum Percolation, Cambridge Tracts in Mathematics, vol. 119. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  11. 11.
    Murray, J.D.: Mathematical Biology. Interdisciplinary Applied Mathematics. Spatial Models and Biomedical Applications, vol. 18, 3rd edn. Springer, New York (2003)Google Scholar
  12. 12.
    Perthame, B.: Parabolic Equations in Biology, Lecture Notes on Mathematical Modelling in the Life Sciences. Growth, Reaction, Movement and Diffusion. Springer, Cham (2015)Google Scholar
  13. 13.
    Pollard, D.: Convergence of Stochastic Processes. Springer Series in Statistics. Springer, New York (1984)CrossRefzbMATHGoogle Scholar
  14. 14.
    Scarsoglio, S., Laio, F., D’Odorico, P., Ridolfi, L.: Spatial pattern formation induced by Gaussian white noise. Math. Biosci. 229(2), 174–184 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Terras, A.: Fourier Analysis on Finite Groups and Applications, London Mathematical Society Student Texts, vol. 43. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  16. 16.
    Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. Ser. B 237(641), 37–72 (1952)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zheng, Q., Wang, Z., Shen, J., Iqbal, A., Muhammad, H.: Turing bifurcation and pattern formation of stochastic reaction–diffusion system. Adv. Math. Phys. (2017). Scholar

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Authors and Affiliations

  1. 1.Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Università degli Studi dell’AquilaL’AquilaItaly
  3. 3.Gran Sasso Science InstituteL’AquilaItaly

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