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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
Article

Abstract

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.

Keywords

Non-equilibrium fluctuations Turing instability Ising Kac potential 

Notes

Acknowledgements

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.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

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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