Turing instability in a model with two interacting Ising lines: linear stability and non-equilibrium fluctuations

TL;DR

This paper investigates conditions for Turing instability in a particle system with two interacting Ising lines, analyzing linear stability and non-equilibrium fluctuations around the null equilibrium, revealing Gaussian fluctuations and mode behaviors near criticality.

Contribution

It provides a detailed analysis of Turing instability conditions and fluctuation behaviors in a two-line Ising particle system, extending previous hydrodynamic results.

Findings

Turing instability occurs under specific conditions around the null equilibrium.

Non-equilibrium fluctuations are Gaussian for long times near the hydrodynamic limit.

Fourier modes remain away from zero at the critical time.

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 quations obtained in \cite{CSL17a}, we find conditions under which Turing instability occurs around the null 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 one at which the process is of finite order, we prove that the $\pm 1$-Fourier modes are uniformly away from zero.