Fisica Matematica Turing instability in a model with two interacting Ising lines M. Capanna
31-01-2018 - 11:30
Largo San Leonardo Murialdo,1 - Pal.C - AULA 211 In [1], the author introduces a reaction-diffusion system to model the pattern formation phenomenon present in morphogenesis. Under the assumption that the reaction part of the system is stable around an equilibrium point, he finds condiditions over the diffusion coefficients under which the hole system is unstable due to the amplification of non-zero Fourier modes. This phenomenon is known as Turing instability.
In this talk, we introduce an interacting particle system at which the latter phenomenon is present. The system is a continuous-time Markov process that has two coupled discrete toruses with Ising spins as state-space. The evolution in each torus responds to macroscopic ferromagnetic Kac's potentials, while the spins in different toruses interact in a local attractive-repulsive way. About this model, we prove hydrodynamic limit, and find conditions that guarantee the occurence of Turing instability.
In the Turing instability regime, we analyze the fluctuations of the density fields around the equilibrium point (0,0) by studying the limiting behaviour of the discrete Fourier modes of the system. More precisely, we prove that, at a time at which the process is infinitesimal, and under the proper spatial scaling, the unstable Fourier modes converge to a normal distribution while the rest of the modes vanish. We finally give a result about pattern formation at a time that converges to the critical one at which the process starts to be finite.
[1] A. M. Turing, The chemical basis of morphogenesis.
org: GIULIANI Alessandro
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