Escape from the unstable equilibrium in a random process with infinitely many interacting particles

We consider a one-dimensional version of the model introduced in Ref. l. At each site of Z there is a particle with spin ± 1. Particles move according to the Stirring Process and spins change according to the Glauber dynamics. In the hydrodynamical limit, with the stirring process suitably speeded up, the local magnetic densitym _t(r) is proven in Ref. 1 to satisfy the reaction-diffusion equation (*) $$partial _t m_t (r) = tfrac{1}{2}partial _r^2 m_t (r) - V'(m_t )$$ (V(m) = - tfrac{1}{2}alpha m^2 + tfrac{1}{4}beta m^4 ) ,α andβ being determined by the parameters of the Glauber dynamics. In the present paper we consider an initial state with zero magnetization,m ₀(r)=0. We then prove that at long times, before taking the hydrodynamical limit, the evolution departs from that predicted by (*) and that the microscopic state becomes a nontrivial mixture of states with different magnetizations.