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

AbstractWe 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 densitymt(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,m0(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.