Escape from the unstable equilibrium in a random process with infinitely many interacting particles. (English) Zbl 0652.60110

Summary: We consider a one-dimensional version of the model introduced by the first author, P. A. Ferrari and J. L. Lebowitz, ibid. 44, 589-644 (1986; Zbl 0629.60107). At each site of Z there is a particle with spin \(\pm 1\). 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 density \(m_ l(r)\) is proven in the paper cited above to satisfy the reaction-diffusion equation \[ (*)\quad \partial_ tm_ t(r)=2^{- 1}\partial^ 2_ rm_ t(r)-V'(m_ t) \] \(V(m)=-2^{-1}\alpha m^ 2+4^{-1}\beta m^ 4\), \(\alpha\) and \(\beta >0\), \(\alpha\) and \(\beta\) 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.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)


Zbl 0629.60107
Full Text: DOI


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