Phase segregation dynamics for the Blume–Capel model with Kac interaction

We consider the Glauber and Kawasaki dynamics for the Blume–Capel spin model with weak long-range interaction on the infinite lattice: a ferromagnetic d-dimensional lattice system with the spin variable σ taking values in {−1,0,1} and pair Kac potential γdJ(γ(|i−j|)), γ>0, i,j∈Zd. The Kawasaki dynamics conserves the empirical averages of σ and σ2 corresponding to local magnetization and local concentration. We study the behaviour of the system under the Kawasaki dynamics on the spatial scale γ−1 and time scale γ−2. We prove that the empirical averages converge in the limit γ→0 to the solutions of two coupled equations, which are in the form of the flux gradient for the energy functional. In the case of the Glauber dynamics we still scale the space as γ−1 but look at finite time and prove in the limit of vanishing γ the law of large number for the empirical fields. The limiting fields are solutions of two coupled nonlocal equations. Finally, we consider a nongradient dynamics which conserves only the magnetization and get a hydrodynamic equation for it in the diffusive limit which is again in the form of the flux gradient for a suitable energy functional.