Phase segregation dynamics for the Blume-Capel model with Kac interaction.
The Blume-Capel model with Kac interaction is a \(d\)-dimensional lattice system with the spin variable \(\sigma \) taking values in \(\{-1,0,1\}\) and the formal Hamiltonian \[ H_\gamma (\sigma ) = {1\over 2} \sum _{i,j\in Z^{d}} J_\gamma (i-j)\,(\sigma (i)-\sigma (j))^{2} - h_1 \sum _{i\in Z^{d}} \sigma (i) - h_2 \sum _{i\in Z^{d}} \sigma ^{2}(i) \] where \(h_1\) and \(h_2\) are real parameters, and the Kac interaction for \(\gamma >0\) is defined as \(J_\gamma (r)=\gamma ^{d} J(\gamma r) \text{ for every } r\in R^{d}\), where \(J\) is a smooth function on a compact support. The limit behaviour for \(\gamma \to 0\) is studied by means of introducing the Glauber and Kawasaki (respectively) dynamics. Namely, two Markov processes in the infinite volume spin configuration space \(\Omega \) with the pregenerators depending on the above energy function are defined. In the Glauber dynamics each spin flips at random to a new value while in the Kawasaki dynamics the neighboring spins exchange their values. For the corresponding parameters the above KBC models are invariant for both the dynamics. The main result of the present paper states that, properly scaled, the models converge for \(\gamma \to 0\) to solutions of some partial differential equations.
scientific article; zbMATH DE number 2098324
scientific article