Method to study relaxation of metastable phases: Macroscopic mean-field dynamics

We propose two different macroscopic dynamics to describe the decay of metastable phases in many-particle systems with local interactions. These dynamics depend on the macroscopic order parameter m through the restricted free energy F(m) and are designed to give the correct equilibrium distribution for m. The connection between macroscopic dynamics and the underlying microscopic dynamic is considered in the context of a projection-operator formalism. Application to the square-lattice nearest-neighbor Ising ferromagnet gives good agreement with droplet theory and Monte Carlo simulations of the underlying microscopic dynamic. This includes quantitative agreement for the exponential dependence of the lifetime 〈τ〉 on the inverse of the applied field H, and the observation of distinct field regions in which Λ≡d ln〈τ〉/d‖H${\mathrm{\Vert }}^{1\mathrm{-}\mathit{d}}$ depends differently on ‖H‖. In addition, at very low temperatures we observe oscillatory behavior of Λ with respect to ‖H‖, which is due to the discreteness of the lattice and in agreement with rigorous results. Similarities and differences between this work and earlier works on finite Ising models in the fixed-magnetization ensemble are discussed.