A nonlocal anisotropic model for phase transitions

Abstract

where u is a scalar density function on a domain of R and takes values in [−1, 1], W is a positive double-well potential which vanishes at ±1, and J is a positive, possibly anisotropic, interaction potential which vanishes at infinity (see paragraph 1.2 for precise definitions). The scalar function u represents the macroscopic density profile of a system which has two equilibrium pure phases described by the profiles u ≡ +1 and u ≡ −1. The integral ∫ W (u) at the right side of (1.1) forces a minimizer of F to take values close to +1 and −1 (phase separation), while the double integral represents an interaction energy which penalizes the spatial inhomogeneity of the system (surface tension). In equilibrium Statistical Mechanics functionals of the form (1.1) arise as free energies of continuum limits of Ising spin systems on lattices; in this setting u plays the role of a macroscopic magnetization density and J is a ferromagnetic Kac potential (see for instance [2] and references therein). We underline the analogy with the more familiar gradient theory for phase transition proposed in [9], where the free energy of the system is of the form