A microscopic derivation of macroscopic sharp interface problems involving phase transitions

Abstract

Macroscopic free boundary problems involving phase transitions (e.g., the classical Stefan problem or its modifications) are derived in a unified way from a Hamiltonian based on a general set of microscopic interactions. A Hamiltonian of the form ℋ + ∑x,x′J(x−x′)ϕ(x)ϕ(x′) leads to differential equations as a result of Fourier transforms. Expanding the Fourier transform ofJ in powers ofq (the wave number), one can truncate the series at anarbitrary orderM, and thereby obtainMth-order differential equations. An asymptotic analysis of these equations in various scalings of the physical parameters then implies limits which are the standard macroscopic models for the dynamics of phase boundaries.