  This paper discusses a relation between the re-initialization equation of the
level-set functions derived by Wac{\l}awczyk [J.Comp.Phys., 299, (2015)] and
the condition for the phase equilibrium provided by the stationary solution to
the modified Allen-Cahn equation [Acta Metall., 27, (1979)]. As a consequence,
the statistical model of the non-flat interface in the state of phase
equilibrium is postulated. This new physical model of the non-flat interface is
introduced based on the statistical picture of the sharp interface disturbed by
the field of stochastic forces, it yields the relation between the sharp and
diffusive interface models. Furthermore, the new techniques required for the
accurate solution of the model equations are proposed. First it is shown, the
constrained interpolation improves re-initialization of the level-set functions
as it avoids oscillatory numerical errors typical for the second-order accurate
interpolation schemes. Next, the new semi-analytical, second order accurate
Lagrangian scheme is put forward to integrate the advection equation in time
avoiding interface curvature oscillations introduced by the second-order
accurate flux limiters. These techniques provide means to obtain complete,
second-order convergence during advection and re-initialization of the
interface in the state of phase equilibrium.
