Scaling Behavior in Phase-Ordering Dynamics of the Allen–Cahn Equation

We investigate phase-ordering dynamics in a two-dimensional system governed by the Allen–Cahn equation through numerical simulations. Starting from random initial conditions, we analyze the evolution of domain structures and characterize the coarsening process via the time-dependent length scale L(t).

Our results show clear evidence of dynamical scaling, with the characteristic domain size following a power-law growth L(t) ~ t^β. A log–log analysis yields an exponent β ≈ 0.58, in reasonable agreement with theoretical predictions for curvature-driven coarsening, with deviations attributable to finite-size effects and numerical discretization.

We further present spatial configurations of the scalar field, illustrating domain morphology and interface dynamics during the evolution. The numerical results provide a consistent picture of phase-ordering kinetics and support the universality of scaling behavior in non-conserved order parameter systems.

These findings contribute to the understanding of coarsening phenomena and offer a computational framework for studying non-equilibrium statistical physics in systems described by reaction–diffusion dynamics.