  We propose a phase prediction method for the pattern formation in the
uniaxial two-dimensional kinetic Ising model with the dipole-dipole
interactions under the time-dependent Ginzburg-Landau dynamics. Taking the
effects of the material thickness into account by assuming the uniformness
along the magnetization axis, the model corresponds to thin magnetic materials
with long-range repulsive interactions. We propose a new theoretical basis to
understand the effects of the material parameters on the formation of the
magnetic domain patterns in terms of the equation of balance governing the
balance between the linear- and nonlinear forces in the equilibrium state.
Based on this theoretical basis, we propose a new method to predict the phase
in the equilibrium state reached after the time-evolution under the dynamics
with a given set of parameters, by approximating the third-order term using the
restricted phase-space approximation [R. Anzaki, K. Fukushima, Y. Hidaka, and
T. Oka, Ann. Phys. 353, 107 (2015)] for the $\phi^4$-models. Although the
proposed method does not have the perfect concordance with the actual numerical
results, it has no arbitrary parameters and functions to tune the prediction.
In other words, it is a method with no a priori assumptions on domain patterns.

Statistical Mechanics