Phase growth with heat diffusion in a stochastic lattice model

When a stable phase is adjacent to a metastable phase with a planar interface, the stable phase grows. We propose a stochastic lattice model describing the phase growth accompanying heat diffusion. The model is based on an energy-conserving Potts model with a kinetic energy term defined on a two-dimensional lattice, where each site is sparse-randomly connected in one direction and local in the other direction. For this model, we calculate the stable and metastable phases exactly using statistical mechanics. Performing numerical simulations, we measure the displacement of the interface $R(t)$. We observe the scaling relation $R(t)=L_x \bar{\mathcal{R}} (Dt/L_x^2)$, where $D$ is the thermal diffusion constant and $L_x$ is the system size between the two heat baths. The scaling function $\bar{\mathcal{R}}(z)$ shows $\bar{\mathcal{R}}(z) \simeq z^{0.5}$ for $z \ll z_c$ and $\bar{\mathcal{R}}(z) \simeq z^{\alpha}$ for $z \gg z_c$, where the cross-over value $z_c$ and exponent $\alpha$ depend on the temperatures of the baths, and $0.5\le\alpha\le 1$. We then confirm that a deterministic phase-field model exhibits the same scaling relation. Moreover, numerical simulations of the phase-field model show that the cross-over value $\bar{\mathcal{R}}(z_c)$ approaches zero when the stable phase becomes neutral.