Dynamic scaling law for a first-order phase transition

Abstract

Giving a statistical-mechanical formulation of structure-function dynamics, we present a formulation of a scaling law for the first-order phase transition. The basic idea proposed, which previously appeared in a specific form, is that any function of the form $F_k(t_1,t_2,\dots )$, where $k$ is the wave number and $t_1$,$t_2$,$\dots$ are times, is scaled as $F_k(t_1,t_2,\dots )={\left[R\right(t_1\left)\right]}^{x_1}{\left[R\right(t_2\left)\right]}^{x_2}\cdots \tilde{F}\left(kR\right(t_1),kR(t_2),\dots ),$ where $R\left(t\right)\mathrm{}$ is a relevant scale length, such as a linear dimension of an average cluster size, which is found to behave as $R\left(t\right)\mathrm{}\propto t^{\frac{1}{z}}$ with $z$ being a constant. The autocorrelation function of the density fluctuation is found to obey the scaling law for the conserved system: $J_k(t,t^{\prime })={\mathrm{}\left[R\right(t\left)\right]}^{\theta }{\left[R\right(t^{\prime }\left)\right]}^{d-\theta }\tilde{J}\mathrm{}\left(kR\right(t),kR(t^{\prime }\left)\right), \theta \le \frac{d}{2}$ where $d$ is the dimensionality. For $\theta =0\mathrm{}$ this scaling law can be naturally derived on the basis of the dynamic-scaling assumption and the conservation law. However, for large $t$ the possibility of an anomalous scaling law ($\theta \ne 0$) is found. On the computer simulation for the three-dimensional spin-exchange kinetic Ising model we examine such a scaling law for the autocorrelation function. A remarkable difference in the temporal behaviors of the autocorrelation function is found. That observation strongly suggests the existence of the spinodal-like line.