Asymptotic forms and scaling properties of the relaxation time near threshold points in spinodal-type dynamical phase transitions

We study critical properties of the relaxation time at a threshold point in switching processes between bistable states under change in external fields. In particular, we investigate the relaxation processes near the spinodal point of the infinitely long-range interaction model (the Husimi-Temperley model) by analyzing the scaling properties of the corresponding Fokker-Planck equation. We also confirm the obtained scaling relations by direct numerical solution of the original master equation, and by kinetic Monte Carlo simulation of the stochastic decay process. In particular, we study the asymptotic forms of the divergence of the relaxation time near the spinodal point and re-examine its scaling properties. We further extend the analysis to transient critical phenomena such as a threshold behavior with diverging switching time under a general external driving perturbation. This models photoexcitation processes in spin-crossover materials. In the ongoing development of nanosize fabrication, such size-dependence of switching processes should be an important issue, and the properties obtained here will be applicable to a wide range of physical processes.

Figure 1

The scaled potential $U\left(\xi \right)$ in Eq. (19). A: the metastable state. B: a point in the basin of attraction of the stable state. C: the unstable state.

Figure 2

(Color online) Comparison of the asymptotic form of the relaxation time with numerical results. Data are plotted in the coordinates $\left[\Lambda ,\text{ln}\left(N^{-1/3}\tau \right)\right]$. The solid lines denote the asymptotic form (41). The symbols are interpreted as follows: closed circles, upward triangles, downward triangles, and closed squares denote the data obtained by the master equation for $N=1000$, 2000, 4000, and $10000$, respectively. Open circles and squares denote the data obtained by the Monte Carlo method for $N=1000$ and $10000$, respectively.

Figure 3

(Color online) Ratio of the standard deviation of the lifetime, ${\sigma }_{\tau }$, to the lifetime, $\tau$, as obtained from the master equation and MC simulations. The data are shown as a scaling plot vs the scaling variable $\Lambda$. The symbols have the same interpretations as in Fig. 2.

Figure 4

Rough sketches of the scaled potential $U\left(\xi \right)$ and the corresponding Schrödinger potential $V\left(\xi \right)$.