Low-Temperature Nucleation in a Kinetic Ising Model with Soft Stochastic Dynamics

We study low-temperature nucleation in kinetic Ising models by analytical and simulational methods, confirming the general result for the average metastable lifetime, $⟨\tau ⟩=A\exp (\beta \Gamma )$ ($\beta =1/k_{\mathrm{B}}T$) [E. Jordão Neves and R. H. Schonmann, Commun. Math. Phys. 137, 209 (1991)]. Contrary to common belief, we find that both $A$ and $\Gamma$ depend significantly on the stochastic dynamic. In particular, for a “soft” dynamic, in which the effects of the interactions and the applied field factorize in the transition rates, $\Gamma$ does not simply equal the energy barrier against nucleation, as it does for the standard Glauber dynamic, which does not have this factorization property.

Figure 1

The states in the one-step Markov chain of clusters of $i=0,\dots ,4$ overturned spins, used to calculate the metastable lifetime $⟨\tau ⟩$ analytically by hand. The right-pointing arrows give the growth rates $g_{i-1}$, and the left-pointing arrows give the shrinkage rates $s_i$ for $i=1$, 2, and 3. The energies $E_i$ (relative to the metastable state, $i=0$) are given at the top of the figure for even $i$ and at the bottom for odd $i$.

Figure 2

Analytical and simulated results for the soft and hard Glauber dynamics for (a) $\Gamma$ and (b) $A$. In the legends, “one-step analytical” refers to the one-step Markov-chain approximation, “$12/9$ analytical” to the computer-aided AMC calculations with 12 transient and 9 absorbing configurations, and “analytical” to results that are identical for all the analytical calculations. The results only differ for $|H|=2$. The inset in (a) shows analytical (lines) and MC (data points) results for $T\ln ⟨\tau ⟩$ vs $T$ for the soft dynamic, from which $\Gamma$ and $A$ are obtained.