Metastability of Synchronous and Asynchronous Dynamics

Abstract

Metastability is a ubiquitous phenomenon in nature, which interests several fields of natural sciences. Since metastability is a genuine non-equilibrium phenomenon, its description in the framework of thermodynamics and statistical mechanics has progressed slowly for a long time. Since the publication of the first seminal paper in which the metastable behavior of the mean field Curie-Weiss model was approached by means of stochastic techniques, this topic has been largely studied by the scientific community. Several papers and books have been published in which many different spin models were studied and different approaches were developed. In this review, we focus on the comparison between the metastable behavior of synchronous and asynchronous dynamics, namely, stochastic processes in discrete time in which, at each time, either all the spins or one single spin is updated. In particular, we discuss how two different stochastic implementations of the very same Hamiltonian give rise to different metastable behaviors.

Keywords: asynchronous dynamics; lattice spin systems; metastability; probabilistic cellular automata; synchronous dynamics.

Figure 1

Left: the Maxwell equal area rule is used to replace part of the van der Waals isotherm with a horizontal segment yielding the isotherm describing the liquid–vapor transition (black curve). The van der Waals isotherm $(P+a/V^2)(V-b)=RT$, was plotted for $a=1.36{\ell }^2$ atm mol${}^{-2}$ and $b=0.0319\ell$ mol${}^{-1}$, at $T={140}^{\circ }$ K, with $R=0.082058\ell$ atm mol${}^{-1}$. The equal area rule yields $33.48$ atm as the value of the pressure at which vapor and liquid coexist. Right: black lines, from the bottom to the top, are van der Waals isotherms for the same a and b at temperatures $T=130,140,150,153.{94}^{\circ }$ K. The red and the blue curves are, respectively, the liquid and the vapor branches of the spinodal curve, which meet at the critical point $V_{\mathrm{crit}}=3b=0.0957\ell$ and $P_{\mathrm{crit}}=a/\left(27b^2\right)=49.50$ atm on the critical isotherm at $T_{\mathrm{crit}}=8a/\left(27Rb\right)=153.{94}^{\circ }$ K. Note that the values of a and b that we used are the ones valid for the oxygen, the experimental value of the critical temperature is ${156}^{\circ }$ K and the vapor pressure at temperature ${140}^{\circ }$ K is $27.50$ atm [6].

Figure 2

Schematic representation of coupling constants (22).

Figure 3

Schematic representation of the height of a path.

Figure 4

Schematic representation of the optimal path between $\mathbf{d}$ and $\mathbf{u}$ for the Ising model with indication of energy differences. Black squares represent pluses.

Figure 5

Schematic representation of the optimal path between $\mathbf{c}$ and $\mathbf{u}$ for the PCA model with indication of the energy cost computed using (16) with I, the set of nearest neighbors and $J_{ij}=1$ for i and j nearest neighbors. Black squares represent pluses, white ones minuses.

Figure 6

Nucleation of the plus phase in the asynchronous Ising model with $h=0.2$ and $1/T=0.78$. From the left to the right, the configurations after 800, 1000, 1400, 2000, and 3200 Monte Carlo full lattice sweeps of the lattice are reported. Black and white spots correspond to plus and minus spins, respectively.

Figure 7

Nucleation of the plus phase in the synchronous nearest neighbor PCA model with $h=0.3$ and $1/T=0.90$. From the left to the right, the configurations after 3000, 3600, 6000, 7600, and 9600 Monte Carlo full lattice sweeps of the lattice are reported. Configurations on $2\times 2$ tiles are reported using grayscale from $-4$ (white) to $+4$ (black).

Figure 8

Magnetization and staggered magnetization versus the number of full Monte Carlo lattice sweeps. Left: magnetization of the asynchronous Ising model on the $512\times 512$ lattice with $h=0.2$ and $1/T=0.70,0.79,0.80,0.85,0.90$ (respectively from the left to the right, i.e., purple, green, blue, orange, and yellow). Center: magnetizations of the synchronous nearest neighbor reversible PCA on the $512\times 512$ lattice with $h=0.3$ and $1/T=0.85,0.90,0.95,1.00$ (respectively from the left to the right, i.e., purple, green, blue, and orange). Right: staggered magnetization for the same model as in the center panel.