Kinetic Flat-Histogram Simulations of Non-Equilibrium Stochastic Processes with Continuous and Discontinuous Phase Transitions

Abstract

As far as we know, there is no flat-histogram algorithm to sample the stationary distribution of non-equilibrium stochastic processes. The present work addresses this gap by introducing a generalization of the Wang-Landau algorithm, applied to non-equilibrium stochastic processes with local transitions. The main idea is to sample macroscopic states using a kinetic Monte Carlo algorithm to generate trial moves, which are accepted or rejected with a probability that depends inversely on the stationary distribution. The stationary distribution is refined through the simulation by a modification factor, leading to convergence toward the true stationary distribution. A visitation histogram is also accumulated, and the modification factor is updated when the histogram satisfies a flatness condition. The stationary distribution is obtained in the limit where the modification factor reaches a threshold value close to unity. To test the algorithm, we compare simulation results for several stochastic processes with theoretically known behavior. In addition, results from the kinetic flat-histogram algorithm are compared with standard exact stochastic simulations. We show that the kinetic flat-histogram algorithm can be applied to phase transitions in stochastic processes with bistability, which describe a wide range of phenomena such as epidemic spreading, population growth, chemical reactions, and consensus formation. With some adaptations, the kinetic flat-histogram algorithm can also be applied to stochastic models on lattices and complex networks.

Figures (13)

• Figure 1: (Color Online) Results for the Glauber model. Panel (a) shows the stationary distribution $P(M,\lambda)$ for a sequence of inverse temperatures $\lambda$. $P(M,\lambda)$ exhibits the expected behavior of a continuous phase transition, characterized by two symmetric maxima in the ferromagnetic phase ($\lambda>1$) and a single maximum in the paramagnetic phase ($\lambda<1$). Panel (b) presents a magnified view of panel (a), highlighting the critical $P(M,\lambda_c)$. The critical threshold $\lambda_c=1$ separates the two phases. Panel (c) displays the absolute value of the magnetization $m$, demonstrating good agreement between the expected value from $P(M,\lambda)$ and the time-series average from SSA simulations. Finally, panel (d) shows the Shannon entropy $S(\lambda)$, which is continuous as expected for a continuous phase transition. The line in panel (d) is only a guide to the eye.
• Figure 2: (Color Online) Results for the Glauber model. Panel (a) shows the density of states $g(E)$ obtained using the original Wang-Landau algorithm. Panel (b) presents the stationary distribution $P(M,\lambda)$ calculated with both the kinetic flat-histogram algorithm and the original Wang-Landau algorithm, for 50 inverse temperatures $\lambda$ ranging from $0.8$ to $1.3$. The results are consistent, confirming the validity of the kinetic flat-histogram algorithm for equilibrium systems. However, the agreement deteriorates when going deep into the paramagnetic phase.
• Figure 3: (Color Online) Results for the MV model. As in Fig. (\ref{['results-glauber']}), the system transitions from the ferromagnetic phase to the paramagnetic phase as $\lambda$ increases, with a threshold at $\lambda_c=0.5$. For this case, we also present statistical error bars for the order parameter $m$ obtained from the kinetic flat-histogram simulations. The error bars, calculated from 10 independent runs, are smaller than the symbol size. The line in panel (d) is shown only as a guide to the eye.
• Figure 4: (Color Online) Results for the MV model on a square lattice of size $N=50\times50$. As in Fig. (\ref{['results-glauber']}), the transition from the two-maxima regime to the single-maximum regime allows us to identify a pseudo-critical point $q^\prime = 0.081$, which is close to the maximum of the Shannon entropy. The line in panel (d) is shown only as a guide to the eye.
• Figure 5: (Color Online) Same as Fig. (\ref{['results-mv-lattice']}), except for $N=100\times100$. We note that the pseudo-critical point $q^\prime = 0.078$ approaches the critical threshold $q_c=0.075$ as the lattice size increases.