Equilibrium microcanonical annealing for first-order phase transitions

A framework is presented for carrying out simulations of equilibrium systems in the microcanonical ensemble using annealing in an energy ceiling. The framework encompasses an equilibrium version of simulated annealing, population annealing, and hybrid algorithms that interpolate between these extremes. These equilibrium, microcanonical annealing algorithms are applied to the thermal first-order transition in the 20-state, two-dimensional Potts model. All of these algorithms are observed to perform well at the first-order transition though for the system sizes studied here, equilibrium simulated annealing is most efficient.

Figure 1

Equilibrium configurations of the 20-state Potts model with $L=40$ and periodic boundary conditions. From left to right and then bottom to top configurations are shown in the (1) homogeneous disordered phase, (2) disordered phase with an ordered droplet, (3) stripe phase, (4) ordered phase with disordered droplet, and (5) homogeneous ordered phase.

Figure 2

Distribution ${\rho }_{\beta }\left(e\right)$ of the energy per spin, $e$ at the transition temperature calculated with population annealing for system sizes $L=$ 30, 40, 50, and 60 (from top to bottom).

Figure 3

Scatter plot of the dimensionless free energy per spin, $\beta f$ and the disordered phase energy per spin $e_d$ for each of $M=40$ runs for SA (dark blue triangles), HA (light green squares), and PA (red circles) at the transition temperature for system size $L=40$.

Figure 4

Scatter plot of the dimensionless free energy per spin, $\beta f$ and disordered phase excess, $X$ for each of $M=40$ runs for SA (dark blue triangles), HA (light green squares), and PA (red circles) at the transition temperature for $L=40$. The exact value in thermodynamic limit, $X=0$, corresponds to peak ratio $r_c=q$.

Figure 5

Integrated autocorrelation time $\tau$ in units of Monte Carlo sweeps, as a function of energy ceiling per spin $e$ for system sizes $L=30$, 40, 50, and 60, with peak heights increasing with $L$.

Figure 6

Disordered cluster size histograms, $h_c$ for ceiling energies per spin $e=-1.56$ (light green), $-1.73$ (red), and $-1.84$ (dark blue) showing the behavior of $h_c$ above, near, and below the evaporation transition energy $e_4$, respectively, for size system $L=40$. Dashed lines represent fitted power laws to the histograms for cluster sizes below a cutoff of 35.

Figure 7

The integrated autocorrelation time $\tau$ (red, left axis) is plotted against ceiling energy per spin $e$ over the entire simulation energy range $-2<e<0$ for $L=40$. The wrapping fraction ${\omega }_w$ (dark blue, right axis) is plotted from $-1.6<e<.7$ and the droplet fraction ${\omega }_c$ (light green, right axis) is plotted from $-1.85<e<-1.5$. The first two peaks in $\tau$, from right to left, coincide with jumps in ${\omega }_w$ that signify the first and second wrapping transitions, respectively. The third peak in $\tau$ coincides with a rapid drop of ${\omega }_c$ signifying the evaporation transition.