Evaporation-condensation transition of the two-dimensional Potts model in the microcanonical ensemble

The evaporation-condensation transition of the Potts model on a square lattice is numerically investigated by the Wang-Landau sampling method. An intrinsically system-size-dependent discrete transition between supersaturation state and phase-separation state is observed in the microcanonical ensemble by changing constrained internal energy. We calculate the microcanonical temperature, as a derivative of microcanonical entropy, and condensation ratio, and perform a finite-size scaling of them to indicate the clear tendency of numerical data to converge to the infinite-size limit predicted by phenomenological theory for the isotherm lattice gas model.

Figure 1

Typical spin configuration of the system with $q=8$ (top) and $q=21$ (bottom) for $L=256$. The different spin states are denoted by different gray scales. The energy is $E/N=0.3750,0.4500,0.5250$, and $0.7125$ (top) and $E/N=0.1500,0.2250,0.2625$, and $0.6750$ (bottom) from left to right. The black region is occupied by spins of the most major state, and the dappled gray region is an assemblage of tiny domains with various states.

Figure 2

Schematic diagram of dividing the energy region into eight threads and spin configuration exchange.

Figure 3

The standard deviation of microcanonical temperature among samples with different realizations of random number. The data without exchange and exchanges in every 16 and 4 MCSs per spin are plotted together. The simulation condition is $q=8$, $L=512$, 16 samples, and the modification constant is down to $\ln g\left(E\right)=2^{-20}$. The energy region, $0.36<E<0.56$, is distributed to 64 threads, and each thread handles the energy points about $666\times 2$. Smoothing is done by averaging for 128 energy points after calculating deviation. For comparison, the difference between the transition temperature and the spinodal temperature, a characteristic scale of our interest, is about 0.001 for this size, as shown in Fig. 4a.

Figure 4

Inverse temperature as a function of internal energy for (a) $q=8$ and (b) $q=21$. Smoothing is performed by averaging over the range $\sqrt{N}/4$ of $E$. The horizontal line indicates $\beta =\ln (1+1/\sqrt{q})$. The vertical lines indicate $E/N={\varepsilon }_c^{-}$ (left) and ${\varepsilon }_c^{+}$ (right). The arrows indicate the direction that $L$ becomes larger.

Figure 5

Finite-size scaling results of the excess temperature. (a) $q=8$ and (b) $q=21$. The arrows indicate the direction that $L$ becomes larger. The thick gray curve is a guide for the eye. The vertical line in the bottom panel indicates $E-E_c^{-}=1.507L^{4/3}$ as well as in Fig. 8.

Figure 6

Probability distribution function of the energy density of subsystems for $q=21$, $L=512$. Each curve corresponds to $E/N$ $=$ 0.12–0.52 (0.02 step).

Figure 7

Internal energy dependence of the volume fraction of the high-energy domain. The insets are the blowups of the edges of the coexisting phase, where the arrows indicate the direction that $L$ becomes larger.

Figure 8

Finite-size scaling result of the volume fraction of a condensed droplet. The data for $E/N<0.40$ are used. The solid gray curve indicates theoretical prediction, $a|E-E_c^{-}{|}^{3/2}/L^2=1/4\sqrt{\lambda }(1-\lambda )$ for $\lambda >2/3$, where we set $a=0.44$. The vertical line indicates $E-E_c^{-}=1.507L^{4/3}$ as well as in Fig. 5.