Exploring different regimes in finite-size scaling of the droplet condensation-evaporation transition

We present a finite-size scaling analysis of the droplet condensation-evaporation transition of a lattice gas (in two and three dimensions) and a Lennard-Jones gas (in three dimensions) at fixed density. Parallel multicanonical simulations allow sampling of the required system sizes with precise equilibrium estimates. In the limit of large systems, we verify the theoretical leading-order scaling prediction for both the transition temperature and the finite-size rounding. In addition, we present an emerging intermediate scaling regime, consistent in all considered cases and with similar recent observations for polymer aggregation. While the intermediate regime locally may show a different effective scaling, we show that it is a gradual crossover to the large-system scaling behavior by including empirical higher-order corrections. This implies that care has to be taken when considering scaling ranges, possibly leading to completely wrong predictions for the thermodynamic limit. In this study, we consider a crossing of the phase boundary orthogonal to the usual fixed temperature studies. We show that this is an equivalent approach and, under certain conditions, may show smaller finite-size corrections.

Figure 1

Sketch of the infinite system-size transition (solid line) together with the finite-size scaling directions in either the density $(T$ fixed) or temperature $(\rho$ fixed). The transition line may be understood as $T_0\left(\rho \right)$ or similarly ${\rho }_0\left(T\right)$. At the crossing point of both schemes a finite system of size $V$ may be constructed for which $(\rho ,T)={({\rho }_c,T_c)}_V$ corresponds to the finite-size condensation-evaporation transition.

Figure 2

Sketch of the particle gas models under investigation: The discrete lattice gas model (left) in two and three dimensions and the continuous Lennard-Jones gas model with domain decomposition (right) in three dimensions.

Figure 3

Example of equilibrium droplets at coexistence from Metropolis simulations for the 3D lattice gas with $N=10000$ (left) and the 3D Lennard-Jones gas with $N=2000$ (right).

Figure 4

Canonical estimates of the fraction $\eta$ of particles in the largest droplet for a 3D Lennard-Jones gas at fixed density $\rho ={10}^{-2}$ (left). Corresponding probability distribution of $\eta$ at $T_c$ for $N=512$ (right).

Figure 5

Canonical estimates of the specific heat from parallel multicanonical simulations of the 3D Lennard-Jones gas at fixed density $\rho ={10}^{-2}$.

Figure 6

Finite-size scaling of the condensation-evaporation transition temperature for the 2D lattice, 3D lattice, and 3D Lennard-Jones gas from top to bottom. The lattice gas is compared to theoretical predictions (see text and also Sec. 2a).

Figure 7

Example of the transition rounding for the specific heat of the 3D Lennard-Jones gas. The $x$ and $y$ axes are rescaled according to the leading-order scaling behavior.

Figure 8

Finite-size rounding of the droplet condensation-evaporation transition for the 2D lattice, 3D lattice, and 3D Lennard-Jones gas from top to bottom. The results are presented for $\rho =0.01$ but are consistent for different densities.

Figure 9

Scaling of the particle fraction in the largest droplet $\eta$ for the 2D lattice, 3D lattice, and 3D Lennard-Jones gas from top to bottom. For large systems, $\eta$ should scale as $N^{-1/(d+1)}$. For small system sizes, the majority of particles is in the droplet $\eta =\mathcal{O}\left(1\right)$ (horizontal dashed line). The Metropolis data are obtained in the droplet phase at the finite-size transition temperature as extrapolated from the linear fit in Fig. 6.