Free-Energy Barrier at Droplet Condensation

Abstract

We discuss several aspects of a Monte Carlo computer simulation study of the condensation of macroscopic droplets emerging in the two-dimensional Ising lattice-gas model. By varying the particle density at fixed temperature we monitor the droplet formation in detail and compare our results with recent analytical predictions in the infinite-volume limit. Three different lattice discretizations are considered which are found to yield very similar results when presented in properly scaled variables. Particular emphasis is placed on the free-energy barrier associated with droplet formation and its implication for multimagnetical simulations. §1. IntroductionThe precise mechanism for the formation of a first large droplet in condensing systems is one of the fundamental problems in statistical physics. Early studies date back to the seminal analytical work by Fisher 1) and computer simulations by Binder, Kalos and Furukawa. 2), 3) Over the years this problem has been taken up and further advanced in several numerical studies. 4) Particularly noteworthy is the careful analysis of Hager and Neuhaus, 5) which stimulated new theoretical 6)-8) and numerical 9)-12) work. Here, we follow the mathematical considerations of Biskup et al., 6), 7) which are based on an equilibrium framework and result in largely model independent scaling predictions for the condensation process in the infinite-volume limit.One purpose of the present study is to investigate by how much these asymptotic predictions are affected by finite-size effects. The second goal is to test the degree of universality suggested by the analytical treatment. Finally, we also present new results on the free-energy barrier associated with the droplet formation. After briefly recalling the Ising model and its lattice-gas interpretation in §2, the theoretical scaling predictions are summarized in §3. In §4 the results of our quite extensive Monte Carlo computer simulations are discussed, and in §5 we close with a summary and a brief outlook to future work. §2. ModelThroughout this paper we consider the Ising model with Hamiltonian1) where i, j denotes short-range interactions and the spins σ i , i = 1, . . . , V , live on the sites of a regular lattice with periodic boundary conditions to be specified below. Denoting in an arbitrary spin configuration the number of spins pointing up (down) at NERL on June 14, 2015 http://ptps.oxfordjournals.org/ Downloaded from

Thermodynamics of amyloid formation and the role of intersheet interactions

“…[53][54][55][56][57][58][59][60][61][62][63] Indeed, at this transition, mid-size droplets are suppressed and the energy distribution is bimodal. Furthermore, the specific latent heat, which we find to decrease with system size (Fig.…”

“…For finite-volume liquid-vapor systems at phase coexistence, the formation of droplets due to a fixed particle excess above the ambient gas concentration has been extensively investigated, [53][54][55][56][57][58][59][60][61][62][63] often by mapping to the Ising model at fixed magnetization. A sharp transition has been shown to occur, below which the particle excess can be accommo- latent heat scale as V 3/4 .…”

Finite-size scaling of Monte Carlo simulations for the fcc Ising antiferromagnet: Effects of the low-temperature phase degeneracy

“…2. These structures probably correspond to remaining free-energy barriers in other "directions" than the energy, similar to the additional free-energy barriers originating from the dropletformation 63,64 and the droplet-strip 65 transitions in the two-dimensional Ising model.…”

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“…Biskup et al [4] and Binder [5] made quantitative analysis of the equilibrium droplet condensation transition, which is supported by a number of numerical studies of Lenard-Jones particles [7][8][9] and lattice gas [10][11][12][13], where particle-densitydriven transitions at given temperature are investigated. In this paper we consider a simpler situation, an internal-energydriven transition in a microcanonical ensemble [14,15], where density (magnetization) is not explicitly taken account of.…”

“…Here we derive the condensation rate of a droplet for a energy-driven phase transition by translating the result for a magnetization-driven transition [4,13]. We consider a canonical ensemble at the bistable point β = β c and its energy distribution function around a peak at E − c .…”