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The Droplet Formation-Dissolution Transition in Different Ensembles: Finite-Size Scaling from Two Perspectives

by Franz Paul Spitzner, Johannes Zierenberg, Wolfhard Janke

Submission summary

As Contributors: Johannes Zierenberg · Franz Paul Spitzner · Wolfhard Janke
Arxiv Link:
Date submitted: 2018-07-03
Submitted by: Spitzner, Franz Paul
Submitted to: SciPost Physics
Domain(s): Computational
Subject area: Statistical and Soft Matter Physics


The formation and dissolution of a droplet is an important mechanism related to various nucleation phenomena. Here, we address the droplet formation-dissolution transition in a two-dimensional Lennard-Jones gas to demonstrate a consistent finite-size scaling approach from two perspectives using orthogonal control parameters. For the canonical ensemble, this means that we fix the temperature while varying the density and vice versa. Using specialised parallel multicanonical methods for both cases, we confirm analytical predictions at fixed temperature (rigorously only proven for lattice systems) and corresponding scaling predictions from expansions at fixed density. Importantly, our methodological approach provides us with reference quantities from the grand canonical ensemble that enter the analytical predictions. Our orthogonal finite-size scaling setup can be exploited for theoretical and experimental investigations of general nucleation phenomena - if one identifies the corresponding reference ensemble and adapts the theory accordingly. In this case, our numerical approach can be readily translated to the corresponding ensembles and thereby proves very useful for numerical studies of equilibrium droplet formation, in general.

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Anonymous Report 2 on 2018-8-30


1. In-depth analysis of evaporation-condensation transition for the 2d Lennard-Jones system.
2. Quality of simulation data.




The manuscript “The Droplet Formation-Dissolution Transition in Different Ensemble: Finite-Size Scaling from Two Perspectives“ by Spitzner et al investigates the system-size dependent evaporation-condensation transition of a single droplet from a homogeneous phase in the two dimensional Lennard-Jones system. The authors take great care to analyze finite-size dependencies of a number of quantities to determine the exact location of the transition points. To this extent they employ state-of-the art parallelized MUCA Monte Carlo simulations in the canonical and grandcanonical ensemble and approach the problem by changing density while keeping temperature fixed and vice versa. Results are compared to theoretical predictions.

The paper is well-written and a pleasure to read and I fully recommend publication of this manuscript in its present form.

Requested changes

If they like, the authors may consider the following minor remarks in their final version:

1. I’d be interested in the computational effort involved in this endeavor. Maybe the authors can comment on this point.

2. The finite-size dependence of the isothermal compressibility (Fig.5c) looks rather peculiar for the smallest systems under investigation considering the small statistical errors.

3. Typo: page 3: “choosing any” instead of “anyone”.

  • validity: top
  • significance: good
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: excellent

Anonymous Report 1 on 2018-7-24


1) Comprehensive finite-size scaling analysis of equilibrium droplet formation
2) Solid numerical work
3) Beautifully presented


1) Generalizing the approach beyond the simple case considered here may not be straightforward


The authors report extensive multicanonical Monte Carlo simulations of droplet formation in a 2d Lennard-Jones gas. They focus on the equilibrium properties of the largest droplet using either density or temperature as control parameter and perform a careful finite-size scaling analysis. The results are compared to the phenomenological theory of Biskup et al. (Ref.10), which is found to describe well the droplet scaling properties for large system sizes. The authors argue that multicanonical simulations combined with their finite-size scaling setup might prove useful in the study of more general nucleation-related problems.

The paper is extremely well written and the results are beautifully presented. The figures are remarkable for clarity and aesthetics. Scientifically, the work is solid and has the merit of discussing finite-size scaling of droplet formation within a coherent framework that treats density and temperature on similar grounds. Most previous works only focused on either one or the other control parameter. However, it is difficult for me to judge on the usefulness of the present approach beyond the specific application studied here (a simple 2d LJ gas). I only have a few remarks that the authors may want to take into account before publication.

Requested changes

1) The physical origin of the deviations from the theoretical finite-size scaling (figs. 8-11) remains unclear (higher-order corrections or other effects?). In particular, the possible artifacts due to periodic boundary conditions are not explicitly discussed. Can the authors give some further insight into this? It would also be useful to show selected samples showing the typical shape of the largest droplets for different system sizes.

2) It would be useful to provide some details on the length of the multicanonical simulations (MC steps, wall time). How long does it take to ensure an equilibrium sampling of the droplet properties at the largest system sizes?

3) At the end of p. 9, the authors mention a potential flaw in grand-canonical MC simulations, which can be fixed by inserting particles at random not only in physical space but also in the allocated area of computer memory. How bad does the simulation fail if this is not enforced? Subtle violations of detailed balance may be difficult to detect.

  • validity: top
  • significance: good
  • originality: good
  • clarity: top
  • formatting: perfect
  • grammar: perfect

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