Density scaling of generalized Lennard-Jones fluids in different dimensions

First we would like to thank both referees for the careful review of our paper and the positive assessment. We followed most suggestions for this minor revision before publication. In particular we agree that figure organisation could be improved, which we tried to do in this resubmission. Answers to the referees' questions are given below in the order in which they appear in the corresponding referees' reports.

Reply to the questions from Referee 1:

1) The following explanation has been included in section 2, right after the definition Eq. (3):

"The main reasons for the normalization factors of the two IPLs are: 1. The minimum of the LJ$(m',n')$ potential is located at $r = \sigma$ where the potential’s value is $−\varepsilon$, independently o $(m',n')$. This facilitates comparison between different potentials in the class and gives a simple low-density picture of particles with a constant effective diameter $\sigma$ with repulsive and attractive forces that becomes steeper and shorter ranged as exponents are increased, see Fig. 1. 2. One recovers the standard three-dimensional 12-6 LJ with a simple rescaling of $(\varepsilon, \sigma)$ in the potential LJ$(12,6)$. 3. As discussed below, for large $d$ the exponents $(m',n')$ must be scaled linearly with $d$. The normalizations are such that this scaling impacts the IPL exponents while the factors independent of $r/\sigma$ are invariant. This allows for a well-defined comparison between different dimensions."

2) Thanks for the opportunity to clarify this point, which we realize was not stated properly. The numerical results presented in the figures are computed through the correct (i.e. non further simplified) expression (13). The Eqs (14)-(16) are simplifications in order to get a very simple function describing the qualitative shape of the $\gamma(R)$ diagram in Eq.(16), notably by taking the simplifying limit $n \to \infty$. The main contributions to the integrals in Eq.(13) come from the vicinity of the saddle point $y=1$ for low enough temperature. For large enough temperature one approaches the single-IPL limit where $n$ dependence is trivial. These precisions have been added to the text below Eq.(13). A new figure (Fig.2) with four panels showing the mild dependence on $n$ has been included in the paper in Sec 3, which also contains the suggested panel. Fixing $X$, panels (a),(b) show $R$ and $\gamma$ as a function of temperature, varying $n$. Similarly, panel (c) shows the theoretical prediction for the $(R,\gamma)$ plot for several values of $n$. Panel (d) is repeated in Fig.4 for an easier comparison to the simulation results in Fig.4(c).

3) We have shrunk the caption of Fig.2 and added the former Fig.6 as a panel. However we left Fig.4 (now Fig.5) out of Fig.2 as it has a different function. Fig.2 compares the functional form obtained from $d \to\infty$ with the simulation results at fixed dimensionality and varying density. Fig.4 (now Fig.5) compares at fixed density how much the simulation results differ from the $d \to\infty$ results when changing dimensionality. We also think that unfortunately, 9 or 11 such panels in a single figure would not help readability.

4) Thanks for the suggestion, we included this reference as a footnote in the introductory paragraph of Sec. 3 : "Note that an analytical treatment via a transfer matrix method is possible in $d=1$ if one truncates the interaction range suitably (see e.g. Refs. [42,43])."

5) To address this comment we included a new appendix (B) "Estimating $R$ and $\gamma$ in computer simulations: influence of simulation length and system size".

Small correction: We cleared this issue in page 4, and the Boltzmann constant appears only in page 6.

Reply to the questions from Referee 2:

1) In general a physical interpretation for the relation between $R$ and $\gamma$ is not available, but we plan to gain more insight through the findings of this paper. For the LJ potential different phases are mapped into different regions in the $\gamma(R)$ diagram (as briefly discussed in Sec. 4). The presence of a peak in the function $\gamma(R)$ for generalized Lennard-Jones potentials is intriguingly connected with the crossing of the threshold $R=0.9$, associated with Roskilde simplicity. However the shape of the $\gamma(R)$ diagram does not seem to be universal, an example being the EXP potential discussed in [Bacher et al 2018] (added as Ref.26 in the revisited manuscript). So far very few systems have been studied with this approach and it is hard to draw general conclusions.

We have added these remarks in the introduction: the entire paragraph below Eq.(1) discusses the relevance of this diagram. Two sentences that we added specifically are: "Although a physical interpretation for such a relation is not yet available, it allows one to better understand density scaling throughout the phase diagram." "While this diagram is a priori potential dependent (see [26, Fig. 3] for the EXP potential), in the case of the LJ potential different thermodynamic phases are mapped into different regions of the diagram."

2) To address this comment we included a new appendix (B) to the manuscript "Estimating $R$ and $\gamma$ in computer simulations: influence of simulation length and system size". There we show that even for small system sizes, the estimation of $R$ and $\gamma$ is accurate. In order to simulate a pair potential with cutoff distance of $2.5\sigma$ the box has to be at least of linear size $L=5\sigma$. In 4d the simulation box needs to contain at least $N\sim 1000$ particles for density $1.5$ (the highest considered in 4d) and our simulations contain $N=2401$.

3) We agree that the presence of these state points could be confusing with respect to the focus of this paper on the isotropic liquid phase. Therefore we have chosen to remove them instead of marking them. Marking these points will also be quite problematic because in a ($R$, $\gamma$) plot LJ many ordered states have an $R$ coefficient very close to $1$, difficult to distinguish from each other and from the IPL value. Many other ordered states previously included were quite off from the analytic prediction assuming liquid behavior. The full data are in any case available on the data repository.

4) We have shrunk the caption and moved part of them in the main text that discusses them.

5) Thanks for noticing this error, it is now fixed in Fig.2 (see below).

6) Report 1 pointed that out as well, the figure is now a panel of Fig.2.

Precise location of modifications corresponding to the referees' points are given in our reply.

1) We added precisions in the text and corrected typos.
2) New figure 2 has been included, and we merged two figures.
3) Citations 42,43 and a related footnote 3 have been included
4) Numerical simulation details have been added as a new appendix B.
5) We removed ordered state points in what is now Figs. 3-4.
6) We reduced figure captions as asked by the referees.