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Summary

The manuscript addresses a well-defined and practically important question for anyone running reaction-ensemble Monte Carlo (RxMC) simulations: how much does the closed-system boundary condition of the Rx ensemble bias the average composition relative to the "true" open, grand-canonical (GC) description of a microscopic subvolume of a macroscopic reactive system? The authors tackle this by working out the ideal (non-interacting) partition functions for both ensembles explicitly, which lets them compute finite-size deviations δ_i^N to arbitrary numerical precision without running any actual Monte Carlo simulations. They validate the approach against the known analytical result for grand-canonical pair insertion (Begun et al.), then apply it to three cases of increasing complexity: a single isomerization reaction (shown analytically to have zero finite-size effect), decomposition reactions (symmetric and asymmetric), and acid-base equilibria, including a comparison against two previously published simulation protocols (Nová et al. 2017; Rathee et al. 2019). The central practical payoff is a revision of the authors' own earlier rule of thumb (Landsgesell et al. 2019, Soft Matter) that each species needs >10 particles to avoid finite-size effects — they show this was often needlessly conservative, and provide an open-source tool (rxmcpy) to let users check a given setup a priori.

This is a methodologically clean and useful contribution. The derivation is careful, the validation against Begun et al. is a good sanity check, and the conclusion that "previous estimates were overestimating the finite-size effects" is exactly the kind of self-correcting, quantitative result the simulation community benefits from. My comments below are aimed at strengthening the presentation and closing a few gaps before this moves toward journal submission.

Major comments

1. Generalization from ideal to interacting systems is asserted rather than tested. The whole analysis is restricted to non-interacting systems, which is what makes the exact/high-precision treatment possible. The authors justify extrapolating these conclusions to real (interacting) RxMC simulations by citing Smith & Qi (ref. 26) for the claim that "ideal contributions are dominating the reaction equilibrium in many situations." This is a reasonable starting hypothesis, but the paper would be considerably strengthened by at least one worked example comparing the ideal-system prediction of δ_i^N against an actual interacting-particle RxMC simulation (e.g., a simple charged or Lennard-Jones system where the acid-base or decomposition equilibrium is perturbed by interactions). Without this, a reader is left to trust the extrapolation rather than see it demonstrated, which matters a lot given that the paper's main use case (weak polyelectrolytes, acid-base equilibria under screening) is precisely where interactions are non-negligible.

2. Practical guidance for the low-⟨N⟩, high-precision-needed regime is incomplete. Section 4.5 correctly identifies that the interesting regime — where finite-size effects turn out to be small even though ⟨N⟩ ≪ 1 — is exactly the regime where standard Metropolis RxMC sampling needs an impractically large number of uncorrelated samples (∝ 1/(r²⟨N⟩)) to resolve ⟨N⟩ accurately. The authors point to the free-energy approach of Smith & Qi as a possible remedy but do not demonstrate it or quantify the actual sampling cost in any of their example systems. Since this efficiency problem seems to partially undercut the practical benefit of "you can use smaller systems," a bit more discussion (or a demonstration) of how one would actually obtain converged results in this regime would help readers translate the theoretical result into a practical simulation protocol.

3. Reproducibility/software details. The rxmcpy tool (https://gitlab.com/kosovan1/rxmcpy) is a valuable contribution and I'd encourage the authors to (a) tag/archive the exact version used to generate the paper's figures (e.g., via Zenodo, with a citable DOI, as is now standard practice for companion software), and (b) briefly describe the numerical convergence behavior of the adaptive summation scheme (Section 8) for cases with more than 2–3 coupled reactions, since the hyper-rectangle expansion algorithm's cost presumably grows quickly with the number of irreducible reactions. A short note on scaling/limitations here would help potential users decide whether the tool is applicable to their (possibly larger) reaction networks.

Minor comments

• The preprint appears to contain a leftover empty subsection header "7. RESULTS / 7.1. Acid-Base Equilibrium" right before Section 8, with no content — likely a template/versioning artifact that should be removed before further submission.

• In Section 4.3, the paragraph beginning "In this example it was sufficient to average over the two possible states..." is duplicated verbatim (once ending "...as described in the next section," and again in near-identical form). Worth trimming in the next revision.

• The document header shows "(Dated: 2020-02-26; Git commit: 112ba7c)," which predates the 2023 posting date by several years — presumably a stale template field, but worth double-checking before journal submission so the metadata is consistent.

• Minor author-order inconsistency: the ChemRxiv cover/abstract page lists the author order as Hebbeker, Blanco, Uhlík, Kosovan, while the manuscript body lists Hebbeker, Uhlík, Blanco, Kosovan. Probably harmless, but worth reconciling for citation consistency.

• A few typos to catch in copyediting: "nesesary" (Section 2.3), "an general manner," "an integer number νi < o'i" (Section 9), "witch corresponds" (Section 8), "nonmonotonic" spacing inconsistencies, and "nonmonotonic trend. They are the biggest" could be tightened.

• Figure 3's relationship K^{1/2}_AB = 0.25 K^{1/2}_AA is stated without derivation; a one-line explanation (or pointer to where it follows from the stoichiometry) would help readers not want to re-derive it themselves.

Questions for the authors

1. For the acid-base "titration mode" case, you note finite-size effects peak at pH_GC − pK ≈ ±2.6 and interpret the pH_Rx + pOH_Rx > pK_w deviation as a practical finite-size diagnostic. Could this diagnostic (comparing the simulated pH + pOH against pK_w) be turned into an online, on-the-fly convergence/validity check within a running RxMC simulation, rather than only as an a priori planning tool? That seems like it could be a nice practical extension.

2. In the reservoir-coupled acid-base case (Section 4.4b, following Rathee et al.), you show finite-size effects stay below 1% even at pH 7, where Rathee et al. reported needing infeasibly large systems. Is the discrepancy purely due to the reformulation of the reaction network (introducing the Na⁺/H⁺ and Cl⁻/OH⁻ pairings), or does it also reflect a difference in how "finite-size effect" was defined/estimated in the original study? A short explicit comparison of the two error estimates would make the correction more transparent to readers of Rathee et al.'s original paper.

3. Do you have a sense of how the root-finding procedure for the reservoir equilibrium composition (Section 3.1) scales/behaves for reaction networks with many (>5) coupled, possibly stiff, equilibria — e.g., does the Villars-Cruise-Smith-like fallback reliably converge, or are there known failure modes users of rxmcpy should be aware of?

Overall assessment

The core theoretical contribution — an efficient, high-precision way to a priori estimate finite-size effects in the reaction ensemble for arbitrary (linearly independent) sets of ideal reactions — is solid, well-validated against known analytical limits, and practically useful. The main things I'd want to see addressed before this is submitted to a journal are (1) at least one demonstration that the ideal-system predictions hold up against real interacting-particle simulations, and (2) a more concrete discussion of the sampling-efficiency trade-off in the very-low-⟨N⟩ regime that the paper's own conclusions push users toward. The manuscript-level issues (duplicated paragraph, stray heading, stale date field) are minor but should be cleaned up.

Competing interests: I have no competing interests to declare.

Competing interests

The author declares that they have no competing interests.

Use of Artificial Intelligence (AI)

The author declares that they did not use generative AI to come up with new ideas for their review.