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Computing Shear Viscosities from Molecular Dynamics Simulation: Comparing the OrthoBoXY Approach with the Green-Kubo Method

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Shear viscosities from the OrthoBoXY approach agree with Green-Kubo results for 15 molecular liquids, and the block length can be shortened by factors of 1/8 to 1/4 for viscous systems, cutting computational cost by up to 24-fold.

desk verdict Genuinely useful OrthoBoXY validation with reproducible data and a practical recipe refinement, but the 'very good agreement' headline overstates the high-viscosity end and the 1/8 block-length scaling rests on a single viscous system. read the letter →

arxiv 2608.05845 v1 pith:43NDTLFE submitted 2026-08-06 cond-mat.stat-mech physics.chem-ph

Marcel Brandt , Ralf Ludwig , Dietmar Paschek This is my paper · ORCID
classification cond-mat.stat-mechphysics.chem-ph MSC 82B8082D15 PACS 02.70.Ns66.20.-d
keywords shearviscositymoleculardynamicsOrthoBoXYGreen-Kubomethodself-diffusioncoefficientfinite-sizeeffectsblocklengthrecipeOPLSforcefield
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that the OrthoBoXY approach, which extracts the shear viscosity from the direction-dependent finite-size correction to self-diffusion in a single molecular dynamics run, produces viscosities that agree with the standard Green-Kubo method for 15 neat molecular liquids covering three orders of magnitude in viscosity. It shows that neither the viscosity nor its statistical error depends on the simulated system size down to 250 molecules, because the system-size weighting in the OrthoBoXY equation cancels the size dependence of the diffusion-coefficient errors. It then argues that the previously recommended block length can be shortened by a factor of 1/8 for highly viscous systems and by 1/4 for moderately viscous ones, while fluid systems should keep the full block length. Combined with a 250-molecule system, this reduces the computational cost by up to 24-fold without sacrificing accuracy. A sympathetic reader would care because these recommendations turn a costly transport-coefficient calculation into a cheaper routine one with a clear accuracy target.

What carries the argument

The load-bearing object is the OrthoBoXY geometry: an orthorhombic simulation box with $L_z/L_x = L_z/L_y = 2.7933\ldots$, for which the direction-dependent Yeh-Hummer correction to self-diffusion vanishes in the $x$ and $y$ directions ($\zeta_{xx}=\zeta_{yy}=0$). This makes $D_0 = (D_{PBC,xx}+D_{PBC,yy})/2$ a direct estimate of the true self-diffusion coefficient, while the residual $z$-direction correction gives the viscosity through $\eta = k_B T \zeta_{zz} / (6\pi L_z(D_0 - D_{PBC,zz}))$ with $\zeta_{zz}=8.1711\ldots$. The statistical argument is carried by the error-propagation formula $\hat{\sigma}_\eta/\eta = \sqrt{\hat{\sigma}_{D_0}^2+\hat{\sigma}_{D_{PBC,zz}}^2}/|D_0-D_{PBC,zz}|$, whose denominator scales as $N^{-1/3}$ and cancels the $N^{-1/3}$ scaling of the numerator, making the viscosity error nearly independent of system size. The block-length recipe $\tau_{\mathrm{block}} = (8.5s)^2/(6D)$, where $s=(V/N)^{1/3}$, is the target of the paper's cost-saving refinement.

What would settle it

Run OrthoBoXY at $\tau_{\mathrm{block}}/8$ for a viscous liquid outside the four test systems, for instance an ionic liquid or polyol with $\tau_{\mathrm{block}}>100$ ns, and compare with the full-length recipe and with Green-Kubo; a disagreement beyond the standard error would disprove the universal scaling recommendation.

Watch

Extended reading notes

Core claim

The central claim is that the OrthoBoXY method is a valid and cheaper alternative to Green-Kubo for computing shear viscosities of molecular liquids. For 15 OPLS-modeled liquids with viscosities from about 0.1 to 230 mPa s, the OrthoBoXY results agree with Green-Kubo values within the reported errors, with the largest deviations appearing for the most viscous systems where the two methods diverge noticeably and it is unclear which method is at fault. The paper also claims that finite-size effects are absent down to 250 molecules, and that the standard error of the viscosity is nearly independent of system size because the $N^{-1/3}$ growth of the weighting factor $|D_0-D_{PBC,zz}|$ in the error formula compensates the $N^{-1/3}$ scaling of the self-diffusion coefficient errors. On this basis it recommends running small systems with long trajectories, and shortens the block-length recipe to $ au_{\mathrm{block}}/8$ for $ au_{\mathrm{block}}>100$ ns and $ au_{\mathrm{block}}/4$ for $1<\tau_{\mathrm{block}}<100$ ns, while keeping the full length for $ au_{\mathrm{block}}<1$ ns.

Load-bearing premise

The shortening of $\tau_{\mathrm{block}}$ by factors of 1/8 and 1/4 is validated on only four liquids, with formaldehyde already failing to give a constant viscosity at short blocks, and the paper assumes these results transfer to other molecular liquids in the same viscosity ranges.

Editorial extensions

If this is right

  • OrthoBoXY can serve as a drop-in alternative to Green-Kubo for viscosity screens, giving comparable values from a single equilibrium run.
  • Simulations with 250 molecules can replace 1000-molecule boxes without introducing finite-size bias, so computing time can be invested in longer trajectories instead.
  • The standard error of the viscosity being nearly independent of system size means that increasing box size is not an effective way to improve precision; longer runs are.
  • The refined block-length rules let users cut $\tau_{\mathrm{block}}$ by 1/8 for viscous liquids and 1/4 for moderately viscous ones, which with the smaller box yields up to 24-fold lower cost.
  • The averaging variant that averages both $D_0$ and $D_{PBC,zz}$ first avoids the skewed distributions and occasional negative viscosities produced by block-wise averaging.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The 1/8 and 1/4 scaling factors are drawn from only four liquids and should be re-validated on other viscous and moderately viscous systems before being adopted as a universal recipe.
  • If the compensation effect holds generally, the optimal allocation of a fixed CPU budget between block length and number of blocks may be derivable from the reported error formula, allowing simulation planning without pilot runs.
  • Because OrthoBoXY yields both the true self-diffusion coefficient and the viscosity from one run, combining the shortened blocks with 250-molecule boxes would make large-scale viscosity screening of force-field libraries much cheaper than Green-Kubo-based approaches.
  • For highly fluid systems the paper advises against shortening; a direct test of where the breakdown begins, for example by scanning $\tau_{\mathrm{block}}$ continuously for a fluid like acetone, would sharpen the boundary between safe and unsafe scaling.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The manuscript reports equilibrium molecular dynamics simulations of 15 OPLS-AA molecular liquids in the OrthoBoXY geometry and compares the resulting shear viscosities with Green-Kubo values taken from Smith and Sega. It also examines three averaging variants for the OrthoBoXY equation, studies system-size effects (250–1000 molecules) for three liquids, and investigates block-length scaling for four liquids. On this basis it recommends reducing the OrthoBoXY block length by factors of 1/8 for viscous and 1/4 for moderately viscous systems, claiming up to a 24-fold reduction in computational cost without loss of accuracy. The paper includes a useful practical discussion of numerical pitfalls and provides data availability links for topologies and input files.

Significance. If the central agreement claim held uniformly, the paper would be a valuable practical validation of the OrthoBoXY method and a useful resource for practitioners, especially the finite-size analysis and the averaging-variant discussion. The compensation argument connecting self-diffusion error scaling to the system-size independence of the viscosity error is interesting and testable. However, the largest deviations between OrthoBoXY and Green-Kubo occur for precisely the systems where the recommended block-length reduction is applied, and the block-length recommendation rests on a small and partly non-constant sample. These issues limit the significance of the cost-saving claim as it stands, but they are addressable with additional analysis or with appropriately qualified conclusions.

major comments (3)
  1. [IV.A, Table II] The abstract and Section IV.A state that OrthoBoXY and Green-Kubo viscosities 'agree very well', but this is not supported for the three most viscous systems. In Table II, glycerol is 230±24 vs 160±2 mPa s (OrthoBoXY 44% higher), 3-methylphenol is 70±8 vs 40±2 mPa s (75% higher), and acetamide is 10.0±0.8 vs 15±1 mPa s (33% lower). Each deviation is several times the reported standard error. These are also the systems used to justify the 1/8 block-length scaling in Section IV.D, so the validation is weakest exactly where the recommended cost-saving measure is applied. The authors should either perform Green-Kubo calculations under the same protocol (thermostat, run length, system size) for these systems, or substantially temper the agreement claim and discuss possible systematic bias in this regime.
  2. [IV.D, Fig. 5] The recommendation to scale tau_block by 1/8 for viscous and 1/4 for moderately viscous systems is based on only four liquids, and formaldehyde, the most fluid of the four, does not show constant viscosities at shortened blocks. It is therefore not established that the scaling transfers to other liquids in the stated tau_block ranges. The paper should provide a physically motivated criterion (for example, a check that the mean-squared displacement is in the diffusive regime) or restrict the recommendation to systems similar to those tested.
  3. [IV.C, Fig. 4] The claim that the standard error of eta is 'nearly independent' of system size and that the N^{-1/3} weighting 'exactly cancels' the error scaling is stronger than the data support. Only three systems are shown, with large scatter, and no quantitative fit statistics are reported for the N^{-1/3} trends shown in Figs. 4(a) and 4(b). Please report fit parameters and uncertainties, and soften the cancellation statement to 'approximately compensates' unless more data are provided.
minor comments (6)
  1. [IV.B] There is a typo in the variant headings: 'V ariant' should be 'Variant'.
  2. [V] In the Conclusions, 'less viscous systemts' should read 'less viscous systems'.
  3. [IV.C] The word 'indepdent' in 'not completely indepdent from one another' should be 'independent'.
  4. [Fig. 4] The legend entries 'N^{-1/3} Fit' are not accompanied by visible fit curves or fit parameters; please clarify what is plotted or cite fitted values in the text.
  5. [III] The choice of the MSD linear-fit window from 0.03×tau_block to 0.4×tau_block is stated without justification; a brief explanation or citation would help readers assess the sensitivity of the reported diffusion coefficients and viscosities.
  6. [Table I and Table II] The Green-Kubo reference values are attributed to Smith and Sega Ref. 9 in Table II but to Refs. 9 and 10 elsewhere; please harmonize the citation usage.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: OrthoBoXY viscosities are computed from an Ewald-derived geometric relation and benchmarked against independent Green-Kubo data; the block-length scaling is an empirical recommendation, not a fitted target.

full rationale

The central comparison is not circular. Equation (5) computes eta from the simulated difference D0 - DPBC,zz and the geometric constant zeta_zz = 8.1711..., which is obtained by Ewald summation and does not depend on the target viscosities. The Green-Kubo reference values are taken from Smith and Sega (Refs. 9-10), an external data set, so the agreement claim is not constructed from the inputs. The self-citations (Refs. 6-8, 21) supply the OrthoBoXY geometry, the zeta values, and the 8.5-times-diameter block-length heuristic; the heuristic is explicitly revisited and modified, and the zeta values are independently computable constants, so these citations are not load-bearing in a circular sense. The system-size error-compensation argument is a derived prediction from Eq. (13) combined with empirically observed error scalings, and it is tested in Fig. 4 rather than assumed. The block-length scaling factors (1/8, 1/4) are empirical fits to four systems, with formaldehyde explicitly failing the short-block test; this is a generalizability and robustness concern about the cost-saving recommendation, not a circular derivation. Large deviations at the high-viscosity end (glycerol, 3-methylphenol, acetamide in Table II) weaken the headline agreement claim but are benchmark-failure or correctness issues, not evidence that a prediction reduces to its own input. Overall, no step in the claimed derivation chain is equivalent to its inputs by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new particles, forces, mediators, or conserved quantities. The central dependencies are standard simulation assumptions plus a prior recipe factor carried from the authors' earlier work, with scaling factors calibrated to four test liquids.

free parameters (3)
  • Block-length scaling factors (1/8, 1/4, 1) = 1/8 for tau_block > 100 ns, 1/4 for 1 ns < tau_block < 100 ns, 1 for tau_block < 1 ns
    Chosen by hand from scaled-block simulations of only four systems; the thresholds are inferred from the same data, not derived from a model.
  • 8.5 displacement factor in tau_block recipe = 8.5 times the linear diameter s
    Defined in Reference 8 and retained here. The paper argues it was generous but does not re-derive or replace it.
  • MSD linear-fit window = 0.03 tau_block to 0.4 tau_block
    Ad hoc choice used to extract all self-diffusion coefficients; no sensitivity analysis is reported for this window.
assumptions (5)
  • domain assumption The Yeh-Hummer equation (Eq. 3) accurately describes the periodic-boundary-condition correction to self-diffusion in orthorhombic simulation boxes.
    This is the basis of Eq. 5, from which all OrthoBoXY viscosities are computed.
  • standard math For the OrthoBoXY box ratio L_z/L_x = L_z/L_y = 2.7933, zeta_xx = zeta_yy = 0 and zeta_zz = 8.1711.
    Taken from prior Ewald summation results (Refs. 6 and 21); not recomputed in this paper.
  • domain assumption The standard error of the self-diffusion coefficient for molecular liquids scales as N^-1/3.
    Used in the cancellation argument leading to Eq. 13 and the claim that the viscosity error is independent of system size; cited to previous studies and checked only on three liquids with large scatter.
  • domain assumption The OPLS-AA force field gives realistic viscosities for the 15 investigated liquids.
    The comparison to experiment inherits force-field error, and large deviations are visible, for example 3-methylphenol.
  • domain assumption Blocks of length tau_block are statistically independent.
    The error estimates are computed as standard errors across blocks, which assumes block independence; no autocorrelation analysis is shown.

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Cite this review

Pith. "Pith review of Computing Shear Viscosities from Molecular Dynamics Simulation: Comparing the OrthoBoXY Approach with the Green-Kubo Method." pith.science (2026). https://pith.science/paper/43NDTLFE

@misc{pith2026260805845,
  author       = {Pith},
  title        = {Pith review of: Computing Shear Viscosities from Molecular Dynamics Simulation: Comparing the OrthoBoXY Approach with the Green-Kubo Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/43NDTLFE}},
  note         = {Machine review of arXiv:2608.05845}
}
abstract

We calculated shear viscosities of 15 neat molecular liquids from equilibrium molecular dynamics (MD) simulations using the OrthoBoXY approach and compare them to viscosities calculated via the Green-Kubo method. Data from both methods agree very well. Here, we show how to avoid pitfalls while computing the OrthoBoXY-data to obtain optimal results. From simulations of multiple system sizes, we could verify that the viscosity of molecular liquids is not influenced by finite size effects down to systems as small as 250 molecules. Moreover, we demonstrate that also the standard error of the viscosity is nearly independent of the system size. This is shown to be a consequence of a compensation effect of an increasing accuracy of the self-diffusion coefficients with increasing systems-size and the system-size dependent weighting according to the OrthoBoXY-equation. As a consequence, we suggest that it is preferable to run simulations of smaller systems with longer simulation times rather than larger systems with shorter simulation runs. In addition, we discuss a refinement of the recently introduced "recipe" for OrthoBoXY simulations block-lengths $\tau_\mathrm{block}$. Based on data from simulations with varying run-lengths, we suggest the following modification: for highly viscous systems, the value of $\tau_\mathrm{block}$ might safely be scaled by a factor of $1/8$, significantly reducing the computational resources needed. For less viscous systems, the value of $\tau_\mathrm{block}$ might safely be scaled by a factor of $1/4$. For systems with high fluidity, the value of $\tau_\mathrm{block}$ should not be scaled down in order to achieve reliable results. When using a smaller system size of 250 molecules, these refinements are leading up to a 24-fold reduction in computational cost compared to the previous recommended set-up without sacrificing numerical accuracy.

Figures

Figures reproduced from arXiv: 2608.05845 by the authors.

Figure 1
Figure 1. FIG. 1. Comparison of OrthoBoXY viscosities with (a) Green-Kubo viscosities and (b) experimental viscosities from literature. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Viscosities of (a) phenol and (b) pyridine for the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Viscosities of acetonitrile (blue), ethanol (orange) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Relative standard errors of (a) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Viscosities of acetamide (blue), formaldehyde (or [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

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