Bayesian Methods for the Investigation of Temperature-Dependence in Conductivity

Andrew R. McCluskey; Benjamin J. Morgan; Samuel W. Coles

arxiv: 2512.17792 · v5 · pith:B3DHQNC2new · submitted 2025-12-19 · ❄️ cond-mat.mtrl-sci

Bayesian Methods for the Investigation of Temperature-Dependence in Conductivity

Andrew R. McCluskey , Samuel W. Coles , Benjamin J. Morgan This is my paper

Pith reviewed 2026-05-25 07:55 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords Bayesian methodstemperature-dependent conductivityArrhenius equationmodel selectionuncertainty quantificationmolecular dynamicssuperionic materialstransport properties

0 comments

The pith

Bayesian methods supply a single framework for estimating parameters in temperature-dependent conductivity models, selecting among them, and extrapolating with uncertainty.

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

The paper demonstrates that Bayesian statistics can handle the common tasks of fitting empirical models such as the Arrhenius equation to transport data from simulations. It treats parameter estimation, model comparison, and prediction outside the measured temperature range as connected problems that are solved together while carrying forward uncertainties. The approach is shown through molecular-dynamics examples on superionic materials. A reader would see this as a practical alternative to separate least-squares fits followed by ad-hoc error estimates. The tutorial therefore supplies both the statistical machinery and concrete code-level illustrations for these analyses.

Core claim

Bayesian methods offer a coherent framework that addresses challenges in parameter estimation, model selection, and extrapolation with uncertainty propagation for temperature-dependent transport data, illustrated with examples from molecular dynamics simulations of superionic materials.

What carries the argument

Bayesian inference applied to empirical functional forms such as the Arrhenius equation, enabling joint treatment of parameter posteriors, model probabilities, and predictive distributions.

If this is right

Fitted activation energies and prefactors come with full posterior distributions rather than point estimates.
Model choice between candidate functional forms is decided by explicit posterior odds that incorporate data quality.
Extrapolated conductivities at unmeasured temperatures include credible intervals that reflect both parameter and model uncertainty.
The same workflow applies directly to diffusion coefficients and other temperature-dependent transport quantities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The framework could be extended to joint analysis of conductivity and diffusion data from the same simulation trajectories.
It would allow systematic comparison of multiple empirical models across a library of materials rather than one at a time.
Adoption would change how simulation papers report extrapolated performance at device operating temperatures.

Load-bearing premise

Empirical models like the Arrhenius equation are suitable functional forms whose adequacy can be assessed from the available simulation data.

What would settle it

A dataset generated from a known non-Arrhenius mechanism where the Bayesian procedure still assigns high probability to the Arrhenius model would falsify the claim that the framework reliably assesses model adequacy.

Figures

Figures reproduced from arXiv: 2512.17792 by Andrew R. McCluskey, Benjamin J. Morgan, Samuel W. Coles.

**Figure 2.** Figure 2: FIG. 2. Comparison of Arrhenius (a, c) and VTF (b, d) mod [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Extrapolation of the LLZO conductivity model to [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

read the original abstract

Temperature-dependent transport data, including diffusion coefficients and ionic conductivities, are routinely analysed by fitting empirical models such as the Arrhenius equation. These fitted models yield parameters such as the activation energy, and can be used to extrapolate to temperatures outside the measured range. Researchers frequently face challenges in this analysis: quantifying the uncertainty of fitted parameters, assessing whether the data quality is sufficient to support a particular empirical model, and using these models to predict behaviour at temperatures outside the measured range. Bayesian methods offer a coherent framework that addresses all of these challenges. This tutorial introduces the use of Bayesian methods for analysing temperature-dependent transport data, covering parameter estimation, model selection, and extrapolation with uncertainty propagation, with illustrative examples from molecular dynamics simulations of superionic materials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A straightforward tutorial applying standard Bayesian methods to fitting Arrhenius-style models on MD conductivity data, with no new methods or results but a clear practical workflow.

read the letter

The key point is that this paper is a tutorial on using Bayesian inference for analyzing temperature-dependent transport data like ionic conductivities from molecular dynamics. It walks through parameter estimation with uncertainties, model selection, and extrapolation via posterior predictives, using examples from superionic materials simulations. The abstract positions it as addressing common challenges in fitting empirical models such as Arrhenius equations. What it does well is lay out a coherent, step-by-step application of established Bayesian tools to a routine task in computational materials science. For someone already running these simulations and doing least-squares fits, the examples could make the shift to proper uncertainty handling and model checking more accessible. The stress-test note is right that the core claim follows directly from Bayesian properties without needing extra assumptions beyond standard ones like appropriate priors and likelihoods. The soft spots are limited and expected for this type of paper: it introduces no new mathematics, no novel derivations, and no new scientific findings about conductivity mechanisms. It is explicitly a methods tutorial rather than an empirical or theoretical advance, so the novelty and significance scores from the reader align with what is shown. The full text would need to include enough code or data to make the examples reproducible, but nothing in the description suggests hidden flaws or circularity. This is for computational materials researchers or simulation groups who fit such data regularly and want a practical reference. A reader seeking groundbreaking results or deep statistical innovation will not find it here. It deserves a serious referee because it targets a narrow but common analysis need with a grounded approach, even if revisions would focus on clarity and reproducibility rather than fixing major issues. I would recommend sending it to peer review in a methods-oriented venue.

Referee Report

0 major / 3 minor

Summary. The manuscript is a tutorial on applying Bayesian inference to temperature-dependent transport data (diffusion coefficients and ionic conductivities) obtained from molecular dynamics simulations of superionic materials. It demonstrates parameter estimation for empirical models such as the Arrhenius equation, model selection to assess data quality and model adequacy, and extrapolation to unmeasured temperatures with full uncertainty propagation via posterior predictive distributions.

Significance. If the tutorial examples are reproducible and correctly illustrate the claimed advantages, the work provides a practical, coherent framework that directly addresses three recurring challenges in materials-science transport analysis: parameter uncertainty, model adequacy testing, and reliable extrapolation. The explicit use of marginal likelihoods or information criteria for model selection and posterior predictive checks for extrapolation constitutes a clear pedagogical contribution that can be adopted without new theoretical machinery.

minor comments (3)

The abstract states that the tutorial covers 'illustrative examples from molecular dynamics simulations' but does not indicate whether the underlying conductivity data, prior specifications, or Stan/JAGS/PyMC code are provided as supplementary material or a public repository; this is essential for a methods tutorial.
In the section describing the Arrhenius model, the functional form and the definition of the likelihood (including any assumed error model on the conductivity values) should be written explicitly with equation numbers so that readers can reproduce the posterior exactly.
The discussion of model selection would benefit from a short table comparing the marginal likelihoods (or WAIC/LOO values) across the candidate models for at least one of the MD datasets, rather than only qualitative statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript as a tutorial on Bayesian methods for temperature-dependent transport data and for recommending minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a tutorial demonstrating standard Bayesian inference (posteriors for parameters, marginal likelihoods for model selection, posterior predictive distributions for extrapolation) applied to Arrhenius-style fits on external MD-derived conductivity data. No equations, derivations, or predictions are shown that reduce by construction to quantities defined or fitted within the paper itself. The central claim follows directly from the general properties of Bayesian methods and requires no domain-specific self-referential steps, self-citation chains, or ansatzes smuggled via prior work by the same authors. The illustrative examples serve only to demonstrate application and do not form load-bearing premises.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are specified in the provided text.

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Reference graph

Works this paper leans on

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A unified formulation of the constant temper- ature molecular dynamics methods.J

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Bayesian Methods for the Investigation of Temperature-Dependence in Conductivity

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[1] [1]

in Energy2, 022001 (2020)

Ohno, S.et al.Materials design of ionic conductors for solid state batteries.Prog. in Energy2, 022001 (2020)

work page 2020

[2] [2]

R.et al.Neutron studies of Na-ion battery materials.J

Shah, A. R.et al.Neutron studies of Na-ion battery materials.J. Phys.: Mater.4, 042008 (2021)

work page 2021

[3] [3]

Energy Mater.5, 14119–14126 (2022)

Kumar, S.et al.Li-Ion Diffusion Correlations in LiAlGeO4: Quasielastic Neutron Scattering and Ab Ini- tio Simulation.ACS Appl. Energy Mater.5, 14119–14126 (2022)

work page 2022

[4] [4]

Commun.6, 7497 (2015)

Eames, C.et al.Ionic transport in hybrid lead iodide perovskite solar cells.Nat. Commun.6, 7497 (2015)

work page 2015

[5] [5]

N., Quirk, J

Forrester, F. N., Quirk, J. A., Famprikis, T. & Daw- son, J. A. Disentangling Cation and Anion Dynamics in Li3PS4 Solid Electrolytes.Chem. Mater.34, 10561– 10571 (2022)

work page 2022

[6] [6]

Mater.35, 6133–6140 (2023)

Krenzer, G.et al.Nature of the Superionic Phase Tran- sition of Lithium Nitride from Machine Learning Force Fields.Chem. Mater.35, 6133–6140 (2023)

work page 2023

[7] [7]

L., Kemp, D., Bonkowski, A

Usler, A. L., Kemp, D., Bonkowski, A. & De Souza, R. A. A general expression for the statistical error in a diffusion coefficient obtained from a solid-state ¡scp¿molecular-dynamics¡/scp¿ simulation.J. Comput. Chem.44, 1347–1359 (2023)

work page 2023

[8] [8]

Pushing Forward Simulation Techniques of Ion Transport in Ion Conductors for Energy Materials

Canepa, P. Pushing Forward Simulation Techniques of Ion Transport in Ion Conductors for Energy Materials. ACS Mater. Au3, 75–82 (2022)

work page 2022

[9] [9]

& Arya, G

Wang, J., Ding, J., Delaire, O. & Arya, G. Atomistic Mechanisms Underlying Non-Arrhenius Ion Transport in Superionic Conductor AgCrSe 2.ACS Appl. Energy Mater.4, 7157–7167 (2021)

work page 2021

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Das Temperaturabhaengigkeitsgesetz der Viskositaet von Fluessigkeiten.Phys

Vogel, H. Das Temperaturabhaengigkeitsgesetz der Viskositaet von Fluessigkeiten.Phys. Z.22, 645 (1921)

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[11] [11]

Fulcher, G. S. Analysis of Recent Measurements of the Viscosity of Glasses.J. Am. Ceram. Soc.8, 339–355 (1925)

work page 1925

[12] [12]

& Hesse, W

Tammann, G. & Hesse, W. Die Abh¨ angigkeit der Viscosit¨ at von der Temperatur bie unterk¨ uhlten Fl¨ ussigkeiten.Z. Anorg. Allg. Chem.156, 245–257 (1926)

work page 1926

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de Souza, V. K. & Wales, D. J. Super-Arrhenius Diffusion in an Undercooled Binary Lennard-Jones Liquid Results from a Quantifiable Correlation Effect.Phys. Rev. Lett. 96, 057802 (2006)

work page 2006

[14] [14]

R., Scanlon, D

Yeandel, S. R., Scanlon, D. O. & Goddard, P. Enhanced Li-ion dynamics in trivalently doped lithium phosphi- dosilicate Li 2SiP2: a candidate material as a solid Li electrolyte.J. Mater. Chem. A7, 3953–3961 (2019)

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A.et al.Rotational stacking faults in the ionic conductor Li 3ScCl6.Chem

Goldmann, B. A.et al.Rotational stacking faults in the ionic conductor Li 3ScCl6.Chem. Mater.(2025)

work page 2025

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R., Squires, A

McCluskey, A. R., Squires, A. G., Dunn, J., Coles, S. W. & Morgan, B. J. kinisi: Bayesian analysis of mass transport from molecular dynamics simulations.J. Open Source Softw.9, 5984 (2024)

work page 2024

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Akaike, H. A new look at the statistical model identifi- cation.IEEE Trans. Autom. Contr.19, 716–723 (1974)

work page 1974

[18] [18]

Estimating the Dimension of a Model.Ann

Schwarz, G. Estimating the Dimension of a Model.Ann. Stat.6, 461–464 (1978)

work page 1978

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McCluskey, A. R. Is There Still a Place for Linearization in the Chemistry Curriculum?J. Chem. Educ.100, 4174–4176 (2023)

work page 2023

[20] [20]

& Howard, J

Mayer, J., Khairy, K. & Howard, J. Drawing an elephant with four complex parameters.Am. J. Phys.78, 648–649 (2010)

work page 2010

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Kass, R. E. & Raftery, A. E. Bayes Factors.J. Am. Stat. Assoc.90, 773–795 (1995)

work page 1995

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Sivia, D. S. & Skilling, J.Data Analysis: A Bayesian Tutorial(Oxford University Press, Oxford, UK, 2006), 2nd edn

work page 2006

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Mean-squared displacements exhibit ballistic behaviour at short times and diffusive behaviour at longer times [26, 27]. Because diffusion coefficients are estimated by linear regression of MSD against time, only data from the dif- fusive regime, where the MSD is approximately linear in time, is used for fitting

work page

[24] [24]

For extrap- olation far from the measured range, parameter uncer- tainty typically dominates, so this distinction is rarely important in practice

This extrapolated distribution captures parameter uncer- tainty but not estimated measurement noise. For extrap- olation far from the measured range, parameter uncer- tainty typically dominates, so this distinction is rarely important in practice

work page

[25] [25]

R., Coles, S

McCluskey, A. R., Coles, S. W. & Morgan, B. J. Accu- rate Estimation of Diffusion Coefficients and their Un- 9 certainties from Computer Simulation.J. Chem. Theory Comput.21, 79–87 (2024)

work page 2024

[26] [26]

He, X., Zhu, Y., Epstein, A. & Mo, Y. Statistical vari- ances of diffusional properties from ab initio molecular dynamics simulations.npj Comput. Mater.4, 18 (2018)

work page 2018

[27] [27]

& De Souza, R

Bonkowski, A. & De Souza, R. A. From setup to analy- sis: A compact guide to performing Molecular Dynamics simulations of ion transport in solids.Solid State Ionics 429, 116967 (2025)

work page 2025

[28] [28]

R., Coles, S

McCluskey, A. R., Coles, S. W. & Morgan, B. J. model-arrhenius-0.0.1. https://github.com/scams- research/model-arrhenius (2025)

work page 2025

[29] [29]

Open Source Softw.5, 2373 (2020)

Marin-Lafl` eche, A.et al.MetalWalls: A classical molec- ular dynamics software dedicated to the simulation of electrochemical systems.J. Open Source Softw.5, 2373 (2020)

work page 2020

[30] [30]

Burbano, M., Carlier, D., Boucher, F., Morgan, B. J. & Salanne, M. Sparse Cyclic Excitations Explain the Low Ionic Conductivity of Stoichiometric Li7La3Xr2O12. Phys. Rev. Lett.116, 135901 (2016)

work page 2016

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A unified formulation of the constant temper- ature molecular dynamics methods.J

Nos´ e, S. A unified formulation of the constant temper- ature molecular dynamics methods.J. Chem. Phys.81, 511–519 (1984)

work page 1984

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Hoover, W. G. Canonical dynamics: Equilibrium phase- space distributions.Phys. Rev. A31, 1695–1697 (1985)

work page 1985

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J., Klein, M

Martyna, G. J., Klein, M. L. & Tuckerman, M. Nos´ e–Hoover chains: The canonical ensemble via contin- uous dynamics.J. Chem. Phys.97, 2635–2643 (1992)

work page 1992

[34] [34]

Coles, S. W. LLZO Simulation Trajectories. https://doi.org/10.5281/zenodo.17702491 (2025)

work page doi:10.5281/zenodo.17702491 2025

[35] [35]

J., Woolf, T

Michaud-Agrawal, N., Denning, E. J., Woolf, T. B. & Beckstein, O. MDAnalysis: A toolkit for the analysis of molecular dynamics simulations.J. Comput. Chem.32, 2319–2327 (2011)

work page 2011

[36] [36]

InProceedings of the 15th Python in Science Conference, 98–105 (2016)

Gowers, R.et al.MDAnalysis: A Python Package for the Rapid Analysis of Molecular Dynamics Simulations. InProceedings of the 15th Python in Science Conference, 98–105 (2016)

work page 2016

[37] [37]

& Arya, G

Wang, J., Ding, J., Delaire, O. & Arya, G. AgCrSe2 Simulation Trajectories. https://doi.org/10.5281/zenodo.17910336 (2025)

work page doi:10.5281/zenodo.17910336 2025

[38] [38]

Nested Sampling

Skilling, J. Nested Sampling. InAIP Conference Pro- ceedings, vol. 735, 395–405 (AIP, 2004)

work page 2004

[39] [39]

Bayesian Methods for the Investigation of Temperature-Dependence in Conductivity

Speagle, J. S. dynesty: a dynamic nested sampling package for estimating bayesian posteriors and evidences. Mon. Not. R. Astron. Soc.493, 3132–3158 (2020). Supplemental Material for “Bayesian Methods for the Investigation of Temperature-Dependence in Conductivity” Andrew R. McCluskey, 1, 2,∗ Samuel W. Coles, 3, 4 and Benjamin J. Morgan 5, 6,† 1Centre for ...

work page 2020