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  • Tutorial
  • Open Access

Bayesian Methods for the Investigation of Temperature Dependence in Conductivity

Andrew R. McCluskey1,2,*, Samuel W. Coles3,4, and Benjamin J. Morgan5,6,†

  • *Contact author: andrew.mccluskey@bristol.ac.uk
  • †Contact author: b.j.morgan@bath.ac.uk
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PRX Energy 5, 022001 – Published 21 May, 2026

DOI: https://doi.org/10.1103/nxgq-lmp6

Abstract

Temperature-dependent transport data, including diffusion coefficients and ionic conductivities, are routinely analyzed 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 behavior 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 analyzing temperature-dependent transport data, covering parameter estimation, model selection, and extrapolation with uncertainty propagation, with illustrative examples from molecular dynamics simulations of superionic materials.

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How ion transport changes with temperature is central to understanding energy materials such as battery electrolytes and fuel cell membranes. To characterize this dependence, researchers typically fit empirical models to data; this requires choosing from possible competing models and deriving meaningful uncertainties in the fitted parameters. These models can then be used to predict behavior at unmeasured temperatures, provided the uncertainties are properly propagated. This Tutorial introduces Bayesian methods as a coherent framework for these tasks. Rather than returning a single best-fit value for each model parameter, Bayesian analysis yields a full probability distribution. This allows researchers to rigorously quantify uncertainties in fitted parameters, objectively compare competing models, and propagate those uncertainties into predictions at new temperatures. Using examples from computer simulations of fast ion-conducting materials, the authors demonstrate the use of Bayesian methods to quantify the evidence for non-Arrhenius behavior and provide meaningful uncertainty estimates when extrapolating to unmeasured temperatures.

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References (34)

  1. P. Canepa, Pushing forward simulation techniques of ion transport in ion conductors for energy materials, ACS Mater. Au 3, 75 (2023).
  2. J. Wang, J. Ding, O. Delaire, and G. Arya, Atomistic mechanisms underlying non-Arrhenius ion transport in superionic conductor AgCrSe2, ACS Appl. Energy Mater. 4, 7157 (2021).
  3. The Nernst-Einstein relation between σ and D∗ introduces a T−1 factor, so the Arrhenius form applies to σT rather than σ.
  4. H. Vogel, Das Temperaturabhaengigkeitsgesetz der Viskositaet von Fluessigkeiten, Phys. Z. 22, 645 (1921).
  5. G. S. Fulcher, Analysis of recent measurements of the viscosity of glasses, J. Am. Ceram. Soc. 8, 339 (1925).
  6. G. Tammann and W. Hesse, Die Abhängigkeit der Viscosität von der Temperatur bie unterkühlten Flüssigkeiten, Z. Anorg. Allg. Chem. 156, 245 (1926).
  7. V. K. de Souza and D. J. Wales, Super-Arrhenius diffusion in anundercooled binary Lennard-Jones liquid results from a quantifiable correlation effect, Phys. Rev. Lett. 96, 057802 (2006).
  8. S. R. Yeandel, D. O. Scanlon, and P. Goddard, Enhanced Li-ion dynamics in trivalently doped lithium phosphidosilicate Li2SiP2: A candidate material as a solid Li electrolyte, J. Mater. Chem. A 7, 3953 (2019).
  9. B. A. Goldmann, C. Rosenbach, H. A. Evans, B. Helm, B. Wankmiller, O. Maus, E. Suard, L. F. Nazar, M. R. Hansen, B. J. Morgan, M. S. Islam, and W. G. Zeier, Rotational stacking faults in the ionic conductor Li3ScCl6, Chem. Mater. 37, 9858 (2025).
  10. A. R. McCluskey, A. G. Squires, J. Dunn, S. W. Coles, and B. J. Morgan, Kinisi: Bayesian analysis of mass transport from molecular dynamics simulations, J. Open Source Softw. 9, 5984 (2024).
  11. Information criteria such as AIC and BIC address this problem [32, 33], but rely on assumptions such as normally distributed parameters that may not hold in practice.
  12. This follows from the principle of maximum entropy [34].
  13. A. R. McCluskey, Is there still a place for linearization in the chemistry curriculum? J. Chem. Educ. 100, 4174 (2023).
  14. J. Mayer, K. Khairy, and J. Howard, Drawing an elephant with four complex parameters, Am. J. Phys. 78, 648 (2010).
  15. R. E. Kass and A. E. Raftery, Bayes factors, J. Am. Stat. Assoc. 90, 773 (1995).
  16. D. S. Sivia and J. Skilling, Data Analysis: A Bayesian Tutorial (Oxford University Press, Oxford, UK, 2006), 2nd ed.
  17. This extrapolated distribution captures parameter uncertainty but not estimated measurement noise. For extrapolation far from the measured range, parameter uncertainty typically dominates, so this distinction is rarely important in practice.
  18. A. R. McCluskey, S. W. Coles, and B. J. Morgan, Accurate estimation of diffusion coefficients and their uncertainties from computer simulation, J. Chem. Theory Comput. 21, 79 (2025).
  19. A temperature-independent Haven ratio would uniformly scale the conductivity values without affecting the temperature dependence; a temperature-dependent ratio would introduce additional curvature, but the analysis procedure would be unchanged.
  20. N. Michaud-Agrawal, E. J. Denning, T. B. Woolf, and O. Beckstein, MDAnalysis: A toolkit for the analysis of molecular dynamics simulations, J. Comput. Chem. 32, 2319 (2011).
  21. R. Gowers, M. Linke, J. Barnoud, T. Reddy, M. Melo, S. Seyler, J. Domański, D. Dotson, S. Buchoux, I. Kenney, and O. Beckstein, in Proceedings of the 15th Python in Science Conference (Austin, US, 2016), pp. 98–105.
  22. A. R. McCluskey, S. W. Coles, and B. J. Morgan, model-arrhenius-1.0.0, 2025, https://github.com/scams-research/model-arrhenius.
  23. A. Marin-Laflèche, M. Haefele, L. Scalfi, A. Coretti, T. Dufils, G. Jeanmairet, S. Reed, A. Serva, R. Berthin, C. Bacon, S. Bonella, B. Rotenberg, P. Madden, and M. Salanne, MetalWalls: A classical molecular dynamics software dedicated to the simulation of electrochemical systems, J. Open Source Softw. 5, 2373 (2020).
  24. M. Burbano, D. Carlier, F. Boucher, B. J. Morgan, and M. Salanne, Sparse cyclic excitations explain the low ionic conductivity of stoichiometric Li7La3Zr2O12, Phys. Rev. Lett. 116, 135901 (2016).
  25. S. Nosé, A unified formulation of the constant temperature molecular dynamics methods, J. Chem. Phys. 81, 511 (1984).
  26. W. G. Hoover, Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A 31, 1695 (1985).
  27. G. J. Martyna, M. L. Klein, and M. Tuckerman, Nosé–Hoover chains: The canonical ensemble via continuous dynamics, J. Chem. Phys. 97, 2635 (1992).
  28. Samuel W. Coles, LLZO Simulation Trajectories, 2025, https://doi.org/10.5281/zenodo.17702491.
  29. J. Wang, J. Ding, O. Delaire, and G. Arya, AgCrSe2 Simulation Trajectories, 2025, https://doi.org/10.5281/zenodo.17910336.
  30. J. Skilling, in AIP Conference Proceedings (AIP, Garching, DE, 2004), Vol. 735, pp. 395–405.
  31. J. S Speagle, dynesty: A dynamic nested sampling package for estimating Bayesian posteriors and evidences, Mon. Not. R. Astron. Soc. 493, 3132 (2020).
  32. H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Control 19, 716 (1974).
  33. G. Schwarz, Estimating the dimension of a model, Ann. Stat. 6, 461 (1978).
  34. E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106, 620 (1957).

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PRX Energy 5, 022001– Published 21 May, 2026
  • Received 19 December 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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