Bayesian Methods for the Investigation of Temperature-Dependence in Conductivity

TL;DR

The paper tackles the challenge of analyzing temperature-dependent transport coefficients (e.g., $D^*$ and $\sigma$) by adopting a Bayesian framework that unifies parameter estimation, model selection, and extrapolation with uncertainty propagation, addressing limitations of traditional Arrhenius fitting. It demonstrates how posterior sampling yields full distributions for parameters like $E_{\mathrm{a}}$ and $A$, reveals correlations, and enables principled model comparison via marginal likelihoods and Bayes factors. Through MD data on LLZO and AgCrSe2, the study illustrates Arrhenius versus non-Arrhenius (VTF) modelling, showing how Bayes factors depend on data quantity and precision, and how extrapolation to unmeasured temperatures produces predictive distributions rather than single point estimates. The work provides a practical, reproducible approach (implemented in $\texttt{kinisi}$) for rigorous uncertainty quantification in temperature-dependent transport, with implications for materials design and interpretation of diffusion and conductivity data.

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 extrapolated temperatures. 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.

Figures (15)

• Figure 1: (a) Lithium-ion conductivity in c-LLZO (black points; error bars show 95 credible intervals). Blue curves show Arrhenius models evaluated using parameter samples drawn from the posterior; darker shading indicates where more curves overlap. (b) Joint posterior distribution showing the correlation between $E_{\mathrm{a}}$ and $A$. (c, d) Marginal posterior distributions for the activation energy and pre-exponential factor.
• Figure 2: Comparison of Arrhenius (a, c) and VTF (b, d) models for silver-ion conductivity in AgCrSe2. Black points show conductivity estimates with 95 credible intervals. Coloured shading shows models evaluated at parameter values sampled from each posterior. Panels (a, b) use a fitting window of 40; panels (c, d) use 140. (e) Bayes factor comparing VTF to Arrhenius as a function of the amount of diffusive-regime data used, $t_\mathrm{diff}$. The dashed line indicates $\ln(B_{\beta\alpha}) = 5$, the threshold for "very strong" evidence.
• Figure 3: Extrapolation of the LLZO conductivity model to 300 (vertical dashed line). Blue shading shows Arrhenius models evaluated at parameter values sampled from the posterior (same data as \ref{['fig:llzo']}). The shaded region widens at lower temperatures, reflecting increased uncertainty in the extrapolated predictions. Inset: distribution of predicted conductivity at 300.
• Figure SI.1: The mean squared displacement data and associated distribution of linear models at temperatures of 500;600;700;800 with 50ps of LLZO simulation.
• Figure SI.2: The mean squared displacement data and associated $\sigma$ distributions at temperatures of 300;350;400;500;600;700 with 40ps of diffusive simulation (a-l), the appropriate Arrhenius (m) and VTF model (n) plots and the resulting distributions of $E_{\mathrm{a}}$ from each modelling approach (o).