Seebeck coefficient of ionic conductors from Bayesian regression analysis

We propose a novel approach to evaluating the ionic Seebeck coefficient in electrolytes from relatively short equilibrium molecular dynamics simulations, based on the Green-Kubo theory of linear response and Bayesian regression analysis. By exploiting the probability distribution of the off-diagonal elements of a Wishart matrix, we develop a consistent and unbiased estimator for the Seebeck coefficient whose statistical uncertainty can be arbitrarily reduced in the long-time limit. To validate the effectiveness of our method, we benchmark it against extensive equilibrium molecular dynamics simulations conducted on molten $\mathrm{C}\mathrm{s}\mathrm{F}$ using empirical force fields. We then employ this procedure to calculate the Seebeck coefficient of molten $\mathrm{N}\mathrm{a}\mathrm{C}\mathrm{l}$, $\mathrm{K}\mathrm{C}\mathrm{l}$ and $\mathrm{L}\mathrm{i}\mathrm{C}\mathrm{l}$ using neural-network force fields trained on ab initio data over a range of pressure-temperature conditions.