# Context¶

The diffusion of ions through solid materials is a fundamental physical process that underpins applications such as lithium-ion batteries, fuel cells, and chemical sensors. One common technique for studying atomic scale diffusion processes in solids is molecular dynamics simulation, which can be used to directly calculate macroscopic transport coefficients, i.e. diffusion coefficients and ionic conductivities, and also to examine the detailed atomic scale mechanisms of ionic transport. While molecular dynamics simulations, in principle, provides a description of all atomic scale dynamical processes that occur on a simulation timescale, analysing the resulting raw data to extract quantitative data about diffusion mechanisms can be challenging.

One approach to obtain mechanistic information from a molecular dynamics simulation is to spatially discretise the atomic trajectories by projecting from a set of three dimensional continuous coordinates onto a set of discrete “sites”. This approach is founded on the idea that in many solids, diffusion proceeds by ions undergoing a sequence of “jumps” between local potential energy minima. These minima typically correspond to a particular arrangement of the mobile ions within some set of crystallographic sites. In a site-projection analysis, we first define a set of bounded volumes within the simulation cell, which represent our “sites”, and then project the coordinates of the mobile ions onto these volumes. This gives two spatially discretised representations of each configuration in a simulation trajectory. From the perspective of the atoms, each atom is assigned to zero, one, or multiple sites (depending on whether out sites are defined to be mutually space-filling and / or non-overlapping). From the perspective of the sites, each site is occupied by zero, one, or more mobile ions.

This site occupation data can be used to build quantiative descriptions of diffusion mechanisms. For example, we can calculate time-averaged site-occupation probabilities, which can be compared to experimental results, such as diffraction data. Because the site-occupation data is time-resolved, we can also use this to analyse the trajectories of mobile atoms. For example, specific sequences of sites that an ion moves through can be statistically analysed, or temporal and spatical correlations between the movements of groups of mobile atoms can be quantified.