\(\renewcommand{\AA}{\text{Å}}\)

# 10.3.8. Calculate diffusion coefficients

The diffusion coefficient \(D\) of a material can be measured in at least 2 ways using various options in LAMMPS. See the `examples/DIFFUSE` directory for scripts that implement the 2 methods discussed here for a simple Lennard-Jones fluid model.

The first method is to measure the mean-squared displacement (MSD) of the system, via the compute msd command. The slope of the MSD versus time is proportional to the diffusion coefficient. The instantaneous MSD values can be accumulated in a vector via the fix vector command, and a line fit to the vector to compute its slope via the variable slope function, and thus extract \(D\).

The second method is to measure the velocity auto-correlation function (VACF) of the system, via the compute vacf command. The time-integral of the VACF is proportional to the diffusion coefficient. The instantaneous VACF values can be accumulated in a vector via the fix vector command, and time integrated via the variable trap function, and thus extract \(D\).