Method to determine the electron-ion temperature relaxation rate from test particle distributions

A method to calculate the electron-ion energy exchange rate from the forces on and velocities of an ensemble of test particles is described. The essence of the method is that the energy exchange rate is related to the covariance between the distribution of velocities of test particles and the forces acting on them. The energy exchange rate is related to the electron-ion temperature relaxation rate in the limit of a low test particle speed. A proof of concept is conducted with first-principles molecular dynamics simulations.