  Time- and angle-resolved photoemission spectroscopy (TR-ARPES) accesses the
electronic structure of solids under optical excitation, and is a powerful
technique for studying the coupling between electrons and collective modes. One
approach to infer electron-boson coupling is through the relaxation dynamics of
optically-excited electrons, and the characteristic timescales of energy
redistribution. A common description of electron relaxation dynamics is through
the effective electronic temperature. Such a description requires that
thermodynamic quantities are well-defined, an assumption that is generally
violated at early delays. Additionally, precise estimation of the non-thermal
window -- within which effective temperature models may not be applied -- is
challenging. We perform TR-ARPES on graphite and show that Boltzmann rate
equations can be used to calculate the time-dependent electronic occupation
function, and reproduce experimental features given by non-thermal electron
occupation. Using this model, we define a quantitative measure of non-thermal
electron occupation and use it to define distinct phases of electron relaxation
in the fluence-delay phase space. More generally, this approach can be used to
inform the non-thermal-to-thermal crossover in pump-probe experiments.
