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.
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