Ultrafast dynamics of electrons and phonons: from the two-temperature model to the time-dependent Boltzmann equation

The advent of pump-probe spectroscopy techniques paved the way to the exploration of the ultrafast dynamics of electrons and phonons in crystalline solids. Following photo-absorption of a pump pulse and the initial electronic thermalization, the dynamics of electronic and vibrational degrees of freedom is dominated by electron-phonon and phonon-phonon scattering processes.The two-temperature model (TTM) and its generalizations -- as, e.g., the non-thermal lattice model (NLM) -- provide valuable tools to describe these phenomena and the ensuing coupled electron-phonon dynamics over timescales ranging between few hundreds of femtoseconds and tens of picoseconds. While more sophisticated theoretical approaches are nowadays available, the conceptual and computational simplicity of the TTM makes it the method of choice to model thermalization processes in pump-probe spectroscopy, and it keeps being widely applied in both experimental and theoretical studies. In the domain of ab-initio methods, the time-dependent Boltzmann equation (TDBE) ameliorates many of the shortcomings of the TTM and it enables a realistic and parameter-free description of ultrafast phenomena with full momentum resolution. After a pedagogical introduction to the TTM and TDBE, in this manuscript we review their application to the description of ultrafast process in solid-state physics and materials science as well as their theoretical foundation.