Stochastic Langevin dynamics has been traditionally used as a tool to describe non-equilibrium processes. When utilized in systems with collective modes, traditional Langevin dynamics relaxes all modes indiscriminately, regardless of their wavelength. We propose a generalization of Langevin dynamics that can capture a differential coupling between collective modes and the bath, by introducing spatial correlations in the random forces. This allows modeling the electronic subsystem in a metal as a generalized Langevin bath endowed with a concept of locality, greatly improving the capabilities of the two-temperature model. The specific form proposed here for the spatial correlations produces physical wavevector- and polarization-dependency of the relaxation produced by the electron-phonon coupling in a solid. We show that the resulting model can be used for describing the path to equilibration of ions and electrons, and also as a thermostat to sample the equilibrium canonical ensemble. By extension, the family of models presented here can be applied in general to any dense system, solids, alloys and dense plasmas. As an example, we apply the model to study the non-equilibrium dynamics of an electron-ion two-temperature Ni crystal.
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