This master thesis examines a coupled electron-phonon system, the Holstein polaron model, as an isolated many-body quantum system far from equilibrium. The elementary question of non-equilibrium dynamics is the problem of equilibration and thermalization. That is to say: why and to what extent are standard statistical descriptions applicable to closed quantum systems after a long enough time? One approach to this question is the so-called eigenstate thermalization hypothesis (ETH), which is a statement about the matrix elements of all physically relevant observables. In this work, we study the Holstein polaron model with a numerical approach with the goal of observing ETH. We apply several numerical methods: exact diagonalization, a general method for many-body systems, and a variational approach to constructing the many-body Hilbert space, developed specifically for the low-energy physics in the Holstein model. We compare the two methods by how well they describe the highly excited states, relevant for nonequilibrium dynamics. We test for indicators of quantum chaos in the spectrum and indicators of thermalization in the eigenstate-to-eigenstate fluctuations in the thermodynamic limit. In finite systems (e.g. spin chains), the thermodynamic limit is usually taken by increasing the number of sites in a periodic chain, but with bosonic degrees of freedom present, it is also possible to increase the maximum occupancy of phononic states, which is the route we choose. In addition to this, we inverstigate the entanglement entropy for the partition along different types of degrees of freedom in the entire spectrum.
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