Our theoretical work is organized in two independent parts: Part I belongs to the field of condensed matter theory and deals with the spectral signatures of collective states in one dimensional (1D) metals: Electrons in 1D metals are expected to fractionalize into collective spin and charge degrees of freedom. A recent candidate material for the realization of a 1D metal are mirror-twin boundaries in monolayer MoS2. Scanning tunneling spectroscopy was used to record the local density of states along these 1D line defects. In our purely theoretical work, we calculate the local density of states as predicted by Tomonaga-Luttinger-liquid theory in order to reveal the nature of the 1D states spectroscopically. The comparison of measured and theoretical spectra allows us to identify the observed doubling of the energy levels as signature of emergent spin and charge excitations. Part II belongs to the field of non-equilibrium physics and deals with the macroscopic description of equilibration: Equilibration of closed systems is hampered by the diffusive transport of locally conserved quantities as described by fluctuating hydrodynamics. After a sudden perturbation, the buildup of equilibrium fluctuations occurs only algebraically slowly, giving rise to hydrodynamic long-time tails. However, the standard tool in transport theory, the Boltzmann equation, fails to describe equilibration. Adding a noise term restores the missing correlations, resulting in a stochastic Langevin-Boltzmann equation. In our work, we derive a simplified version: a fluctuating relaxation-time approximation. We also set up a stable integration scheme for this type of equation and demonstrate that the numerical solution is in agreement with the predictions of fluctuating hydrodynamics. As an addition, we discuss slow changes of state. We show that the entropy production vanishes algebraically slowly in the adiabatic limit due to the presence of hydrodynamic slow modes
