Dephasing-induced relaxation in tight-binding chains with linear and nonlinear defects

We investigate thermalization in a tight-binding chain with an on-site defect subject to local dephasing noise implemented as random phase kicks. For a single linear defect of strength $\epsilon$, we obtain an exact analytical description of the system spectrum and formulate the dephasing-induced dynamics in the eigenstate basis. We derive an approximate kinetic equation for mode populations that describes a continuous-time random walk in action space. The walk transition rates are defined by the overlap matrix encoding the spatial structure of eigenstates that can be computed exactly. Analyzing the spectral properties of the equation, we show that defect-induced localized modes act as bottlenecks that strongly slow down relaxation, with rates scaling as $\epsilon^{-2}$ for strong defects. Using large-deviation theory, we characterize rare dynamical trajectories and identify distinct relaxation pathways associated with low- and high-activity regimes in action space. We provide numerical evidence that the large-deviation function exhibits a dynamical phase transition in the limit $\epsilon \to \infty$. We then extend our analysis to the nonlinear case, considering a single nonlinear defect embedded in either a linear or a fully nonlinear discrete Schr\"odinger equation. Numerical simulations reveal a qualitatively faster approach to equilibrium driven by the amplitude-dependent weakening of the defect. Our results provide a unified framework for understanding thermalization, rare fluctuations, and relaxation pathways in stochastic tight-binding systems.