Non-stationary current fluctuations in 1D boundary-driven diffusive systems via Macroscopic Fluctuation Theory

While Macroscopic Fluctuation Theory (MFT) has been highly successful in analyzing non-equilibrium steady states, its application to non-steady-state processes remains limited. In this study, we apply MFT to the relaxation process of one-dimensional boundary-driven diffusive systems coupled to particle reservoirs at both ends. We exactly derive the current variance for systems with a constant diffusion coefficient and arbitrary mobility, as well as the cumulant generating function for the current in Reflective Brownian Motion (RBM). Our results demonstrate that non-steady current fluctuations during the approach to a steady state can be quantitatively described within the MFT framework.