This project focuses on understanding the long-time dynamics of Bose gases driven out of equilibrium, a fundamental challenge in quantum many-body physics. When such gases are quenched within their superfluid phase, the relaxation dynamics, governed by phononic excitations, may exhibit universal scaling behavior described by the Kardar-Parisi-Zhang (KPZ) equation. However, uncertainties about the phonon lifetime in low-dimensional systems remain, where standard perturbative methods are inadequate. This project aims to develop a non-perturbative framework using the Functional Renormalization Group (FRG) to derive the phonon lifetime in 1D and 2D Bose superfluids, clarifying the KPZ conjecture in 1D and generating novel predictions for 2D systems. The student will begin by learning the FRG method and its application to close-to and out of -equilibrium quantum systems, deriving the phonon scattering rate and analyzing its scaling behavior. The results will be benchmarked against numerical simulations and existing theories to validate the approach and refine the theoretical understanding of phonon dynamics. If time allows, the project will explore the connection between phonon lifetime scaling and the KPZ universality class, comparing FRG-derived results with experimental data. This project is in collaboration with the theory group of Nicolas Cherroret at Laboratoire Kastler-Brossel, Sorbonne Université.