This paper studies the rapid thermalization of dissipative many-body dynamics for commuting Hamiltonians. The authors show that for a large class of geometrically-2-local models with commuting Hamiltonians, the thermalization time is much shorter than previously estimated, scaling logarithmically with system size. This is referred to as rapid mixing. The result is particularly relevant for 1D systems, where rapid thermalization with a system size independent decay rate is achieved from a positive gap in the generator. The authors also prove that systems on hypercubic lattices and exponential graphs (such as trees) exhibit rapid mixing at high temperatures. They introduce a novel notion of clustering called "strong local indistinguishability" based on max-relative entropy, which implies a lower bound on the modified logarithmic Sobolev inequality (MLSI) for nearest neighbour commuting models. This has implications for the rate of thermalization towards Gibbs states and their relevant Wasserstein distances and transportation cost inequalities. The authors also show that several measures of correlation decay on Gibbs states of commuting Hamiltonians are equivalent. At the technical level, they establish a direct relation between properties of Davies and Schmidt dynamics, allowing results of thermalization to be transferred between both. The paper provides a detailed analysis of the mixing times of Davies generators associated with systems with nearest neighbour commuting interactions, ordered by system lattice and assumptions required. The results show that for 1D systems, the optimal scaling for the MLSI constant is achieved, and for 2D systems, a square-root improved mixing time is obtained under sufficient correlation decay. For high temperature regimes, the authors show that nearest neighbour commuting potentials satisfy a MLSI with system-size independent or log-decreasing constants depending on the graph structure. The paper also discusses the implications of these results for the study of correlation properties of thermal fixed points and the stability of dissipative evolutions under perturbations.This paper studies the rapid thermalization of dissipative many-body dynamics for commuting Hamiltonians. The authors show that for a large class of geometrically-2-local models with commuting Hamiltonians, the thermalization time is much shorter than previously estimated, scaling logarithmically with system size. This is referred to as rapid mixing. The result is particularly relevant for 1D systems, where rapid thermalization with a system size independent decay rate is achieved from a positive gap in the generator. The authors also prove that systems on hypercubic lattices and exponential graphs (such as trees) exhibit rapid mixing at high temperatures. They introduce a novel notion of clustering called "strong local indistinguishability" based on max-relative entropy, which implies a lower bound on the modified logarithmic Sobolev inequality (MLSI) for nearest neighbour commuting models. This has implications for the rate of thermalization towards Gibbs states and their relevant Wasserstein distances and transportation cost inequalities. The authors also show that several measures of correlation decay on Gibbs states of commuting Hamiltonians are equivalent. At the technical level, they establish a direct relation between properties of Davies and Schmidt dynamics, allowing results of thermalization to be transferred between both. The paper provides a detailed analysis of the mixing times of Davies generators associated with systems with nearest neighbour commuting interactions, ordered by system lattice and assumptions required. The results show that for 1D systems, the optimal scaling for the MLSI constant is achieved, and for 2D systems, a square-root improved mixing time is obtained under sufficient correlation decay. For high temperature regimes, the authors show that nearest neighbour commuting potentials satisfy a MLSI with system-size independent or log-decreasing constants depending on the graph structure. The paper also discusses the implications of these results for the study of correlation properties of thermal fixed points and the stability of dissipative evolutions under perturbations.