The paper "Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians" by Jan Kochanowski, Álvaro M. Alhambra, Ángela Capel, and Cambysse Rouzé explores the rapid thermalization of quantum systems coupled to a thermal environment. The authors focus on geometrically-2-local models of Davies generators with commuting Hamiltonians and show that the thermalization time is much shorter than what one might naively estimate from the spectral gap. Specifically, they prove that the thermalization time is at most logarithmic in the system size, achieving rapid mixing of dissipative dynamics. Key results include: 1. **Optimal Rapid Thermalization in 1D**: For 1D systems with Davies generators of commuting, local Hamiltonians, having a positive gap implies the existence of a system-size independent positive modified logarithmic Sobolev inequality (MLSI) constant, leading to optimal rapid mixing at all positive temperatures. 2. **Sub-linear Thermalization in 2D**: For 2D systems with Davies generators of commuting, nearest-neighbour Hamiltonians, a positive gap and a sufficiently small correlation length imply the existence of a strictly positive square root-decreasing MLSI constant, resulting in a mixing time that scales at worst with the square root of the system size. 3. **Rapid Thermalization at High Temperature**: Nearest-neighbour, commuting potentials at sufficiently high temperatures satisfy an MLSI with either a system-size independent constant or a log-decreasing constant, depending on the graph structure. The authors introduce a novel notion called "strong local indistinguishability" based on max-relative entropy, which is linked to the strategies for proving rapid mixing. They also show that several measures of correlation decay on Gibbs states of commuting Hamiltonians are equivalent, providing independent interest in the study of correlation properties. The paper has implications for the rate of thermalization towards Gibbs states, Wasserstein distances, and transportation cost inequalities. The results are particularly relevant for 1D systems and higher-dimensional lattices, including exponential graphs like trees.The paper "Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians" by Jan Kochanowski, Álvaro M. Alhambra, Ángela Capel, and Cambysse Rouzé explores the rapid thermalization of quantum systems coupled to a thermal environment. The authors focus on geometrically-2-local models of Davies generators with commuting Hamiltonians and show that the thermalization time is much shorter than what one might naively estimate from the spectral gap. Specifically, they prove that the thermalization time is at most logarithmic in the system size, achieving rapid mixing of dissipative dynamics. Key results include: 1. **Optimal Rapid Thermalization in 1D**: For 1D systems with Davies generators of commuting, local Hamiltonians, having a positive gap implies the existence of a system-size independent positive modified logarithmic Sobolev inequality (MLSI) constant, leading to optimal rapid mixing at all positive temperatures. 2. **Sub-linear Thermalization in 2D**: For 2D systems with Davies generators of commuting, nearest-neighbour Hamiltonians, a positive gap and a sufficiently small correlation length imply the existence of a strictly positive square root-decreasing MLSI constant, resulting in a mixing time that scales at worst with the square root of the system size. 3. **Rapid Thermalization at High Temperature**: Nearest-neighbour, commuting potentials at sufficiently high temperatures satisfy an MLSI with either a system-size independent constant or a log-decreasing constant, depending on the graph structure. The authors introduce a novel notion called "strong local indistinguishability" based on max-relative entropy, which is linked to the strategies for proving rapid mixing. They also show that several measures of correlation decay on Gibbs states of commuting Hamiltonians are equivalent, providing independent interest in the study of correlation properties. The paper has implications for the rate of thermalization towards Gibbs states, Wasserstein distances, and transportation cost inequalities. The results are particularly relevant for 1D systems and higher-dimensional lattices, including exponential graphs like trees.