When describing the effective dynamics of an observable in a many-body system, the repeated randomness assumption, which states that the system returns in a short time to a maximum entropy state, is a crucial hypothesis to guarantee that the effective dynamics is classical and Markovian and obeys local detailed balance. While the latter behavior is frequently observed in naturally occurring processes, the repeated randomness assumption is in blatant contradiction to the microscopic reversibility of the system. Here we show that the use of the repeated randomness assumption can be justified in the description of the effective dynamics of an observable that is both slow and coarse, two properties we will define rigorously. Then our derivation will invoke essentially only the eigenstate thermalization hypothesis and typicality arguments. While the assumption of a slow observable is subtle, as it provides only a necessary but not sufficient condition, it also offers a unifying perspective applicable to, e.g., open systems as well as collective observables of many-body systems. All our ideas are numerically verified by studying density waves in spin chains.

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• Collective effects in atomic physics
• Decoherence in quantum gases
• Eigenstate thermalization
• Entropy
• Irreversible processes
• Nonequilibrium statistical mechanics
• Open quantum systems
• Quantum statistical mechanics
• Quantum stochastic processes
• Stochastic thermodynamics
• Quantum many-body systems
• Quantum spin models
• Coarse graining