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I’ve been told that the fact that quantum systems tend to become more entangled over time shows how statistical mechanics can arise from quantum mechanics, but how exactly does this work?

Is it just the idea of particles being equally likely to be in any of the microstates or is there something more fundamental?

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It is something more fundamental, and the answer is the eigenstate thermalization hypothesis (ETH). Closed quantum systems that fulfill this ansatz will thermalize (so the ETH is a sufficient condition, but it is not proved if it is necessary. So far all systems that thermalize, fulfill the ETH), and this thermalization can be described with the tools of statistical mechanics. Then, it has been shown in numerical experiments that the entanglement entropy (for a pure non-entangle product state) follows a ballistic growth ($S(t)\sim t$) until it saturates. So using this tool, you can try to see if your system thermalizes, being described by statistical mechanics, or if it violates the ETH and it cannot be described by statistical mechanics. However, it is not just a matter of observing the growth of entanglement, because you can find examples, like the many-body localization phenomenon, where there is a growth of entanglement entropy (logarithmic in this case) but the system violates the ETH. If you are interested in the topic, check the following review about ETH: https://www.tandfonline.com/doi/full/10.1080/00018732.2016.1198134

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