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Boltzmann's ergodic hypothesis furnishes a possible explanation for theemergence of statistical mechanics in the framework of classical physics. Inquantum mechanics, the Eigenstate Thermalization Hypothesis (ETH) is insteadgenerally considered as a possible route to thermalization. This is because thenotion of ergodicity itself is vague in the quantum world and it is oftensimply taken as a synonym for thermalization. Here we show, in an elementaryway, that when quantum ergodicity is properly defined, it is, in fact,equivalent to ETH. In turn, ergodicity is equivalent to thermalization, thusimplying the equivalence of thermalization and ETH. This result previouslyappeared in [De Palma et al., Phys. Rev. Lett. 115, 220401 (2015)], but becomesparticularly clear in the present context. We also show that it is possible todefine a classical analogue of ETH which is implicitly assumed to be satisfiedwhen constructing classical statistical mechanics. Classical and quantumstatistical mechanics are built according to the familiar standardprescription. This prescription, however, is ontologically justified only inthe quantum world.

Ergodicity, eigenstate thermalization, and the foundations of statistical mechanics in quantum and classical systems