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

Understand

Boltzmann's ergodic hypothesis furnishes a possible explanation for the emergence of statistical mechanics in the framework of classical physics.

• In quantum mechanics, the Eigenstate Thermalization Hypothesis (ETH) is instead generally considered as a possible route to thermalization.
• This is because the notion of ergodicity itself is vague in the quantum world and it is often simply taken as a synonym for thermalization.
• Here we show, in an elementary way, that when quantum ergodicity is properly defined, it is, in fact, equivalent to ETH.