Quantum Thermalization: Understanding the Dynamical Foundation of Quantum Thermodynamics

There is a long-standing question of connections between chaos and thermalization. In classical systems, chaos is usually defined through the exponential sensitivity of trajectories to small perturbations. It is believed that in many-particle systems chaos leads to ergodicity or thermalization where after long times the isolated systems reach the thermodynamic equilibrium. In single-particle systems the notions of chaos and ergodicity are, however, not equivalent and chaos does not always lead to thermalization. In the last three decades it was realized that thermalization in quantum systems can be understood through the eigenstate thermalization hypothesis, which is an extension of the random matrix theory to physical systems. Sometimes this hypothesis is associated with the notion of quantum chaos. I will argue that this is not accurate and the situation in quantum and classical systems is similar such that chaos should be distinguished from thermalization. In both quantum and classical systems chaos can be defined through the long-time response of physical observables. In this way the random matrix theory and the eigenstate thermalization hypothesis are connected to fast, typically exponential, relaxation of observables to equilibrium. Conversely the strongest chaos occurs close to integrable points, where thermalization is very slow such that the systems are very unstable to various perturbations and can be trapped in non-thermal states like turbulent flows for long times.