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

by Dr Lorenzo Campos Venuti (Univerisity of Souther California)




Boltzmann ergodic hypothesis is a possible way to explain the emergence of statistical mechanics in the classical world. In quantum mechanics instead, the so called Eigenstate Thermalization Hypothesis (ETH) is generally considered to be a possible route to thermalization. The notion of ergodicity itself is less clear in the quantum world and often it is simply taken as a synonym for thermalization. Here I will show, in an elementary way, that when quantum ergodicity is properly defined, ETH is in fact equivalent to the latter. In turn ergodicity is equivalent to thermalization thus implying equivalence between thermalization and ETH. I will also show that it is possible to define a classical analogue of ETH which is implicitly assumed to be satisfied when constructing classical statistical mechanics. Classical and quantum statistical mechanics are built according to the familiar standard prescription. This prescription, however, is ontologically justified only in the quantum world.