Macroscopic Irreversibility in Quantum Systems: ETH and Equilibration in a Free Fermion Chain

Hal Tasaki

Published 2024-01-27Version 1

We consider a free fermion chain with a uniform nearest-neighbor hopping and a macroscopic number of particles. Fix any subset of the chain. For any initial state, we prove that, at a sufficiently large and typical time, the (measurement result of the) number of particles in the subset almost certainly equals its equilibrium value (corresponding to the uniform particle distribution). This establishes the emergence of irreversible behavior in a system governed by the quantum mechanical unitary time evolution. It is conceptually important that irreversibility is proved here without introducing any randomness to the initial state of the Hamiltonian, while the derivation of irreversibility in classical systems relies on certain randomness. The essential new ingredient in the proof is the justification of the strong ETH (energy eigenstate thermalization hypothesis) in the large-deviation form.