  If we prepare an isolated, interacting quantum system in an eigenstate and
perturb a local observable at an initial time, its expectation value will relax
towards a thermal expectation value, even though the time evolution of the
system is deterministic. The eigenstate thermalization hypothesis (ETH) of
Deutsch and Srednicki suggests that this is possible because each eigenstate of
the full quantum system acts as a thermal bath to its subsystems, such that the
reduced density matrices of the subsystems resemble thermal density matrices.
Here, we use the observation that the eigenvalue distribution of interacting
quantum systems is a Gaussian under very general circumstances, and Dyson
Brownian motion random matrix theory, to derive the ETH and thereby elevate it
from hypothesis to theory. Our analysis provides a derivation of statistical
mechanics which neither requires the concepts of ergodicity or typicality, nor
that of entropy. Thermodynamic equilibrium follows solely from the
applicability of quantum mechanics to large systems and the absence of
integrability.
