  It has been recently shown that small subsystems of finite quantum systems
generically equilibrate. We extend these results to infinite-dimensional
Hilbert spaces of field theories and matrix models. We consider a quench setup,
where initial states are chosen from a microcanonical ensemble of finite energy
in free theory, and then evolve with an arbitrary non-perturbative Hamiltonian.
Given a dynamical assumption on the expectation value of particle number
density, we prove that small subsystems reach equilibrium at the level of
quantum wave-function, and with respect to all observables. The picture that
emerges is that at higher energies, larger subsystems can reach equilibrium.
For bosonic fields on a lattice, in the limit of large number of bosons per
site, all subsystem smaller than half equilibrate. In the Hermitian matrix
model, by contrast, this occurs in the limit of large energy per matrix
element, emphasizing the importance of the $O(N^2)$ energy scale for the fast
scrambling conjecture. Applying our techniques to continuum field theories on
compact spaces, we show that the density matrix of small momentum-space
observables equilibrate. Finally, we discuss the connection with scrambling,
and provide a sufficient condition for a time-independent Hamiltonian to be a
scrambler in terms of the entanglement entropy of its energy eigenstates.
