Timescale for Macroscopic Equilibration in Isolated Quantum Systems: A Rigorous Derivation for Free Fermions

For a class of translation-invariant free-fermion systems including those with uniform nearest-neighbor hopping) on a d-dimensional L×⋯×L hypercubic lattice, we prove that, starting from an arbitrary pure initial state, the system equilibrates with respect to the coarse-grained density within a timescale of order L. This scaling is optimal, since there exist initial states whose equilibration requires time of order L. Our result establishes O(L) as the equilibration timescale, as is expected in normal macroscopic systems with a conserved quantity, such as total number of particles.