Gaussian Equilibration

A finite quantum system evolving unitarily equilibrates in a probabilistic fashion. In the general many-body setting the time-fluctuations of an observable \mathcal{A} are typically exponentially small in the system size. We consider here quasi-free Fermi systems where the Hamiltonian and observables are quadratic in the Fermi operators. We first prove a novel bound on the temporal fluctuations \Delta\mathcal{A}^{2} and then map the equilibration dynamics to a generalized classical XY model in the infinite temperature limit. Using this insight we conjecture that, in most cases, a central limit theorem can be formulated leading to what we call Gaussian equilibration: observables display a Gaussian distribution with relative error \Delta\mathcal{A}/\bar{\mathcal{A}}=O(L^{-1/2}) where L is the dimension of the single particle space. The conjecture, corroborated by numerical evidence, is proven analytically under mild assumptions for the magnetization in the quantum XY model and for a class of observables in a tight-binding model. We also show that the variance is discontinuous at the transition between a quasi-free model and a non-integrable one.