  We give rigorous analytical results on the temporal behavior of two-point
correlation functions --also known as dynamical response functions or Green's
functions-- in closed many-body quantum systems. We show that in a large class
of translation-invariant models the correlation functions factorize at late
times $\langle A(t) B\rangle_\beta \rightarrow \langle A \rangle_\beta \langle
B \rangle_\beta$, thus proving that dissipation emerges out of the unitary
dynamics of the system. We also show that for systems with a generic spectrum
the fluctuations around this late-time value are bounded by the purity of the
thermal ensemble, which generally decays exponentially with system size. For
auto-correlation functions we provide an upper bound on the timescale at which
they reach the factorized late time value. Remarkably, this bound is only a
function of local expectation values, and does not increase with system size.
We give numerical examples that show that this bound is a good estimate in
non-integrable models, and argue that the timescale that appears can be
understood in terms of an emergent fluctuation-dissipation theorem. Our study
extends to further classes of two point functions such as the symmetrized ones
and the Kubo function that appears in linear response theory, for which we give
analogous results.
