  We propose a Gaussian ensemble as a description of the long-time dynamics of
isolated quantum integrable systems. Our approach extends the Generalized Gibbs
Ensemble (GGE) by incorporating fluctuations of integrals of motion. It is
asymptotically exact in the classical limit irrespective of the system size
and, under appropriate conditions, in the thermodynamic limit irrespective of
the value of Planck's constant. Moreover, it captures quantum corrections near
the classical limit, finite size corrections near the thermodynamic limit, and
is valid in the presence of non-local interactions. The Gaussian ensemble
bridges the gap between classical integrable systems, where a generalized
microcanonical ensemble is exact even for few degrees of freedom and GGE, which
requires thermodynamic limit. We illustrate our results with examples of
increasing complexity.
