  Is there really such a thing as weak turbulence? Here we analyze turbulence
of weakly interacting waves using the tools of information theory. It offers a
unique perspective for comparing thermal equilibrium and turbulence: the mutual
information between modes is shown to be stationary and small in equilibrium
but grows linearly with time in weak turbulence. We trace this growth to the
concentration of probability on the resonance surfaces, which can go all the
way to a singular measure. The surprising conclusion is that no matter how
small is the nonlinearity and how close to Gaussian is the statistics of any
single amplitude, a stationary phase-space measure is far from Gaussian, as
manifested by a large relative entropy. Though it might be upsetting to
practitioners of weak turbulence approach, this is a rare piece of good news
for turbulence modeling: the resolved scales carry significant information
about the unresolved scales. The mutual information between large and small
scales is the information capacity of turbulent cascade, setting the limit on
the representation of subgrid scales in turbulence modeling.
