  Never is the difference between thermal equilibrium and turbulence so
dramatic, as when a quadratic invariant makes the equilibrium statistics
exactly Gaussian with independently fluctuating modes. That happens in two very
different yet deeply connected classes of systems: incompressible hydrodynamics
and resonantly interacting waves. This work presents the first case of a
detailed information-theoretic analysis of turbulence in such strongly
interacting systems. The analysis elucidates the fundamental roles of space and
time in setting the cascade direction and the changes of the statistics along
it. We introduce a beautifully simple yet rich family of discrete models with
neighboring triplet interactions and show that it has families of quadratic
conservation laws defined by the Fibonacci numbers. Depending on the single
model parameter, three types of turbulence were found: single direct cascade,
double cascade, and the first ever case of a single inverse cascade. We
describe quantitatively how deviation from thermal equilibrium all the way to
turbulent cascades makes statistics increasingly non-Gaussian and find the
self-similar form of the one-mode probability distribution. We reveal where the
information (entropy deficit) is encoded and disentangle the communication
channels between modes, as quantified by the mutual information in pairs and
the interaction information inside triplets.
