  When the symmetries of homogenous isotropic turbulent flows are broken,
different sets of modes with different physical roles emerge. In particular,
choosing a forcing which puts more weight on one or the other of these sets may
result in different statistics for the energy transfers. We use the general
method of computing a partition function restricted to a portion of phase space
to study analytically these different statistics. We illustrate this method in
the case of parity symmetry breaking, measured by helicity. It is shown that
when helicity is sign definite at all scales, an inverse cascade is expected
for the energy. When sign-definiteness is lost, even for a small set of modes,
this cascade disappears and there is a sharp phase transition to the standard
helical equipartition spectra.
