  The dissipation of turbulent magnetic fields is an appealing scenario to
explain the origin of non-thermal particles in high-energy astrophysical
sources. However, it has been suggested that the particle distribution may
effectively thermalise when the radiative (synchrotron and/or Inverse Compton)
losses are severe. Inspired by recent PIC simulations of relativistic
turbulence, which show that electrons are impulsively heated in intermittent
current sheets by a strong electric field aligned with the local magnetic
field, we instead argue that in plasmas where the particle number density is
dominated by the pairs (electron-positron and electron-positron-ion plasmas):
(i) as an effect of fast cooling and of different injection times, the electron
energy distribution is ${\rm d}n_e/{\rm d}\gamma\propto\gamma^{-2}$ for
$\gamma\lesssim\gamma_{\rm heat}$ (the Lorentz factor $\gamma_{\rm heat}$ being
close to the equipartition value), while the distribution steepens at higher
energies; (ii) since the time scales for the turbulent fields to decay and for
the photons to escape are of the same order, the magnetic and the radiation
energy densities in the dissipation region are comparable; (iii) if the mass
energy of the plasma is dominated by the ion component, the pairs with a
Lorentz factor smaller than a critical one (of the order of the
proton-to-electron mass ratio) become isotropic, while the pitch angle remains
small otherwise. The outlined scenario is consistent with the typical
conditions required to reproduce the Spectral Energy Distribution of blazars,
and allows one to estimate the magnetisation of the emission site. Finally, we
show that turbulence within the Crab Nebula may power the observed gamma-ray
flares if the pulsar wind is nearly charge-separated at high latitudes.
