A new analytic solution for 2nd-order Fermi acceleration is presented .
In particular , we consider time-dependent rates for stochastic acceleration , diffusive and convective escape as well as adiabatic losses .
The power law index q of the turbulence spectrum is unconstrained and can therefore account for Kolmogorov ( q = 5 / 3 ) and Kraichnan ( q = 3 / 2 ) turbulence , Bohm diffusion ( q = 1 ) as well as the hard-sphere approximation ( q = 2 ) .
This considerably improves beyond solutions known to date and will prove a useful tool for more realistic modelling of 2nd-order Fermi acceleration in a variety of astrophysical environments .