  This paper is devoted to $\phi$-entropies applied to Fokker-Planck and
kinetic Fokker-Planck equations in the whole space, with confinement. The
so-called $\phi$-entropies are Lyapunov functionals which typically interpolate
between Gibbs entropies and L2 estimates. We review some of their properties in
the case of diffusion equations of Fokker-Planck type, give new and simplified
proofs, and then adapt these methods to a kinetic Fokker-Planck equation acting
on a phase space with positions and velocities. At kinetic level, since the
diffusion only acts on the velocity variable, the transport operator plays an
essential role in the relaxation process. Here we adopt the H1 point of view
and establish a sharp decay rate. Rather than giving general but quantitatively
vague estimates, our goal here is to consider simple cases, benchmark available
methods and obtain sharp estimates on a key example. Some $\phi$-entropies give
rise to improved entropy -- entropy production inequalities and, as a
consequence, to faster decay rates for entropy estimates of solutions to
non-degenerate diffusion equations. Our main result is to prove that faster
entropy decay also holds at kinetic level and that optimal decay rates are
achieved only in asymptotic regimes.
