  We review and complete the kinetic theory of spatially inhomogeneous stellar
systems when collective effects (dressing of the stars by their polarization
cloud) are neglected. We start from the BBGKY hierarchy issued from the
Liouville equation and consider an expansion in powers of 1/N in a proper
thermodynamic limit. For $N\rightarrow +\infty$, we obtain the Vlasov equation
describing the evolution of collisionless stellar systems like elliptical
galaxies. At the order 1/N, we obtain a kinetic equation describing the
evolution of collisional stellar systems like globular clusters. This equation
does not suffer logarithmic divergences at large scales since spatial
inhomogeneity is explicitly taken into account. Making a local approximation,
and introducing an upper cut-off at the Jeans length, it reduces to the
Vlasov-Landau equation which is the standard kinetic equation of stellar
systems. Our approach provides a simple and pedagogical derivation of these
important equations from the BBGKY hierarchy which is more rigorous for systems
with long-range interactions than the two-body encounters theory. Making an
adiabatic approximation, we write the generalized Landau equation in
angle-action variables and obtain a Landau-type kinetic equation that is valid
for fully inhomogeneous stellar systems and is free of divergences at large
scales. This equation is less general than the Lenard Balescu-type kinetic
equation recently derived by Heyvaerts (2010) since it neglects collective
effects, but it is substantially simpler and could be useful as a first step.
We discuss the evolution of the system as a whole and the relaxation of a test
star in a bath of field stars. We derive the corresponding Fokker-Planck
equation in angle-action variables and provide expressions for the diffusion
coefficient and friction force.
