  Starting from the Liouville equation, we derive the exact hierarchy of
equations satisfied by the reduced distribution functions of the single species
point vortex gas in two dimensions. Considering an expansion of the solution in
powers of 1/N in a proper thermodynamic limit $N\to +\infty$, and neglecting
some collective effects, we derive a kinetic equation satisfied by the smooth
vorticity field which is valid at order $O(1/N)$. This equation was obtained
previously [P.H. Chavanis, Phys. Rev. E, 64, 026309 (2001)] from a more
abstract projection operator formalism. If we consider axisymmetric flows and
make a markovian approximation, we obtain a simpler kinetic equation which can
be studied in great detail. We discuss the properties of these kinetic
equations in regard to the $H$-theorem and the convergence (or not) towards the
statistical equilibrium state. We also study the growth of correlations by
explicitly calculating the time evolution of the two-body correlation function
in the linear regime. In a second part of the paper, we consider the relaxation
of a test vortex in a bath of field vortices and obtain the Fokker-Planck
equation by directly calculating the second (diffusion) and first (drift)
moments of the increment of position of the test vortex. A specificity of our
approach is to obtain general equations, with a clear physical meaning, that
are valid for flows that are not necessarily axisymmetric and that take into
account non-Markovian effects. A limitations of our approach, however, is that
it ignores collective effects.
