  We complete the literature on the statistical mechanics of point vortices in
two-dimensional hydrodynamics. Using a maximum entropy principle, we determine
the multi-species Boltzmann-Poisson equation and establish a form of virial
theorem. Using a maximum entropy production principle (MEPP), we derive a set
of relaxation equations towards statistical equilibrium. These relaxation
equations can be used as a numerical algorithm to compute the maximum entropy
state. We mention the analogies with the Fokker-Planck equations derived by
Debye and H\"uckel for electrolytes. We then consider the limit of strong
mixing (or low energy). To leading order, the relationship between the
vorticity and the stream function at equilibrium is linear and the maximization
of the entropy becomes equivalent to the minimization of the enstrophy. This
expansion is similar to the Debye-H\"uckel approximation for electrolytes,
except that the temperature is negative instead of positive so that the
effective interaction between like-sign vortices is attractive instead of
repulsive. This leads to an organization at large scales presenting
geometry-induced phase transitions, instead of Debye shielding. We compare the
results obtained with point vortices to those obtained in the context of the
statistical mechanics of continuous vorticity fields described by the
Miller-Robert-Sommeria (MRS) theory. At linear order, we get the same results
but differences appear at the next order. In particular, the MRS theory
predicts a transition between sinh and tanh-like \omega-\psi relationships
depending on the sign of Ku-3 (where Ku is the Kurtosis) while there is no such
transition for point vortices which always show a sinh-like \omega-\psi
relationship. We derive the form of the relaxation equations in the strong
mixing limit and show that the enstrophy plays the role of a Lyapunov
functional.
