  A basic statistical mechanics analysis of many-body systems with
non-reciprocal pair interactions is presented. Different non-reciprocity
classes in two- and three-dimensional binary systems (relevant to real
experimental situations) are investigated, where the action-reaction symmetry
is broken for the interaction between different species. The asymmetry is
characterized by a non-reciprocity parameter $\Delta$, which is the ratio of
the non-reciprocal to reciprocal pair forces. It is shown that for the
"constant" non-reciprocity (when $\Delta$ is independent of the interparticle
distance $r$) one can construct a pseudo-Hamiltonian and such systems, being
intrinsically non-equilibrium, can nevertheless be described in terms of
equilibrium statistical mechanics and exhibit detailed balance with distinct
temperatures for the different species. For a general case (when $\Delta$ is a
function of $r$) the temperatures grow with time, approaching a universal
power-law scaling, while their ratio is determined by an effective constant
non-reciprocity which is uniquely defined for a given interaction.
