Statistical mechanics for non-reciprocal forces

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.