Equivalence of various methods in the statistical mechanics of irreversible processes

It is shown that the nonequilbrium statistical methods proposed in the authors investigations [1, 3] are equivalent in the sense that they lead to the same transport equations for macroscopic nonequilibrium systems if the flux operators satisfy conditions of correlation weakening. It is shown that the Robertson procedure for constructing the nonequilibrium statistical operator [4] is equivalent to the procedure of [3] if one chooses the lower limit of integration in Robertson's equation for the nonequilibrium statistical operator to be $t_0=-\infty$ and not $t_0=0$ and one introduces an infinitesimally small damping. It is shown that the Peletminskii–Yatsenko procedure for constructing the nonequilibrium statistical operator [5] corresponds to the procedure of [3] if memory effects are ignored in the latter. However, if one alters the boundary condition in the method of [5], replacing the “mixing” condition by evolution along the phase trajectory, the method of [5] becomes equivalent to the other methods [1–4].