Equations of motion, Green's functions, and thermodynamic relations in theories of linear relaxation with various sets of macroscopic variables

A generalized scheme of a theory of linear relaxation in a macroscopic nonequilibrium system is investigated in the case when the set of macrovariables $\operatorname{Sp}P\rho(t)$ is enlarged by the average values of the first, second . . ., and $\alpha$-th time derivatives of the operators $P$ It is shown that for all values of $\alpha$ the same dispersion equation holds for the spectrum of normal modes of the system and also the same infinite system of linear equations. This system contains a finite number of equations of motion of the macrovariables and a hierarchy of equations for the two-time correlation functions which arise in the calculation of the memory function or Green's function.