Equations of motion in nonequilibrium statistical mechanics of open systems

In this paper the method of Robertson and that of Zubarev, which have been applied to isolated systems, are modified by using a special projection operator or a special density operator so that they become applicable to open systems. As a result, exact equations are obtained in the form of coupled integro-differential equations with expectation values corresponding to a set of operators of an open system $S$ as the only unknowns. The variables of the system $R$, which interacts with the system $S$, are completely eliminated up to the expectation values taken over the initial state of $R$. The differences between the above mentioned two methods are discussed. It is shown that for special choice of the initial conditions, sets of operators and properties of system $R$, significant simplifications of the equations of motion can be made. Moreover, an expansion of the equations of motion in powers of the interaction and an approximation of the second order are made. Finally, a Kawasaki-Gunton modification of our projection operator is made in the Appendix.