Hierarchy of equations for reduced density matrices in the case of thermodynamic equilibrium

A hierarchy of equations for s-particle density matrices at thermodynamic equilibrium is obtained, with the equation for the nonequilibrium density matrix as the starting point. When deducing the hierarchy the hypothesis of maximum statistical independence for the density matrices is used. The hierarchy obtained is an analogue of the classical equilibrium BBGKY hierarchy and goes over into it when . It is shown that thermodynamic quantities can be expressed in terms of functions that enter only into the first hierarchy equations. The hierarchy is analysed in detail in the case of a uniform fluid. As an example in which the equations can be solved easily enough, a hard-sphere system wherein triplet correlations are neglected is considered. Different approximations that can be used when solving the equations derived are discussed. Comparisons are made with the results of other theoretical treatments.