Funktionalmittelwerte in der statistischen Theorie von quantenmechanischen und klassischen Systemen vieler Teilchen

The grand-canonical partition function of an interacting many-particle-system is represented as a functional integral with Gaussian random variables. The representation can be regarded as a Gaussian average over the partition function of free particles in an external fluctuating potential. The latter partition function is studied by means of diagrammatical techniques. The set of diagrams of a particularly simple structure is summed up by introducing the full scattering amplitude for the scattering in the external potential. The thermodynamicalGibbs' potential proves to be stationary with respect to the true particle density. It is shown that a variational procedure leads directly to an approximation which may be regarded as the renormalized form of the well-known Random-Phase-Approximation (RPA). The main feature of the approximation is thatGibbs' potential is stationary with respect to the two-particle-density correlation function. The classical limit of the renormalized RPA yields the results of the Debye-Hückel theory. In case of an hard-core potential the approximation applies only to the long-range part of the potential. The results are similar to recent developments in the theory of the Ising model and of real gases.