Stationary Entropy Principle and Renormalization in Normal and Superfluid Systems. II. Diagrammatic Formulation

Abstract

Quantum statistical mechanics is renormalized, that is, the thermodynamical functions and the ``bare'' ν-body potentials vν occurring in the Hamiltonian are expressed as functionals of the ν-body distribution functions Gν (the expectation values of 2ν field operators). The field operators are considered as having two components (creation and annihilation), and Gν thus has (2)2ν components. The renormalization is carried out for superfluid Bose systems for which it is necessary to consider the expectation values of 2ν = 1, 2, 3, 4 operators. Superfluid Fermi systems and normal systems appear as particular cases, described by expectation values of 2ν = 2, 4 operators. This problem, dealt with by algebraic methods in part I, is attacked here by diagrammatic methods. These methods are more dependent on convergence properties but the resulting functionals are obtained explicitly as power series of the Gν′s, the general term being represented by a class of diagrams characterized by their topological structure. In the final result, the entropy is exhibited as an explicit functional of the distribution functions Gν, (2ν = 1, 2, 3, 4), or more precisely, of functions directly related to them. This functional no longer involves the potentials (nor the equilibrium parameters). It is stationary under independent variations of the Gν′s, subject to the constraints of constant energy and particle number. The four equations of stationarity exhibit each function vν as a functional of the Gν′s, self-consistently defining these distribution functions.