Single-particle approximation to the canonical density operator of a system of N interacting fermions
A necessary condition (nc) for the minimum of free energy \(F(\rho)=Tr(\rho H^ N)+\beta^{-1}Tr(\rho \ell n\rho)\) is derived for a system of N interacting fermions (with the Hamiltonian \(H^ N)\) on the set of all density matrices of the form \(\rho =Z^{-1}\exp (-\beta N\Gamma_ 1^ Nh^ 1)\) where \(\Gamma_ 1^ N\) is Kummer's (1,N)- expansion map \((\Gamma^ N_ 1h^ 1=A^ N(h^ 1\otimes I^{\otimes (N-1)})A^ N,\) I being the identity, \(A^ N\) the antisymmetric projection) \(Z=Tr \exp (-\beta N\Gamma_ 1^ Nh^ 1),\) \(\beta >0\), and \(h^ 1\) is a finite-dimensional 1-particle operator. The nc is derived using the well-known variational principle of statistical mechanics which states that the free energy as a functional of the state of the system at fixed temperature \(T=(k\beta)^{-1},\) \(k>0\), and with a fixed number of particles N, reaches its minimum on the canonical density operator. This implies that the necessary condition takes the form \(\delta F(\rho)/\delta h^ 1=0,\) where \(\delta /\delta \rho\) is Fréchet's derivative. It is shown that for values of \(\beta\) in an open circle \(C(0,\beta_ 0)\) in the complex plane there exists a unique bounded analytic in \(\beta\) 1-particle operator \(h^ 1\) satisfying this condition nc. \(h^ 1\) has the form of a power series in \(\beta\) which is uniformly convergent in \(C(0,\beta_ 0)\). The consecutive coefficients of this series are single-particle projections of some definite N- particle operators. The method of calculating these projections, due to S. Pruski, is also presented.
variational principle of statistical mechanics
single-particle projections of some definite N-particle operators
scientific article; zbMATH DE number 3899592