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Progress of Theoretical Physics

The Evolution Equation of Quantum Gases

Lesser Blum

Published Feb 1, 1963• 0 citations
Publisher

Abstract

The terms of the series obtained by iteration of the von Neumann's equation for a quantum gas are summed by a method which groups only connected and disconnected diagrams. The main hypothesis made is “random a priori phases” at the initial time t0 = 0; if we assume that it is diagonal at a time t, too (repeated random a priori phases) we get the Pauli equation, provided this time t is much longer than the collision time.

DOI
10.1143/ptp.29.213

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The terms of the series obtained by iteration of the von Neumann's equation for a quantum gas are summed by a method which groups only connected and disconnected diagrams. The main hypothesis made is “random a priori phases” at the initial time t0 = 0; if we assume that it is diagonal at a time t, too (repeated random a priori phases) we get the Pauli equation, provided this time t is much longer than the collision time.
We discuss the derivation of the Pauli master equation, based on a repeated random phase assumption, and of van Hove's result, based on an initial random phase assumption. For the former we indicate a derivation which is closer to the general approach of stochastic theory than Pauli's original method. For the van Hove result, we show that the diagonal and nondiagonal parts of the evolution operator of the Schrödinger or von Neumann equation are readily obtained by Zwanzig's projection operator method.
It is shown by a perturbation diagram method that partial Green's functions satisfy closed evolution equations exact in the thermodynamic limit except for the neglect of the contribution of initial particle correlations. This contribution is negligible for a system at a time t ≫ τc (average collision duration) when the main interaction processes are localized in space and time. In particular this makes unnecessary the often imposed assumption of the absence of initial particle correlations for the derivation of the usual Boltzmann equation or its generalizations.
We present a method for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The main differences with respect to previous approaches are underlined and discussed. In particular, no use of renormalized operators is made. By means of the method of linearization of equations of motion, we derive a set of RPA-like equations depending only on the one-body density matrix. The latter is no more assumed to be diagonal and it is expressed in terms of the XXX and YYY amplitudes of the particle-hole phonon operators. This set of nonlinear equations is solved via an iterative procedure, which allows us to calculate the energies and the wave functions of the excited states. After presenting our approach in its general formulation, we test its quality by using metal clusters as a good test laboratory for a generic many-body system. Comparison to RPA shows significant improvements. We discuss also how the present approach can be further extended beyond the particle-hole configuration space, getting an approximation scheme in which energy weighted sum rules are exactly preserved, thus solving a problem common to all extension of RPA proposed until now. The implementation of such approach will be afforded in a future work.