The rigorous evolution equations for the one-body Green's functions ( ${g}^{>},{g}^{<}$ ) describing a system of interacting fermions are not closed (an analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for many-body distribution functions). The analysis of these equations is made in terms of proper connected diagrams. It is established that in the bulk limit, $N$ (particle number) $\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}$ (volume) $\ensuremath{\rightarrow}\ensuremath{\infty}$ while $\frac{N}{\ensuremath{\Omega}}$ remains finite, these equations can rigorously be transformed into closed equations with respect to ( ${g}^{>},{g}^{<}$ ).The rigorous evolution equations for the one-body Green's functions (${g}^{>},{g}^{<}$) describing a system of interacting fermions are not closed (an analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for many-body distribution functions). The analysis of these equations is made in terms of proper connected diagrams. It is established that in the bulk limit, $N$ (particle number) $\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}$ (volume) $\ensuremath{\rightarrow}\ensuremath{\infty}$ while $\frac{N}{\ensuremath{\Omega}}$ remains finite, these equations can rigorously be transformed into closed equations with respect to (${g}^{>},{g}^{<}$).

Physical Review ASep 1, 197128 citations

Computational methods for the one-particle green's function

Wolfgang von Niessen, J. Schirmer, Lorenz S. Cederbaum

Computer Physics ReportsApr 1, 19841,127 citations

Thermodynamic Evolution Equation for a Quantum Statistical Gas

Shigeji Fujita

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.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.

Journal of Mathematical PhysicsDec 1, 196521 citations

Resolution of the Hierarchy of Many-Body Green's Functions

Shigeji Fujita

The rigorous evolution equations for the one-body Green's functions ( g > , g < ) describing an imperfect gas in a finite volume are not closed (an analogue of B–B–G–K–Y hierarchy for the many-body distribution functions). The analysis of these equations are made in terms of proper connected diagrams. It is shown that in the bulk limit: N (particle number)→∞, $\varOmega$ (volume)→∞ such that N /Ω=constant, these equations can be reduced to the closed equations, which are however, non-linear and non-Markoffian, in contrast to the original hierarchy equations.The rigorous evolution equations for the one-body Green's functions ( g > , g < ) describing an imperfect gas in a finite volume are not closed (an analogue of B–B–G–K–Y hierarchy for the many-body distribution functions). The analysis of these equations are made in terms of proper connected diagrams. It is shown that in the bulk limit: N (particle number)→∞, $\varOmega$(volume)→∞ such that N /Ω=constant, these equations can be reduced to the closed equations, which are however, non-linear and non-Markoffian, in contrast to the original hierarchy equations.

Journal of the Physical Society of JapanNov 1, 196912 citations

The rigorous evolution equations for the one-body Green's functions (

{g}^{>},{g}^{<}

) describing a system of interacting fermions are not closed (an analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for many-body distribution functions). The analysis of these equations is made in terms of proper connected diagrams. It is established that in the bulk limit,

N

(particle number) $\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}$ (volume) $\ensuremath{\rightarrow}\ensuremath{\infty}$ while $\frac{N}{\ensuremath{\Omega}}$ remains finite, these equations can rigorously be transformed into closed equations with respect to (

{g}^{>},{g}^{<}

\varOmega

(volume)→∞ such that N /Ω=constant, these equations can be reduced to the closed equations, which are however, non-linear and non-Markoffian, in contrast to the original hierarchy equations.