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Physical Review Letters

Causality of the Nonequilibrium Correlation Functions

F. X. Murphy, Guido V. Sandri

Published Aug 16, 1971• 3 citations
Publisher

Abstract

It is shown that the correct lowest-order two-particle correlation function solution to the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy is given via a Green's function whose Fourier transform satisfies the analyticity requirements for causality. The Bogoliubov adiabatic approximation is shown to imply a solution which is acausal. Extension to the $s$-particle correlation function is discussed.

DOI
10.1103/physrevlett.27.381

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It is shown that the correct lowest-order two-particle correlation function solution to the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy is given via a Green's function whose Fourier transform satisfies the analyticity requirements for causality. The Bogoliubov adiabatic approximation is shown to imply a solution which is acausal. Extension to the sss-particle correlation function is discussed.
We consider the problem of the rigorous description of nonequilibrium quantum correlations. Within the framework of an alternative approach to the description of the evolution of states of finitely many particles in terms of correlation operators governed by the von Neumann hierarchy, we derive the nonlinear quantum Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy for marginal correlation operators that is adopted to the description of quantum infinite‐particle systems as well. A nonperturbative solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy for marginal correlation operators is constructed. Copyright © 2013 John Wiley & Sons, Ltd.
The rigorous evolution equations for the one-body Green's functions (g>,g<{g}^{>},{g}^{<}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, NNN (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<{g}^{>},{g}^{<}g>,g<).
The nonequilibrium description of quantum systems requires, for more than two or three particles, the use of a reduced description to be numerically tractable. Two possible approaches are based on either reduced density matrices or nonequilibrium Green functions (NEGF). Both concepts are formulated in terms of hierarchies of coupled equations—the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for the reduced density operators and the Martin-Schwinger-hierarchy (MS) for the Green functions, respectively. In both cases, similar approximations are introduced to decouple the hierarchy, yet still many questions regarding the correspondence of both approaches remain open. Here we analyze this correspondence by studying the generalized Kadanoff–Baym ansatz (GKBA) that reduces the NEGF to a single-time theory. Starting from the BBGKY-hierarchy we present the approximations that are necessary to recover the GKBA result both, with Hartree-Fock propagators (HF-GKBA) and propagators in second Born approximation. To test the quality of the HF-GKBA, we study the dynamics of a 4-electron Hubbard nanocluster starting from a strong nonequilibrium initial state and compare to exact results and the Wang-Cassing approximation to the BBGKY hierarchy presented recently by Akbari et al. [1].
We consider the dynamics of |NNN| bosons in three dimensions. We assume that the pair interaction is given by |N3β−1V(Nβ⋅)N^{3\beta -1}V(N^{\beta }\cdot )N3β−1V(Nβ⋅)|⁠. By studying an associated many-body wave operator, we introduce a Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy which takes into account all of the interparticle singular correlation structures developed by the many-body evolution from the beginning. Assuming energy conditions on the |NNN|-body wave function, for |$\beta \in ( 0,1] ∣⁠,wederivetheGross–Pitaevskiihierarchywith2−bodyinteraction.Inparticular,weestablishthat,inthe∣|⁠, we derive the Gross–Pitaevskii hierarchy with 2-body interaction. In particular, we establish that, in the |∣⁠,wederivetheGross–Pitaevskiihierarchywith2−bodyinteraction.Inparticular,weestablishthat,inthe∣N\rightarrow \infty ∣limit,all∣| limit, all |∣limit,all∣k∣−bodyscatteringprocessesvanishif∣|-body scattering processes vanish if |∣−bodyscatteringprocessesvanishif∣k\geqslant 3∣andthusprovideadirectanswertoaquestionraisedbyErdo¨setal.[30].Moreover,thisnewBBGKYhierarchysharesthelimitpointswiththeordinaryBBGKYhierarchystronglyfor∣| and thus provide a direct answer to a question raised by Erdös et al. [30]. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for |∣andthusprovideadirectanswertoaquestionraisedbyErdo¨setal.[30].Moreover,thisnewBBGKYhierarchysharesthelimitpointswiththeordinaryBBGKYhierarchystronglyfor∣\beta \in ( 0,1) ∣andweaklyfor∣| and weakly for |∣andweaklyfor∣\beta =1∣⁠.SincethisnewBBGKYhierarchyconvertstheproblemfromatwo−bodyestimatetoaweakerthree−bodyestimateforwhichwehavetheestimatestoachieve∣|⁠. Since this new BBGKY hierarchy converts the problem from a two-body estimate to a weaker three-body estimate for which we have the estimates to achieve |∣⁠.SincethisnewBBGKYhierarchyconvertstheproblemfromatwo−bodyestimatetoaweakerthree−bodyestimateforwhichwehavetheestimatestoachieve∣\beta <1∣⁠,itthenallowsustoprovethatalllimitpointsoftheordinaryBBGKYhierarchysatisfythespace–timeboundconjecturedbyKlainermanandMachedon[48]for∣|⁠, it then allows us to prove that all limit points of the ordinary BBGKY hierarchy satisfy the space–time bound conjectured by Klainerman and Machedon [48] for |∣⁠,itthenallowsustoprovethatalllimitpointsoftheordinaryBBGKYhierarchysatisfythespace–timeboundconjecturedbyKlainermanandMachedon[48]for∣\beta \in (0,1)$|⁠.