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Physical Review Letters

Higher Order Behavior in the Boltzmann Expansion of the Bogoliubov-Born-Green-Kirkwood-Yvon Hierarchy

R. Goldman

Published Oct 24, 1966• 5 citations
Publisher

Abstract

Equations governing divergent many-body behavior within BBGKY hierarchy in Boltzmann approximation for cut-off potentials, noting nondivergent nature of solutions
DOI
10.1103/physrevlett.17.910

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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 |$\beta \in ( 0,1] ∣⁠,wederivetheGross–Pitaevskiihierarchywith2−bodyinteraction.Inparticular,weestablishthat,inthe∣|⁠, we derive the Gross–Pitaevskii hierarchy with 2-body interaction. In particular, we establish that, in the |∣⁠,wederivetheGross–Pitaevskiihierarchywith2−bodyinteraction.Inparticular,weestablishthat,inthe∣N\rightarrow \infty ∣limit,all∣| limit, all |∣limit,all∣k∣−bodyscatteringprocessesvanishif∣|-body scattering processes vanish if |∣−bodyscatteringprocessesvanishif∣k\geqslant 3∣andthusprovideadirectanswertoaquestionraisedbyErdo¨setal.[30].Moreover,thisnewBBGKYhierarchysharesthelimitpointswiththeordinaryBBGKYhierarchystronglyfor∣| and thus provide a direct answer to a question raised by Erdös et al. [30]. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for |∣andthusprovideadirectanswertoaquestionraisedbyErdo¨setal.[30].Moreover,thisnewBBGKYhierarchysharesthelimitpointswiththeordinaryBBGKYhierarchystronglyfor∣\beta \in ( 0,1) ∣andweaklyfor∣| and weakly for |∣andweaklyfor∣\beta =1∣⁠.SincethisnewBBGKYhierarchyconvertstheproblemfromatwo−bodyestimatetoaweakerthree−bodyestimateforwhichwehavetheestimatestoachieve∣|⁠. Since this new BBGKY hierarchy converts the problem from a two-body estimate to a weaker three-body estimate for which we have the estimates to achieve |∣⁠.SincethisnewBBGKYhierarchyconvertstheproblemfromatwo−bodyestimatetoaweakerthree−bodyestimateforwhichwehavetheestimatestoachieve∣\beta <1∣⁠,itthenallowsustoprovethatalllimitpointsoftheordinaryBBGKYhierarchysatisfythespace–timeboundconjecturedbyKlainermanandMachedon[48]for∣|⁠, it then allows us to prove that all limit points of the ordinary BBGKY hierarchy satisfy the space–time bound conjectured by Klainerman and Machedon [48] for |∣⁠,itthenallowsustoprovethatalllimitpointsoftheordinaryBBGKYhierarchysatisfythespace–timeboundconjecturedbyKlainermanandMachedon[48]for∣\beta \in (0,1)$|⁠.
We establish a theoretical framework for exploring the quantum dynamics of finite ultracold bosonic ensembles based on the Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy for equations of motion for few-particle reduced density matrices (RDMs). The theory applies to zero as well as low temperatures and is formulated in a highly efficient way by utilizing dynamically optimized single-particle basis states and representing the RDMs in terms of permanents with respect to those. An energy, RDM compatibility, and symmetry conserving closure approximation is developed on the basis of a recursively formulated cluster expansion for these finite systems. In order to enforce necessary representability conditions, two minimally invasive and energy-conserving correction algorithms are proposed, involving the dynamical purification of the solution of the truncated BBGKY hierarchy and the correction of the equations of motion themselves, respectively. For gaining conceptual insights, the impact of two-particle correlations on the dynamical quantum depletion is studied analytically. We apply this theoretical framework to both a tunneling and an interaction-quench scenario. Due to our efficient formulation of the theory, we can reach truncation orders as large as twelve and thereby systematically study the impact of the truncation order on the results. While the short-time dynamics is found to be excellently described with controllable accuracy, significant deviations occur on a longer timescale in sufficiently far off-equilibrium situations. Theses deviations are accompanied by exponential-like instabilities leading to unphysical results. The phenomenology of these instabilities is investigated in detail and we show that the minimally invasive correction algorithm of the equation of motion can indeed stabilize the BBGKY hierarchy truncated at the second order.
We establish a theoretical framework for exploring the quantum dynamics of finite ultracold bosonic ensembles based on the Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations of motion for few-particle reduced density matrices (RDMs). The theory applies to zero as well as low temperatures and is formulated in a highly efficient way by utilizing dynamically optimized single-particle basis states and representing the RDMs in terms of permanents with respect to those. An energy, RDM compatibility and symmetry conserving closure approximation is developed on the basis of a recursively formulated cluster expansion for these finite systems. In order to enforce necessary representability conditions, two novel, minimal-invasive and energy-conserving correction algorithms are proposed, involving the dynamical purification of the solution of the truncated BBGKY hierarchy and the correction of the equations of motion themselves, respectively. For gaining conceptual insights, the impact of two-particle correlations on the dynamical quantum depletion is studied analytically. We apply this theoretical framework to both a tunneling and an interaction-quench scenario. Due to our efficient formulation of the theory, we can reach truncation orders as large as twelve and thereby systematically study the impact of the truncation order on the results. While the short-time dynamics is found to be excellently described with controllable accuracy, significant deviations occur on a longer time-scale in sufficiently far off-equilibrium situations. Theses deviations are accompanied by exponential-like instabilities leading to unphysical results. The phenomenology of these instabilities is investigated in detail and we show that the minimal-invasive correction algorithm of the equation of motion can indeed stabilize the BBGKY hierarchy truncated at the second order.