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arXiv:1409.1425 · International Mathematics Research Notices

Correlation Structures, Many-Body Scattering Processes, and the Derivation of the Gross–Pitaevskii Hierarchy

Xuwen Chen, Justin Holmer

Published Aug 5, 2015• 56 citations
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Abstract

We consider the dynamics of |$N$| bosons in three dimensions. We assume that the pair interaction is given by |$N^{3\beta -1}V(N^{\beta }\cdot )$|⁠. 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 |$N$|-body wave function, for |$\beta \in ( 0,1] $|⁠, we derive the Gross–Pitaevskii hierarchy with 2-body interaction. In particular, we establish that, in the |$N\rightarrow \infty $| limit, all |$k$|-body scattering processes vanish if |$k\geqslant 3$| and thus provide a direct answer to a question raised by Erdös et al. [30]. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for |$\beta \in ( 0,1) $| and weakly for |$\beta =1$|⁠. Since this new BBGKY hierarchy converts the problem from a two-body estimate to a weaker three-body estimate for which we have the estimates to achieve |$\beta <1$|⁠, it then allows us to prove that all limit points of the ordinary BBGKY hierarchy satisfy the space–time bound conjectured by Klainerman and Machedon [48] for |$\beta \in (0,1)$|⁠.

DOI
10.1093/imrn/rnv228

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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 |β∈(0,1]\beta \in ( 0,1] β∈(0,1]|⁠, we derive the Gross–Pitaevskii hierarchy with 2-body interaction. In particular, we establish that, in the |N→∞N\rightarrow \infty N→∞| limit, all |kkk|-body scattering processes vanish if |k⩾3k\geqslant 3k⩾3| 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 |β∈(0,1)\beta \in ( 0,1) β∈(0,1)| and weakly for |β=1\beta =1β=1|⁠. 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 |β<1\beta <1β<1|⁠, 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 |β∈(0,1)\beta \in (0,1)β∈(0,1)|⁠.
We study quantum and classical many-body Hamiltonian systems that combine integrable contact interactions with generic long-range two-body potentials. Starting from an ansatz for the state at time t , which we call the correlated fluid-cell ensemble, we show that the dynamics of local observables at macroscopic times and length scales can be cast into a generalized Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy formulated in terms of the quasiparticle densities of the underlying integrable model and their correlations. We derive this hierarchy and validate these predictions against microscopic molecular-dynamics simulations, finding perfect agreement. At late times, the one-particle distribution relaxes via a Boltzmann-type scattering integral encoding the interplay between integrable contact processes and long-range collisions, whereas higher-point correlations remain strongly nonthermal on thermalization timescales, indicative of a form of incomplete or generalized thermalization. Focusing on long-range dipolar quantum gases, where the relevant matrix elements can be obtained explicitly, we show that our collision integral reduces exactly to the Fermi golden rule result and provide a complete theoretical account of the experimental observations of Tang []. More broadly, our framework extends the BBGKY program to regimes with strong local interactions and applies to a wide class of experimentally relevant systems, from one-dimensional dipolar cold-atom gases to Lennard-Jones molecular fluids.
Abstract Consider a system of N bosons on the three‐dimensional unit torus interacting via a pair potential N2V(N(xi − xj)) where x = (x1, …, xN) denotes the positions of the particles. Suppose that the initial data ψN, 0 satisfies the condition where HN is the Hamiltonian of the Bose system. This condition is satisfied if ψN, 0 = WNϕN, 0 where WN is an approximate ground state to HN and ϕN, 0 is regular. Let ψN, t denote the solution to the Schrödinger equation with Hamiltonian HN. Gross and Pitaevskii proposed to model the dynamics of such a system by a nonlinear Schrödinger equation, the Gross‐Pitaevskii (GP) equation. The GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k‐particle density matrices ⊗k|ut〉 〈ut| solves the GP hierarchy. We prove that as N → ∞ the limit points of the k‐particle density matrices of ψN, t are solutions of the GP hierarchy. Our analysis requires that the N‐boson dynamics be described by a modified Hamiltonian that cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical interparticle distance. Our proof can be extended to a modified Hamiltonian that only forbids at least n particles from coming close together for any fixed n. © 2006 Wiley Periodicals, Inc.
We establish a theoretical framework for the exploration of 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, if applicable, symmetry conserving closure approximation is developed on the basis of a cluster expansion for these finite systems, using suitably normalized RDMs and symmetrization operators. The underlying cluster expansion is formulated recursively by means of super-operators, which allows for going to large truncation orders. 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. The theoretical framework provided here is extensively applied and evaluated in our immediate follow-up work [1], involving truncation orders as large as twelve.
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<).
An accurate description of the nonequilibrium dynamics of systems with coupled spin and bosonic degrees of freedom remains theoretically challenging, especially for large system sizes and in higher than one dimension. Phase-space methods such as the truncated Wigner approximation (TWA) have the advantage of being easily scalable and applicable to arbitrary dimensions. In this work we adapt the TWA to generic spin-boson models by making use of recently developed algorithms for discrete phase spaces [J. Schachenmayer, A. Pikovski, and A. M. Rey, Phys. Rev. X 5, 011022 (2015)]. Furthermore we go beyond the standard TWA approximation by applying a scheme based on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations to our coupled spin-boson model. This allows us, in principle, to study how systematically adding higher-order corrections improves the convergence of the method. To test various levels of approximation we study an exactly solvable spin-boson model, which is particularly relevant for trapped-ion arrays. Using TWA and its BBGKY extension we accurately reproduce the time evolution of a number of one- and two-point correlation functions in several dimensions and for an arbitrary number of bosonic modes.