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arXiv:1509.05911 · Communications in Partial Differential Equations

Pair excitations and the mean field approximation of interacting Bosons, II

M. Grillakis, M. Machedon

Published Nov 28, 2016• 67 citations
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Abstract

We consider a large number of Bosons with interaction potential . In our earlier papers (Grillakis et al. in Comm. Math. Phys. (2010) and in Adv. Math. (2011), as well as Grillakis and Machedon in Comm. Math. Phys., (2013)) we considered a set of equations for the condensate ϕ and pair excitation function k and proved that they provide a Fock space approximation to the exact evolution of a coherent state for . In Grillakis and Machedon, J. Fixed Point Theory Appl., (2013), in the hope of treating higher values of β<1, we introduced a coupled refinement of our original equations. In that paper, we showed the coupled equations conserve the number of particles and energy. In the current paper, we prove that the coupled equations do indeed provide a Fock space approximation for , at least locally in time. In order to do that, we reformulate the coupled equations in a way reminiscent of BBGKY and apply harmonic analysis techniques in the spirit of those used by Chen and Holmer in J. Euro. Math. Soc. (2016) to prove the necessary estimates. In turn, these estimates provide bounds for the pair excitation function k. While our earlier papers provide background material, the methods of this paper paper are mostly new, and the presentation is self-contained.

DOI
10.1080/03605302.2016.1255228

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We consider a large number of Bosons with interaction potential . In our earlier papers (Grillakis et al. in Comm. Math. Phys. (2010) and in Adv. Math. (2011), as well as Grillakis and Machedon in Comm. Math. Phys., (2013)) we considered a set of equations for the condensate ϕ and pair excitation function k and proved that they provide a Fock space approximation to the exact evolution of a coherent state for . In Grillakis and Machedon, J. Fixed Point Theory Appl., (2013), in the hope of treating higher values of β<1, we introduced a coupled refinement of our original equations. In that paper, we showed the coupled equations conserve the number of particles and energy. In the current paper, we prove that the coupled equations do indeed provide a Fock space approximation for , at least locally in time. In order to do that, we reformulate the coupled equations in a way reminiscent of BBGKY and apply harmonic analysis techniques in the spirit of those used by Chen and Holmer in J. Euro. Math. Soc. (2016) to prove the necessary estimates. In turn, these estimates provide bounds for the pair excitation function k. While our earlier papers provide background material, the methods of this paper paper are mostly new, and the presentation is self-contained.
This thesis proves certain results concerning an important question in non-equilibrium quantum statistical mechanics which is the derivation of effective evolution equations approximating the dynamics of a system of large number of bosons initially at equilibrium (ground state at very low temperatures). The dynamics of such systems are governed by the time-dependent linear many-body Schroedinger equation from which it is typically difficult to extract useful information due to the number of particles being large. We will study quantitatively (i.e. with explicit bounds on the error) how a suitable one particle non-linear Schroedinger equation arises in the mean field limit as number of particles N → ∞ and how the appropriate corrections to the mean field will provide better approximations of the exact dynamics. In the first part of this thesis we consider the evolution of N bosons, where N is large, with two-body interactions of the form N³ᵝv(Nᵝ⋅), 0≤β≤1. The parameter β measures the strength and the range of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [18,19] by Grillakis-Machedon-Margetis. We extend the results for 0 ≤ β < 1/3 in [19, 20] to the case of β < 1/2 and obtain an error bound of the form p(t)/Nᵅ, where α>0 and p(t) is a polynomial, which implies a specific rate of convergence as N → ∞. In the second part, utilizing estimates of the type discussed in the first part, we compare the exact evolution with the mean field approximation in the sense of marginals. We prove that the exact evolution is close to the approximate in trace norm for times of the order o(1)√N compared to log(o(1)N) as obtained in Chen-Lee-Schlein [6] for the Hartree evolution. Estimates of similar type are obtained for stronger interactions as well.
We study a physically motivated model for a trapped dilute gas of Bosons with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states of this system by accounting for the scattering of atoms in pairs from the macroscopic state. We start with an approximate many-body Hamiltonian, H a p p \mathcal {H}_{\mathrm {app}} , in the Bosonic Fock space. This H a p p \mathcal {H}_{\mathrm {app}} conserves the total number of atoms. Inspired by Wu [J. Math. Phys. 2 (1961), 105–123], we apply a non-unitary transformation to H a p p \mathcal {H}_{\mathrm {app}} . Key in this procedure is the pair-excitation kernel, which obeys a nonlinear integro-partial differential equation. In the stationary case, we develop an existence theory for solutions to this equation by a variational principle. We connect this theory to a system of partial differential equations for one-particle excitation (“quasiparticle”-) wave functions derived by Fetter [Ann. Phys. 70 (1972), 67–101], and prove existence of solutions for this system. These wave functions solve an eigenvalue problem for a J J -self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter’s energy spectrum. We also analytically provide an explicit construction of the excited eigenstates of the reduced Hamiltonian in the N N -particle sector of Fock space.
This thesis is about the derivation of effective mean field equations and their next-order corrections starting from nonrelativistic many-body quantum theory. Mean field equations provide an approximate ansatz for the description of interacting many-particle systems. In this ansatz the interaction between the particles is replaced by a self-consistent external potential leading to an effective one-body description of the many-particle system. Next-order corrections provide an approximation which goes one step further and tries to capture also subleading effects that are not resolved by the mean field ansatz. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a many-body Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its next-order corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three self-contained chapters. In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective one-particle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of Bose-Einstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single one-particle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the Bose-Einstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the N-particle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the N-particle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space. Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding lll particles out of their total number N not in the condensate is exponentially small in the number lll. Using this bound, we then prove that the ground state wave function of the microscopic model satisfies certain quasifree type properties. The exponential decay is moreover used to provide an alternative proof for the validity of Bogoliubov's approximation for the low-lying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energy-momentum dispersion relation which is linear for small momentum. The linearity of the effective dispersion relation is an essential ingredient for the explanation of the superfluid character of the Bose gas. In Chapter four we study the time evolution of a single particle coupled through a pair potential to a dense and homogeneous ideal Fermi gas in two spatial dimensions. This type of model is well known in plasma physics where it is used to describe the energy loss of ions moving through a dense and degenerate electron gas. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift. To leading order, this energy shift is given by the spatially homogeneous mean field potential produced by the gas particles, whereas at next-to-leading order, one has to consider an additional correction to the mean field energy which is due to so-called recollision processes.
A large system of N N integer-spin atoms, called Bosons, manifests one of the most coherent macroscopic quantum states known to date, the “Bose-Einstein condensate”, at extremely low temperatures. As N → ∞ N\to \infty , this system is usually described by a mean-field limit: a single-particle wave function, the condensate wave function , that satisfies a nonlinear Schrödinger-type equation. In this expository paper, we review kinetic aspects of the mean-field Boson evolution. Furthermore, we discuss recent advances in the rigorous study of second-order corrections to this mean-field limit. These corrections originate from the quantum-kinetic mechanism of pair excitation , which lies at the core of pioneering works in theoretical physics including ideas of Bogoliubov, Lee, Huang, Yang and Wu. In the course of our exposition, we revisit the formalism of Fock space, which is indispensable for the analysis of pair excitation.
Abstract In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of N ddd d -dimensional bosons for large N. The bosons initially form a Bose–Einstein condensate and interact with each other via a pair potential of the form (N−1)−1Ndβv(Nβ⋅)(N-1)^{-1}N^{d\beta }v(N^\beta \cdot )(N−1)−1Ndβv(Nβ⋅) (N-1)-1Ndβv(Nβ·) for β∈[0,14d)\beta \in [0,\frac{1}{4d})β∈[0,4d1​) β∈[0,14d) . We derive a sequence of N-body functions which approximate the true many-body dynamics in L2(RdN)L^2({\mathbb {R}}^{dN})L2(RdN) L2(RdN) -norm to arbitrary precision in powers of N−1N^{-1}N−1 N-1 . The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.
In a first part we give a brief presentation of general Fock space setting to describe quantum field theory. Bosons are quantum particles with integer spin and have symmetric wave functions; Fermions are quantum particles with half-integer spin and are represented by anti-symmetric wave functions. The functional setting is given by a symmetric or anti-symmetric tensor product of Hilbert spaces. We describe these spaces and transformations between these spaces. We shall follow the references [23, 42]. Coherent states are defined by translating the vacuum states with the Weyl operators. This is easily done here for bosons. We shall see in the next chapter how to deal with fermions. In a second part we give an interesting application of bosonic coherent states to the study of the classical limit as ℏ↘0\hbar \searrow 0ℏ↘0 of non-relativistic boson systems with two body interaction in the neighborhood of a solution of the classical system (here the Hartree equation). The classical limit corresponds here to the mean-field limit as the number of particles goes to infinity. As we have done for finite systems, we use here Hepp’s method, which is a linearization procedure of the quantum Hamiltonian around the classical field. The fluctuations around this solution are controlled by a purely quadratic Hamiltonian. In a series of several important papers [114–116] Ginibre and Velo have proven an asymptotic expansion and remainder estimates for these quantum fluctuations. Finally, following the paper [225] one can show that, in the limit ℏ→0\hbar \rightarrow 0ℏ→0, the marginal distribution of the time-evolved coherent states tends in trace-norm to the projector onto the solution of the classical fields equation (Hartree equation) with a uniform remainder estimates in time.