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arXiv:1511.00487 · Differential and Integral Equations

Exact Evolution versus mean field with second-order correction for Bosons interacting via short-range two-body potential

Elif Kuz

Published Jul 1, 2017• 26 citations
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Abstract

We consider the evolution of $N$ bosons, where $N$ is large, with two-body interactions of the form $N^{3\beta}v(N^{\beta}\mathbf{\cdot})$, $0\leq\beta\leq 1$. The parameter $\beta$ measures the strength 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 [25, 26]. For $0\leq \beta < 1/2$, we derive an error bound of the form $p(t)/N^\alpha$, where $\alpha>0$ and $p(t)$ is a polynomial, which implies a specific rate of convergence as $N\rightarrow\infty$.

DOI
10.57262/die/1493863396

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We consider the evolution of NNN bosons, where NNN is large, with two-body interactions of the form N3βv(Nβ⋅)N^{3\beta}v(N^{\beta}\mathbf{\cdot})N3βv(Nβ⋅), 0≤β≤10\leq\beta\leq 10≤β≤1. The parameter β\betaβ measures the strength 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 [25, 26]. For 0≤β<1/20\leq \beta < 1/20≤β<1/2, we derive an error bound of the form p(t)/Nαp(t)/N^\alphap(t)/Nα, where α>0\alpha>0α>0 and p(t)p(t)p(t) is a polynomial, which implies a specific rate of convergence as N→∞N\rightarrow\inftyN→∞.
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 the present work we consider the evolution of N bosons, where N is large, and 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 (compared to log (o(1)N) as obtained in Chen et al. (J. Stat. Phys. 144: 872–903, 2011)) for the Hartree evolution. Estimates of similar type are obtained for stronger interactions as well. The precise estimates are stated in Theorem 5 in Section 3 and Theorem 9 in Section 4.
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.
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.
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.
The dynamics of a quantum system with a large number NNN of identical bosonic particles interacting by means of weak two-body potentials can be simplified by using mean-field equations in which all interactions to any one body have been replaced with an average or effective interaction in the mean-field limit N→∞N \rightarrow \inftyN→∞. In order to show these mean-field equations are accurate, one needs to show convergence of the quantum NNN-body dynamics to these equations in the mean-field limit. Previous results on convergence in the mean field limit have been derived for certain initial conditions in the case of one species of bosonic particles, but no results have yet been shown for multi-species. In this thesis, we look at a quantum bosonic system with two species of particles. For this system, we derive a formula for the rate of convergence in the mean-field limit in the case of an initial coherent state, and we also show convergence in the mean-field limit for the case of an initial factorized state. The analysis for two species can then be extended to multiple species.