Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

This article is meant as a summary and introduction to the ideas of effective field theory as applied to gravitational systems, ideas which provide the theoretical foundations for the modern use of general relativity as a theory from which precise predictions are possible.

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Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory

In this review we describe our current understanding of the properties of open string tachyons on an unstable D-brane or brane–antibrane system in string theory. The various string theoretic methods used for this study include techniques of two-dimensional conformal field theory, open string field theory, boundary string field theory, noncommutative solitons, etc. We also describe various attempts to understand these results using field theoretic methods. These field theory models include toy models like singular potential models and p-adic string theory, as well as more realistic version of the tachyon effective action based on Dirac–Born–Infeld type action. Finally we study closed string background produced by the "decaying" unstable D-branes, both in the critical string theory and in the two-dimensional string theory, and describe the open string completeness conjecture that emerges out of this study. According to this conjecture the quantum dynamics of an unstable D-brane system is described by an int...

Tachyon Dynamics in Open String Theory

Purpose – The purpose of this paper is to review the literature on maintenance management and suggest possible gaps from the point of view of researchers and practitionersDesign/methodology/approach – The paper systematically categorizes the published literature and then analyzes and reviews it methodicallyFindings – The paper finds that important issues in maintenance management range from various optimization models, maintenance techniques, scheduling, and information systems etc Within each category, gaps have been identified A new shift in maintenance paradigm is also highlightedPractical implications – Literature on classification of maintenance management has so far been very limited This paper reviews a large number of papers in this field and suggests a classification in to various areas and sub areas Subsequently, various emerging trends in the field of maintenance management are identified to help researchers specifying gaps in the literature and direct research efforts suitablyOriginali

Maintenance management: literature review and directions

In this paper, we mainly review recent results on mathematical theory and
numerical methods for Bose-Einstein condensation (BEC),
based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then
carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for
spin-1 BEC, Bogoliubov excitation and BEC at finite temperature.

http://osada.csp.escience.cn/dct/attach/Y2xiOmNsYjpwZGY6NjI5NzY=

Mathematical theory and numerical methods for Bose-Einstein condensation

Smooth Feshbach map and operator-theoretic renormalization group methods

The quintic NLS as the mean field limit of a boson gas with three-body interactions

We derive the defocusing cubic Gross–Pitaevskii (GP) hierarchy in dimension d = 3, from an N-body Schrodinger equation describing a gas of interacting bosons in the GP scaling, in the limit N → ∞. The main result of this paper is the proof of convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies (Chen and Pavlovic in Discr Contin Dyn Syst 27(2):715–739, 2010; http://arxiv.org/abs/0906.2984
 ; Proc Am Math Soc 141:279–293, 2013), which are inspired by the solution spaces based on space-time norms introduced by Klainerman and Machedon (Comm Math Phys 279(1):169–185, 2008). We note that in d = 3, this has been a well-known open problem in the field. While our results do not assume factorization of the solutions, consideration of factorized solutions yields a new derivation of the cubic, defocusing nonlinear Schrodinger equation (NLS) in d = 3.

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Derivation of the Cubic NLS and Gross–Pitaevskii Hierarchy from Manybody Dynamics in d = 3 Based on Spacetime Norms

We investigate the dynamics of a boson gas with three-body interactions in dimensions $d=1,2$. We prove that in the limit of infinite particle number, the BBGKY hierarchy of $k$-particle marginals converges to a limiting (Gross-Pitaevskii (GP)) hierarchy for which we prove existence and uniqueness of solutions. Factorized solutions of the GP hierarchy are shown to be determined by solutions of a quintic nonlinear Schrodinger equation. Our proof is based on, and extends, methods of Erdos-Schlein-Yau, Klainerman-Machedon, and Kirkpatrick-Schlein-Staffilani.

The quintic NLS as the mean field limit of a Boson gas with three-body interactions

We consider the
dynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$, $d\geq1$,
for cubic, quintic, focusing and defocusing interactions.
For both the focusing and defocusing case, and any $d\geq1$,
we prove local
existence and uniqueness of solutions in certain
Sobolev type spaces $\H_\xi^\alpha$ of sequences of marginal
density matrices which satisfy the space-time bound conjectured
by Klainerman and Machedon for the cubic GP hierarchy in $d=3$.
The regularity is accounted for by
 $ \alpha $ > 1/2 if d=1 $ \alpha > \frac d2-\frac{1}{2(p-1)} if d\geq2 and (d,p)\neq(3,2) $ $ \alpha \geq 1 if (d,p)=(3,2) $
 

where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy;
the parameter $\xi>0$ is arbitrary and determines the energy scale of the problem.
For focusing GP hierarchies, we prove lower bounds on the blowup rate.
Moreover, pseudoconformal invariance is established in the cases corresponding to $L^2$ criticality,
both in the focusing and defocusing context.
All of these results hold without the assumption of factorized initial conditions.

https://www.aimsciences.org/article/exportPdf?id=f4ef88c1-0a1e-4e45-bdd7-fcaedb3ea32a

Thomas Chen

Papers

Smooth Feshbach map and operator-theoretic renormalization group methods

The quintic NLS as the mean field limit of a boson gas with three-body interactions

Derivation of the Cubic NLS and Gross–Pitaevskii Hierarchy from Manybody Dynamics in d = 3 Based on Spacetime Norms

The quintic NLS as the mean field limit of a Boson gas with three-body interactions

On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies