A new method is proposed to obtain Gross–Pitaevskii equation by the chain of Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) quantum kinetic equations. In that sense, we investigate the dynamics of a quantum system including infinite number of identical particles which interact via a (special) pair potential on the form of Dirac delta-function.
A new method is proposed to obtain Gross–Pitaevskii equation by the chain of Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) quantum kinetic equations. In that sense, we investigate the dynamics of a quantum system including infinite number of identical particles which interact via a (special) pair potential on the form of Dirac delta-function.
The Bogoliubov method for the excitation spectrum of a Bose-condensed gas is generalized to apply to a gas with an exact large number N of particles. This generalization yields a description of the Schr\"odinger picture field operators as the product of an annihilation operator A for the total number of particles and the sum of a ``condensate wave function'' $\ensuremath{\xi}(\mathbf{x})$ and a phonon field operator $\ensuremath{\chi}(\mathbf{x})$ in the form $\ensuremath{\psi}(\mathbf{x})\ensuremath{\approx}A{\ensuremath{\xi}(\mathbf{x})+\ensuremath{\chi}(\mathbf{x})/\sqrt{N}}$ when the field operator acts on the N particle subspace. It is then possible to expand the Hamiltonian in decreasing powers of N, and thus obtain solutions for eigenvalues and eigenstates as an asymptotic expansion of the same kind. It is also possible to compute all matrix elements of field operators between states of different N. The excitation spectrum can be obtained by essentially the same method as Bogoliubov only if $\ensuremath{\xi}(\mathbf{x})$ is a solution of the time-independent Gross-Pitaevskii equation for N particles and any chemical potential $\ensuremath{\mu}$, which yields a valid and stable solution of the Gross-Pitaevskii equation. The treatment within a subspace of fixed N is identical in form to that usually used, but the interpretation of the operators is slightly different. A time-dependent generalization is then made, yielding an asymptotic expansion in decreasing powers of N for the equations of motion. In this expansion the condensate wave function has the time-dependent form $\ensuremath{\xi}(\mathbf{x},t)$, and the condition for the validity of the expansion is that $\ensuremath{\xi}(\mathbf{x},t)$ satisfies the time-dependent Gross-Pitaevskii equation $\ensuremath{\partial}\ensuremath{\xi}/\ensuremath{\partial}t=\ensuremath{-}({\ensuremath{\Elzxh}}^{2}/2m){\ensuremath{\nabla}}^{2}\ensuremath{\xi}+V\ensuremath{\xi}+Nu|\ensuremath{\xi}{|}^{2}\ensuremath{\xi}$. The physics is then described in a kind of interaction picture, called the condensate picture, in which the phonon operator can be expressed as $\ensuremath{\chi}(\mathbf{x},t)={\ensuremath{\sum}}_{k}{\ensuremath{\xi}}_{k}(\mathbf{x},t){\ensuremath{\alpha}}_{k},$ where the operators ${\ensuremath{\alpha}}_{k}$ are time-independent annihilation operators, and the state vector has a time evolution described by a Schr\"odinger equation in which the Hamiltonian is a time-dependent quadratic form in the phonon creation and annihilation operators, whose coefficients are explicitly determined in terms of the time-dependent condensate wave function $\ensuremath{\xi}(\mathbf{x},t)$.
We consider the dynamics of |N| bosons in three dimensions. We assume that the pair interaction is given by |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 |N|-body wave function, for |$\beta \in ( 0,1] ∣,wederivetheGross–Pitaevskiihierarchywith2−bodyinteraction.Inparticular,weestablishthat,inthe∣N\rightarrow \infty ∣limit,all∣k∣−bodyscatteringprocessesvanishif∣k\geqslant 3∣andthusprovideadirectanswertoaquestionraisedbyErdo¨setal.[30].Moreover,thisnewBBGKYhierarchysharesthelimitpointswiththeordinaryBBGKYhierarchystronglyfor∣\beta \in ( 0,1) ∣andweaklyfor∣\beta =1∣.SincethisnewBBGKYhierarchyconvertstheproblemfromatwo−bodyestimatetoaweakerthree−bodyestimateforwhichwehavetheestimatestoachieve∣\beta <1∣,itthenallowsustoprovethatalllimitpointsoftheordinaryBBGKYhierarchysatisfythespace–timeboundconjecturedbyKlainermanandMachedon[48]for∣\beta \in (0,1)$|.
We clarify that an ideal gas obeying infinite statistics cannot undergo condensation. Then we derive the dynamic equation for an identical particle system obeying infinite statistics under external potential and inter-particle interaction. The derivation utilizes the Hamiltonian written in terms of the number operators and the transition number operators. At a very low temperature, where one can discard the dynamics of the excited occupation level, the dynamic of an infinite statistics system can be described by the Gross Pitaevskii equation, similar to the Bose-Einstein case.
Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N2V(N(xi−xj)), where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let HN denote the Hamiltonian of the system and let ψN,t be the solution to the Schrodinger equation. Suppose that the initial data ψN,0 satisfies the energy condition ≤CN>. and that the one-particle density matrix converges to a projection as N→∞. Then, we prove that the k-particle density matrices of ψN,t factorize in the limit N→∞. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrodinger equation with the coupling constant proportional to the scattering length of the potential V. In \cite{ESY}, we proved the same statement under the condition that the interaction potential V is sufficiently small; in the present work we extend this result to large potentials.
Consider a system of N bosons in three dimensions interacting via a repulsive short range pair potential N2V(N(xi−xj)), where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let HN denote the Hamiltonian of the system and let ψN,t be the solution to the Schrödinger equation. Suppose that the initial data ψN,0 satisfies the energy condition <ψN,0,HNkψN,0>≤CkNk for $k=1,2,... $. We also assume that the k-particle density matrices of the initial state are asymptotically factorized as N→∞. We prove that the k-particle density matrices of ψN,t are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential V. We also prove the same conclusion if the energy condition holds only for k=1 but the factorization of ψN,0 is assumed in a stronger sense.