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arXiv:1412.1775 · Kinetic and Related Models

On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation

Xuwen Chen, Yan Guo

Published Jan 1, 2015• 12 citations
arXivHTMLPDF

Abstract

The rigorous derivation of the Uehling-Uhlenbeck equation from morefundamental quantum many-particle systems is a challenging open problem inmathematics. In this paper, we exam the weak coupling limit of quantum $N$-particle dynamics. We assume the integral of the microscopic interaction iszero and we assume $W^{4,1}$ per-particle regularity on the coresspondingBBGKY sequence so that we can rigorously commute limits and integrals. Weprove that, if the BBGKY sequence does converge in some weak sense, thenthis weak-coupling limit must satisfy the infinite quantum Maxwell-Boltzmannhierarchy instead of the expected infinite Uehling-Uhlenbeck hierarchy,regardless of the statistics the particles obey. Our result indicates that,in order to derive the Uehling-Uhlenbeck equation, one must work withper-particle regularity bound below $W^{4,1}$.

DOI
10.3934/krm.2015.8.443

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We consider the quantum many-body dynamics at the weak-coupling scaling. We derive rigorously the quantum Boltzmann equation, which contains the classical hard sphere model and, effectively, the inverse power law model, from the many-body dynamics assuming a physical and optimal regularity bound. The regularity bound we find, on the one hand, is satisfied by quasi-free solutions and comes from calculations regarding the local Maxwellian solution, in which we also prove that 2-body molecular chaos never happens unless N=+∞N=+\inftyN=+∞; on the other hand, it arises from the well-posedness threshold of the limiting Boltzmann equation below which we prove ill-posedness. That is, the regularity cannot be higher at the NNN-body level, cannot be lower in the limit, and is hence a double criticality. To work with this borderline case, we analyze all four sides, with respect to the Fourier transform, of the BBGKY hierarchy sequence with new tools and techniques. We prove well-definedness, compactness, convergence, and uniqueness of hierarchies right at the criticality to complete an optimal derivation. In particular, we have proved that, for physical NNN-particle solutions, the Boltzmann equation emerges as the mean-field limit and time is hence irreversible, from first principles of quantum mechanics.
We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is attractive and given by a3β−1V(aβ⋅)a^{3β-1}V(a^β\cdot)a3β−1V(aβ⋅) where ∫V⩽0\int V\leqslant 0∫V⩽0 and aaa matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D NNN-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to $\infty $. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D to 1D coupling constant.
The quantum Boltzmann equation (QBE), originally formulated as the quantum extension of Boltzmann’s classical transport theory by Uehling and Uhlenbeck and subsequently developed within the nonequilibrium Green-function formalisms of Kadanoff–Baym and Keldysh, provides the central kinetic framework for describing relaxation and transport in weakly interacting quantum many-body systems. Although its formal structure is well established, conventional derivations often obscure the microscopic equivalence between operator and nonequilibrium Green-function approaches. Wick factorization is typically applied without explicitly enumerating real-time contractions, while Green-function treatments encode the same physics in Born-level self-energies that conceal their operator origin. In this paper, we present a pedagogical microscopic derivation of the quantum Boltzmann equation that makes this equivalence fully explicit. Starting from the ladder-operator formulation, we enumerate all fifteen Wick pairings of the second-order six-field correlator, identifying three Hartree–Fock contributions and twelve real-time Born scattering channels generated by a quartic, number-conserving interaction Hamiltonian. We then establish a one-to-one correspondence between these twelve operator contractions and the Keldysh components of the second-order Born self-energy, demonstrating explicitly how the Uehling–Uhlenbeck collision integral emerges after Wigner transformation and quasiparticle projection. Under controlled Born–Markov and quasiparticle approximations, this work offers a transparent synthesis of operator and Green-function formulations, and clarifies how kinetic fluctuations can be understood as emergent under coarse-graining.
In the paper we discuss possible approaches to the problem of the rigorous derivation of quantum kinetic equations from underlying many-particle dynamics. For the description of a many-particle evolution we construct solutions of the Cauchy problems of the BBGKY hierarchy and the dual BBGKY hierarchy in suitable Banach spaces. In the framework of the conventional approach to the description of kinetic evolution the mean-field asymptotics of the quantum BBGKY hierarchy solution is constructed. We develop also alternative approaches. One method is based on the construction of the solution asymptotics of the initial-value problem of the quantum dual BBGKY hierarchy. One more approach is based on the generalized quantum kinetic equation that is a consequence of the equivalence of the Cauchy problems of such evolution equation and the BBGKY hierarchy with initial data determined by the one-particle density operator.
We derive rigorously, for both R2{\Bbb R}^2R2 and [−L,L]×2[{-}L,L]^{\times 2}[−L,L]×2, the cubic nonlinear Schr\"odinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.
Abstract. We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is attractive and given by a 3β−1 V (a β ·) where ∫ V � 0 and a matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy which contains a diverging coefficient as the strength of the confining potential tends to ∞. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and