Some Considerations on the Derivation of the Nonlinear Quantum Boltzmann Equation II: The Low Density Regime
Some Considerations on the Derivation of the Nonlinear Quantum Boltzmann Equation II: The Low Density Regime
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非线性量子玻尔兹曼方程推导的一些思考（二）：低密度状态
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
M. Pulvirenti
中科院分区：
文献类型：
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作者：
D. Benedetto;F. Castella;R. Esposito;M. Pulvirenti
本文分析了低密度状态下N个相同量子粒子系统的渐近动力学。我们的方法遵循作者在之前的工作(2)中引入的策略来处理更简单的弱耦合状态。用微扰级数表示了单粒子约简密度矩阵的维格纳变换的时间演化。展开是在迭代Duhamel公式的基础上得到的，在Lanford论文的精神中。（32）对于短时间和小的相互作用势，我们严格地证明了完全微扰级数的一个子级数收敛于非线性玻尔兹曼方程的解，该方程在物理上是相关的。重要的一点是，我们完全确定了进入极限玻尔兹曼方程的横截面，作为量子散射的玻恩级数展开。在文献2中，我们的收敛结果只是部分的，因为我们仅仅描述了完全原始微扰展开式的一个子序列的渐近行为。我们只有似是而非的论证来证明我们忽略的项，当从原始级数到它的相关子级数时，确实在极限中消失。本研究在d≥3的任何维度都成立。
In this paper we analyse the asymptotic dynamics of a system of N identical quantum particles in a low-density regime. Our approach follows the strategy introduced by the authors in a previous work,(2) to treat the simpler weak coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula, in the spirit of the paper by Lanford.(32) For short times and small interaction potential, we rigorously prove that a subseries of the complete perturbative series, converges to the solution of the nonlinear Boltzmann equation that is physically relevant in the context. An important point is that we completely identify the cross-section entering the limiting Boltzmann equation, as being the Born series expansion of quantum scattering.As in ref. 2, our convergence result is only partial, in that we merely characterize the asymptotic behaviour of a subseries of the complete original perturbative expansion. We only have plausibility arguments in the direction of proving that the terms we neglect, when going from the original series to its associated subseries, are indeed vanishing in the limit.The present study holds in any dimension d ≥ 3.