A short review on the derivation of the nonlinear quantum Boltzmann equations
A short review on the derivation of the nonlinear quantum Boltzmann equations
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DOI：
10.4310/cms.2007.v5.n5.a5
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发表时间：
2007-02
影响因子：
1
通讯作者：
D. Benedetto;F. Castella;R. Esposito;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
在这篇综述论文中，我们描述了在适当的尺度限制下推导 N 个相互作用的量子粒子系统的玻尔兹曼方程的问题。我们主要遵循作者在之前的作品中开发的方法。从严格的角度来看，即使时间很短，也只能得到部分结果，因此完整的问题仍然悬而未决。 1. 简介 由 N 个相同的相互作用粒子组成的大型量子系统通常可以用玻尔兹曼方程来描述。这是一个渐近模型：由第一原理给出的方程是N体薛定谔方程。因此，玻尔兹曼描述仅在合适的情况下成立，即当粒子数量很大且粒子对之间的相互作用势影响很小时。关于最后一点，有两个完全不同的设置是相关的。在所谓的弱耦合极限下，相互作用势本身很小，而气体却很稠密：粒子之间的典型距离是一阶的。在低密度方差区，基本相互作用势为一阶，而气体是稀疏的：粒子之间的典型距离很大，因此成对相互作用的影响很小。
In this review paper we describe the problem of deriving a Boltzmann equation for a system of N interacting quantum particles, under the appropriate scaling limits. We mainly follow the approach developed by the authors in previous works. From a rigorous viewpoint, only partial results are available, even for short times, so that the complete problem is still open. 1. Introduction A large quantum system of N identical interacting particles can often be described in terms of a Boltzmann equation. This is an asymptotic model: the equation given from flrst principles is the N body Schrodinger equation. As such, the Boltzmann description only holds in suitable regimes, namely when the number of particles is large, and when the interaction potential between pairs of particles has a small efiect. Concerning this last point, two quite difierent settings are relevant. In the so-called weak-coupling limit, the interaction potential itself is small, while the gas is dense: the typical distance between particles is of order one. In the low-density regime at variance, the elementary interaction potential is of order one, while the gas is rarefled: the typical distance between particles is large, hence the efiect of the pairwise interactions is small.