Semiclassical theory of many-particle systems

自然界中发现的绝大多数系统都是完全或部分混乱的。粗略地说，这意味着只要对初始条件稍作改变，粒子的轨迹就会完全改变。对于由许多粒子组成的系统，例如凝聚态系统或冷原子气体，混沌起着特别重要的作用。它可以通过几种不同的方式发生：如果只有一个粒子存在，一些系统已经表现出混乱的行为。对于其他系统，粒子之间的强烈相互作用(特别是电荷的库仑相互作用)会导致混沌。在冷原子气体中也可能出现另一种形式的混乱。在低温下，这些气体表现出玻色-爱因斯坦凝聚，即所有原子都处于相同的量子力学状态。这种状态可以用一个所谓的宏观波函数来描述，它的动力学由一个非线性微分方程(非线性薛定谔方程)给出。由于非线性，该波函数的行为可以敏感地依赖于初始条件，就像混沌系统中粒子的轨迹一样。混沌的典型例子是Bose-Hubbard模型(多体和冷原子理论中最重要的模型之一)，以及最近关于处于混沌势中的冷原子的实验。发展数学方法来处理多粒子系统中的混沌是当务之急。对冷原子的实验现在正达到一个阶段，在这个阶段，混沌动力学和强相互作用发挥着至关重要的作用。此外，还有一个有趣的预测，即重要的量子力学性质，如强相互作用的多粒子系统的电导，实际上应该是通用的，而不取决于系统的细节。对能级(允许量子系统中的能量承担的离散值)的统计也应该具有普遍的行为。无论从基础还是从技术的角度来看，理解多粒子系统以普遍方式运行的确切条件都很重要：可以以有效的方式对普遍特征进行建模，而非普遍特征可以进行调整，以在技术应用中产生理想的行为。最近在量子混沌方面的突破(我自己也参与了)现在使解决这些问题成为可能。这将是拟议项目的目标。一个重要的步骤将是系统地研究量子多粒子系统的完整理论如何与两种近似描述相联系：经典的多粒子理论和基于宏观波函数的有效的量子力学描述。为了阐明这种联系，我将推广量子混沌的一个中心结果(古兹维勒迹公式)，将量子多粒子系统的性质表示为宏观波函数解的总和。有了这些工具，就有可能澄清多粒子系统在哪些制度下表现出普遍的行为，以及在哪些制度下重要的系统特有的影响是可以预期的。

The vast majority of systems found in nature are fully or partly chaotic. Roughly speaking, this means that the trajectory of a particle will be changed completely by just a small change of the initial conditions. Chaos plays a particularly important role for systems consisting of many particles, such as condensed-matter systems or gases of cold atoms. It can come about in several different ways: Some systems already show chaotic behaviour if only one particle is present. For other systems strong interactions between the particles (in particular the Coulomb interaction of electric charges) give rise to chaos. Yet another form of chaos is possible in cold atom gases. At low temperatures these gases display Bose-Einstein condensation, i.e., all atoms are in the same quantum-mechanical state. This state can be described by a so-called macroscopic wavefunction whose dynamics is given by a nonlinear differential equation (the nonlinear Schroedinger equation). Due to the nonlinearity the behaviour of this wavefunction can depend sensitively on the initial conditions just like the trajectory of a particle in a chaotic system. Paradigmatic examples for chaos are the Bose-Hubbard model (one of the most important models in many-body and cold atom theory) as well as recent experiments with cold atoms in chaotic potentials.Developing mathematical methods to deal with chaos in many-particle systems is an urgent priority. Experiments with cold atoms are now reaching a stage where chaotic dynamics and strong interactions play a crucial role. Moreover there is an interesting prediction that important quantum-mechanical properties such as the conductance of strongly interacting many-particle systems should in fact be universal and not depend on the details of the system.Universal behaviour is also expected for the statistics of the energy levels (the discrete values the energy is allowed to assume in a quantum system). Understanding the precise conditions under which many-particle systems behave in a universal way is important both from a fundamental and from a technological point of view: Universal features can be modelled in an efficient way, whereas non-universal features can be tuned to generate desirable behaviour in technological applications.Recent breakthroughs in quantum chaos (that I was involved in myself) now make it possible to tackle these questions. This will be the aim of the proposed project. An important step will be to systematically study how the full theory of quantum many-particle systems is connected to two approximate descriptions: classical many-particle theory, and an effective quantum mechanical description based on the macroscopic wavefunction. To elucidate this connection I will generalize a central result from quantum chaos (the Gutzwiller trace formula) to express properties of the quantum many-particle system as sums over solutions for the macroscopic wavefunction. With these tools in place it will be possible to clarify in which regimes many-particle systems display universal behaviour, and in which regimes important system specific effects are to be expected.