Limiting dynamics in large quantum systems
Limiting dynamics in large quantum systems
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大型量子系统中的极限动力学
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发表时间：
2009
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通讯作者：
Antti Knowles
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作者：
Antti Knowles
本论文的目的是严格研究量子系统的动力学，其中描述系统大小的一些限制参数（如粒子数或单个粒子的质量）是大的。对于各种系统和限制制度，我们证明了微观量子时间演化近似描述的一个更简单，有效的时间演化。沿着，我们还讨论了一些相关的数学问题。在第一部分中，我们研究了有限个量子粒子与量子化辐射场相互作用的动力学。假设粒子是重的，光子数是大的，我们证明了量子时间演化成为经典的意义下，它是由牛顿-麦克斯韦方程组。这提供了量子系统中经典行为出现的一个例子。我们的极限动力学分析是基于一个半经典的论点，由于赫普。在第二部分中，我们研究了各种量子晶格模型的极限时间演化。开始，我们考虑了一个一般的相互作用量子自旋模型，并研究了两种极限状态：大自旋极限和连续极限。在这两种情况下，我们确定的限制动力学的经典系统的自旋的哈密顿动力学。这提供了一个严格的推导朗道-Lifschitz型方程从量子动力学。我们将这些结果推广到无限大的区域，并作为特例讨论了相干自旋态的极限动力学。我们的证明是基于动力学的微扰展开。在大自旋极限下，我们还证明了含时关联函数在某个正温度下的收敛性。对于足够高的温度，我们将这个结果扩展到一个无限的晶格使用量子集团展开。最后，我们研究了格点玻色气体含时关联函数的平均场极限的相关问题。在第三部分中，我们考虑了具有库仑相互作用势和弱外势的量子气体的平均场动力学。我们的方法是基于微扰图扩展计划的动态观测。我们控制库仑奇异计数图和利用色散性质的自由时间演化。首先，我们考虑玻色气体的平均场极限，并证明了该极限的时间演化是由Hartree方程控制的。其次，我们考虑的费米子系统的平均场极限，例如在一个大的原子或分子中的电子描述，并证明其限制的时间演化是由Hartree-Fock方程。本文的最后一部分研究了玻色气体中相干态的平均场动力学。使用基于Grönwall型参数的非微扰方法，我们在两个方向上加强和推广了许多以前已知的结果。首先，我们考虑一个大类的奇异相互作用势以及强，可能依赖于时间，外部潜力。这使我们能够处理，例如，临界相互作用势|X|对于非相对论性玻色子，以及强限制的时间依赖陷阱，为-2。其次，我们得到估计的速度收敛到平均场极限。这样，我们就可以控制玻色子星星平均场近似的误差。我们还表明，如果平均场动力学满足散射条件，所有的误差估计是均匀的时间。此外，我们得到的粒子的收敛到平均场极限的分数可以控制的最佳界限。
The aim of this thesis is a rigorous study of the dynamics of quantum systems in which some limiting parameter describing the size of the system (such as the number of particles or the mass of a single particle) is large. For a variety of systems and limiting regimes, we prove that the microscopic quantum time evolution is approximately described by a simpler, effective time evolution. Along the way, we also discuss some related mathematical problems. In a first part, we study the dynamics of a finite number of quantum particles interacting with the quantized radiation field. Assuming that the particles are heavy and the number of photons is large, we prove that the quantum time evolution becomes classical in the sense that it is governed by the Newton-Maxwell equations. This provides an example of the emergence of classical behaviour in a quantum system. Our analysis of the limiting dynamics is based on a semiclassical argument due to Hepp. In a second part, we study the limiting time evolution of various quantum lattice models. To begin with, we consider a general model of interacting quantum spins on a lattice, and study two limiting regimes: the large-spin limit and the continuum limit. In both cases, we identify the limiting dynamics as the Hamiltonian dynamics of a classical system of spins. This provides a rigorous derivation of Landau-Lifschitz-type equations from quantum dynamics. We extend these results to domains of infinite size and discuss as a special case the limiting dynamics of coherent spin states. Our proof is based on a perturbative expansion of the dynamics. In the large spin limit, we also prove the convergence of time-dependent correlation functions at some positive temperature. For high enough temperatures, we extend this result to an infinite lattice using a quantum cluster expansion. Finally, we study the related problem of the mean-field limit of time-dependent correlation functions of a lattice Bose gas. In a third part, we consider the mean-field dynamics of quantum gases with a Coulomb interaction potential and a weak external potential. Our method is based on a perturbative graph expansion scheme for the dynamics of observables. We control the Coulomb singularity by counting graphs and by exploiting the dispersive nature of the free time evolution. First, we consider the mean-field limit of a Bose gas, and prove that the limiting time evolution is governed by the Hartree equation. Second, we consider the mean-field limit of a system of fermions describing for instance electrons in a large atom or molecule, and prove that their limiting time evolution is governed by the Hartree-Fock equation. The last part of this thesis is devoted to the mean-field dynamics of coherent states in a Bose gas. Using a nonperturbative method based on a Grönwall-type argument, we strengthen and generalize many previously known results in two directions. First, we consider a large class of singular interaction potentials as well as strong, possibly time-dependent, external potentials. This allows us to deal for instance with the critical interaction potential |x|−2 for nonrelativistic bosons, as well as strongly confining time-dependent traps. Second, we derive estimates on the rate of convergence to the mean-field limit. Thus we can for instance control the error in the mean-field approximation of a boson star. We also show that, if the mean-field dynamics satisfies a scattering condition, all error estimates are uniform in time. Moreover, we derive optimal bounds on the fraction of particles whose convergence to the mean-field limit can be controlled.