小量子混沌系统的热化及温度性质研究

Thermalization of quantum systems and properties of small systems, both are fields in physics, which have received wide attention in recent years. The contents of this proposal lie in the overlap of these two fields, as stated below: From the angle of micro-dynamics, we study thermalization processes of small quantum chaotic systems, make clear the role played by chaotic dynamics in these processes, reveal properties of internal temperature of such systems, and find method for temperature measurement. The fast development of the field of quantum thermalization benefited from the introduction of a new research angle and renewed understanding of some old angles, where the former is related to typical states and the latter involves long-time average of physical quantities and a generalization of a property of eigenfunctions of quantum chaotic systems (the so-called eigenstate thermalization hypothesis). This proposal plan to combine the three angles, in order to find a new ``angle’’ from which one can study thermalization of small quantum chaotic systems. Specifically, we plan to start from dynamics of small quantum chaotic systems, improve ETH to fit more realistic chaotic models, make use of properties of typical states, and determine steady states of subsystems by means of long-time average.

量子系统的热化与小系统的性质都是近年来受到广泛关注的领域。本申请属这两个领域的交叠，具体内容如下：从微观动力学角度，研究小量子混沌系统的热化过程，理清量子混沌动力学在该过程中所起的作用，确定这类系统的内在温度的性质、及其测量方法。量子热化领域在前些年的快速发展，得益于新研究角度的引入、以及对已有研究角度之内涵的重新认识，其中，前者与典型态的性质有关，而后者涉及对物理量长时间平均值的考虑、以及对量子混沌系统本征函数的一个性质的推广（所谓eigenstate thermalization hypothesis (ETH)）。本申请拟有机地结合上述三个角度、形成一定的“合力”效应，以研究小量子混沌系统的热化。具体而言，我们拟以小量子混沌系统的动力学为基础，改进ETH以适用于更为现实的混沌模型，在初态的选取上充分利用典型态的特性，并利用长时间平均的手段来确定子系统的稳态。

我们从动力学角度研究了量子混沌系统的热化过程，尤其是与温度有关的性质。主要取得以下研究成果：（1）完善了半扰动论的理论框架；该理论为研究一般的复杂量子系统的本征函数的结构性性质提供了一个合适的框架；针对量子混沌系统之本征函数在经典能量允许区内、也即半扰动论的非微扰区内的系数，证明其在重新标度之后具有高斯分布。（2）研究了混沌环境下的小量子系统的稳态性质，给出了互作用下的优选基的数学表示式，以及约化密度矩阵具有吉布斯形式的一个充分条件；尤其是，解决了在整体薛定谔演化下的子系统的温度表示式问题，发现玻尔兹曼温度是适用的、而吉布斯温度不适用。（3）利用数值与解析分析研究了热力学第零定律对小量子系统的适用性；当互作用满足一定的条件时，证实不同温度物体在接触之后可以趋于具有共同温度的平衡态。

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