中文摘要：
统计力学成功地把一个系统的宏观热平衡状态与其微观状态联系起来,做为统计力学核心的系综理论明确给出微观状态的概率分布,系综理论在物理、化学、生物等自然科学的各个领域中都获得了广泛应用。然而,系综理论的基本原理至今仍存在争议:在实验室里我们仅仅只观测一个系统,为什么它的性质和系综平均的结果一致?在经典力学的框架下,各态历经假说是普遍接受的一种解释,该假说认为一个经典系统在随时间演化的过程中会历经各个微观状态,因此,系统性质对时间的平均等同于它的系综平均。然而,量子力学的薛定谔方程是一个线性方程,各态历经假说不成立。在量子力学的框架下,系综理论为何仍能预言实验结果?这个问题在量子力学诞生后就获得了人们的关注,重要的进展发生在1955年,Wigner提出随机矩阵模型,并利用该模型解释了原子序数较大的原子核中的能级分布问题,这揭开了研究量子混沌的序幕。在此后的几十年时间里,量子混沌理论得到快速发展。1994年,Srednicki在随机矩阵模型的基础上提出本征态热化假说(ETH),解释了为什么量子系统会达到热平衡状态,以及为什么量子热平衡状态同样可用系综理论来描述。随后,大量的数值模拟证实了 ETH的准确性。另一方面,从1980年代末期开始,纳米技术的发展促使人们研究介观尺寸系统中的电子输运性质,为了解释实验结果,需要将量子力学应用于非平衡状态。例如,Landauer和Buttiker推导出无相互作用情况下的电导公式;而在1990年代初期,研究者们开始利用非平衡格林函数方法计算电导。既然系综理论可以导出热平衡态的密度矩阵,从实用的角度出发,研究者们自然也希望获得一个非平衡态的密度矩阵表达式。1992年,Hershfield从平衡态的密度矩阵出发,通过求解时间演化问题,获得了非平衡稳态的密度矩阵表达式。McLennan和Zubarev尝试从刘维尔方程出发推导密度矩阵的一般形式。2003年,Bokes和Godby通过把流做为限定条件,利用熵最大化原理得到了密度矩阵。2013年,Ness证明上述三个结果相互之间是等价的,密度矩阵被表达成了扩展Gibbs系综的形式。自然而然,在非平衡统计力学中我们也可以提出如下问题:为什么实验室中观测到的非平衡稳态可以用扩展Gibbs系综理论描述?无论是利用Hershfield等人得到的公式或是利用非平衡格林函数方法计算电导,一个普遍的前提假设是非平衡稳态需要从热平衡系综出发历经无限长时间演化得到。因此,上述问题也等价于:为什么实验室观测到的非平衡稳态可以由热平衡系综经历无限长时间演化得到?回答这个问题同样需要运用量子混沌理论。但是,仅仅用ETH无法解释非平衡稳态。为此,我们引入一个新的假说——非平衡稳态假说(NESSH):在混沌系统中,量子态的密度矩阵与可观测量算符的密度矩阵类似,都拥有一个普适的结构。密度矩阵的非对角元可以表示成动力学特征函数f(E,ω)与随机变量的乘积。其中,动力学特征函数决定系统的动力学性质。特别地,在非平衡稳态中,f(E,ω)在ω→0时呈1/ω形式发散。我们利用数值对角化的方法研究多个不同的模型,验证了我们的假说,并利用动力学特征函数解释这些模型的动力学过程。本文第一章介绍了经典混沌和量子混沌的背景知识,着重介绍随机矩阵理论、本征态热化假说、多体物理的实验背景、现有的数值算法以及其优缺点。第二章介绍非平衡稳态的概念,讨论了单体和多体共振能级模型的严格解。通过对该问题的求解,我们给出形成稳定流的条件:在对应的积分中存在1/ω形式发散的因子。第三章介绍NESSH的理论框架。我们发现,初态密度矩阵具有和可观测物理量算符一样的性质,在哈密顿量本征基下,其矩阵元具有普适结构。矩阵元分为对角元素和非对角元素,其中,对角元素是高斯函数与随机变量的和,而非对角元素是动力学特征函数与随机变量的乘积。第四章,我们在一维无序XXZ模型和二维Ising模型中对ETH和NESSH进行验证。我们用数值对角化方法求解模型,数值结果支持ETH和NESSH两个假说。第五章,我们构造了一个共振能级模型,对NESSH假说中的流公式进行验证。该模型包含两个随机矩阵导线和它们之间的一个共振能级。我们的数值结果显示,该模型的Ⅰ-Ⅴ曲线与安德森杂质模型相似。我们利用NESSH的流公式计算了稳定电流,得到了与从头计算完全一致的结果。第六章是全文的总结和展望。
英文摘要：
Statistical mechanics successfully links the macroscopic thermal equilibrium state of a system with its microscopic state.The ensemble theory at the core of statistical mechanics clearly gives the probability distribution of microscopic states.It has been widely used in various fields of natural sciences such as physics,chemistry,and biology.However,the basic principles of ensemble theory are still controversial:Why does the long time average consistent with the ensemble average results,while we only observing one system in laboratory.In the framework of classical mechanics,the ergodic hypothesis is a generally accepted interpretation.This hypothesis holds that a classical system will go through various micro-states in the process of evolution over time.Therefore,the average of system properties over time is equivalent to its Ensemble average.However,the Schrodinger equation for quantum mechanics is a linear equation,and the ergodic hypothesis of states does not hold.Under the framework of quantum mechanics,why can ensemble theory still predict experimental results?This problem has attracted people's attention since the birth of quantum mechanics.Important progress occurred in 1955.Wigner proposed a random matrix model and used this model to explain the problem of energy level distribution in nuclei with large atomic numbers.The prelude to the study of quantum chaos.In the decades that followed,quantum chaos theory developed rapidly.In 1994,Srednicki proposed the eigenstate thermalization hypothesis(ETH)on the basis of a random matrix model,explaining why quantum systems reach thermal equilibrium,and why quantum thermal equilibrium can also be described by ensemble theory.Subsequently,numerous numerical simulations confirmed the accuracy of the ETH.On the other hand,from the late 1980s,the development of nanotechnology prompted people to study the properties of electron transport in mesoscopic size systems.In order to explain experimental results,quantum mechanics needs to be applied to non-equilibrium states.For example,Landauer and Buttiker deduced the conductance formula in the absence of interactions;in the early 1990s,researchers began to use the non-equilibrium Green's function method to calculate conductance Since the ensemble theory can derive the density matrix of the thermal equilibrium state,from a practical point of view,researchers naturally hope to obtain an expression of the density matrix of the non-equilibrium state.In 1992,Hershfield started from the density matrix of the equilibrium state and solved the time evolution problem to obtain the expression of the density matrix of the non-equilibrium steady state.McLennan and Zubarev tried to derive the general form of the density matrix from the Liouville equation.In 2003,Bokes and Godby used the principle of entropy maximization to obtain the density matrix by using the current as a limiting condition.In 2013,Ness proved that the above three results are equivalent to each other,and the density matrix is expressed in the form of an extended Gibbs ensembleNaturally,we can also ask the following questions in non-equilibrium statistical mechanics:Why can the non-equilibrium steady state observed in the laboratory be described by the extended Gibbs ensemble theory?Whether the conductance is calculated by the formula obtained either by Hershfield et.al.or by non-equilibrium Green's function method.Therefore,the above question is also equivalent to:Why can the non-equilibrium steady state observed in the laboratory be derived from the thermal equilibrium ensemble undergoing infinite time evolution?Answering this question also requires the use of quantum chaos theory.However,non-equilibrium steady state cannot be explained by the ETH alone.To this end,we introduce a new hypothesis:the non-equilibrium steady-state hypothesis.We believe that in a chaotic system,the density matrix of quantum states is similar to the density matrix of observable operators,and both have a universal structure.The non-diagonal elements of the density matrix can be expressed as the product of dynamic characteristic functions and random variables.Among them,the nature of the dynamic characteristic function determines the dynamic properties of the system.In particular,in the non-equilibrium steady state,the dynamic characteristic function f(E,ω)diverges in the form of 1/ω at ω→0.We used numerical diagonalization to study a number of different models,verified our hypothesis in these models,and explained the dynamic process of these models using dynamic characteristic functionsThe first chapter of this paper introduces the background knowledge of quantum chaotic systems.From classical chaos to quantum chaos,the random matrix theory,the eigenstate heating hypothesis,the experimental background of many-body physics,the advantages and disadvantages of numerical algorithms are introducedChapter 2 introduces the concept of non-equilibrium steady state.We give rigorous solutions to the single-particle and many-body integrable models respectively.Through the solution of the single-particle problem,We know that there is a condition to form a stable current,that is,there is a factor of form divergence in the corresponding integral By calculating the current in the scattering state,we know that the factor does not exist in the matrix corresponding to the current operatorThe third chapter is the theoretical knowledge of the non-equilibrium steady-state hypothesis(NESSH).We give that the initial density matrix is the same as the local observable operator in the quantum chaotic system.The matrix elements under the Hamiltonian eigenbasis have a universal form.It can be understood as two parts.The diagonal elements are the summation of Gaussian function and random variables,and non-diagonal elements are the product of dynamical characteristic function and random variablesChapter 4 considers one-dimensional disordered XXZ model and two-dimensional Ising model.We numerically analyzed the ETH and the NESSH.The numerical results support two hypotheses,and display the long-time behavior of a local observable operator being unrelated to off-diagonal elementsChapter 5 presents a resonant level model with random coupling to verify ETH and NESSH.This model’s leads are described by random matrix theory.Our numerical results indicate that the real-time dynamics and curves of the current show similar functionality to conventional leads.According to the ETH and the NESSH,we can obtain the formula of stable current.We verify the various approximations and compare the result with those of the time-dependent Schrodinger equationFinally,the summary and outlook of the full text.
