The goodness of ergodic adiabatic invariants

For a slowly time-dependent Hamiltonian system exhibiting chaotic motion that ergodically covers the energy surface, the phase space volume enclosed inside this surface is an adiabatic invariant. In this paper we examine, both numerically and theoretically, how the error in this ergodic adiabatic invariant scales with the slowness of the time variation of the Hamiltonian. It is found that under certain circumstances, the error is diffusive and scales likeT ^–1/2, whereT is the characteristic time over which the Hamiltonian changes. On the other hand, for other cases (where motion in the Hamiltonian has a long-time 1/t tail in a certain correlation function), the error scales like [T ^–1 ln(T)]^1/2. Both of these scalings are verified by numerical experiments. In the situation where invariant tori exist amid chaos, the motion may not be fully ergodic on the entire energy surface. The ergodic adiabatic invariant may still be useful in this case and the circumstances under which this is so are investigated numerically (in particular, the islands have to be small enough).     

Extending the Bethe Ansatz: The Quantum Three-Particle Ring

The quantum problem of three impenetrable particles of arbitrary mass confined to a ring is solved by the Bethe ansatz. The solution of this problem is intimately related to the solution a Helmholtz equation in the interior of an arbitrary acute triangle, a problem thought insoluble by Bethe ansatz methods.     

量子绝热定理(II):近似和适用条件

文章首先概述了近年国际文献中关于量子绝热近似和绝热条件的不自洽性的研究.叙述了文献中关于绝热近似不自洽性的论证和争论.然后引入文章作者的观点,从不同于国外文献的角度出发指出应当正确地理解瞬时本征函数的相位问题,从这个相位的正确处理,得出结论:(1)MS不自洽的存在,不因计及几何相位而消除;(2)量子几何相位不自洽是不存在;(3)现时的标准的绝热近似条件不是充分条件.     

量子绝热定理(Ⅱ):近似和适用条件

文章首先概述了近年国际文献中关于量子绝热近似和绝热条件的不自洽性的研究．叙述了文献中关于绝热近似不自洽性的论证和争论．然后引入文章作者的观点，从不同于国外文献的角度出发指出应当正确地理解瞬时本征函数的相位问题，从这个相位的正确处理，得出结论：（1）MS不自洽的存在，不因计及几何相位而消除；（2）量子几何相位不自洽是不存在；（3）现时的标准的绝热近似条件不是充分条件．     

量子绝热定理（II）：近似和适用条件

文章首先概述了近年国际文献中关于量子绝热近似和绝热条件的不自洽性的研究.叙述了文献中关于绝热近似不自洽性的论证和争论.然后引入文章作者的观点,从不同于国外文献的角度出发指出应当正确地理解瞬时本征函数的相位问题,从这个相位的正确处理,得出结论：（1）MS不自洽的存在,不因计及几何相位而消除;（2）量子几何相位不自洽是不存在;（3）现时的标准的绝热近似条件不是充分条件.     

An empirical approach to the theory of particle and nuclear phenomena: Review and some new ideas

Experimental data on masses and lifetimes of unstable particles falls into a pattern, a brief review of some interesting consequences is presented here. From the experience in semiclassical methods and recent advances in quantum chromodynamics, it is proposed that an appropriate generalization of the Gutzwiller trace formula for field theories may lead to a systematic semiclassical chromodynamics theory. The theory can be developed to get appropriate dynamics leading to an explanation of pattern discovered in the empirical data.     

The Van Vleck formula,Maslov theory,and phase space geometry

The Van Vleck formula is an approximate, semiclassical expression for the quantum propagator. It is the starting point for the derivation of the Gutzwiller trace formula, and through this, a variety of other expansions representing eigenvalues, wave functions, and matrix elements in terms of classical periodic orbits. These are currently among the best and most promising theoretical tools for understanding the asymptotic behavior of quantum systems whose classical analogs are chaotic. Nevertheless, there are currently several questions remaining about the meaning and validity of the Van Vleck formula, such as those involving its behavior for long times. This article surveys an important aspect of the Van Vleck formula, namely, the relationship between it and phase space geometry, as revealed by Maslov's theory of wave asymptotics. The geometrical constructions involved are developed with a minimum of mathematical formalism.     

The periodically kicked rotator: Recurrence and/or energy growth

We explore the properties of the quantum kicked rotator, its classical equivalent being the standard map. Its behavior, as found by computer studies, depends very much on the strength of the external forcing. At low strength it is seemingly recurrent in the sense of Hogg and Huberman. However, its energy increases with time at large forcings. For quantum systems, a unitary map defines the evolution over one period of time. The spectrum of this map in an infinite space does not seem to change continuously when one approaches the ratio of the frequencies of the external and of the unperturbed system by rational approximations of the golden mean.     

Real Space Triplets in Quantum Condensed Matter: Numerical Experiments Using Path Integrals,Closures, and Hard Spheres

Path integral Monte Carlo and closure computations are utilized to study real space triplet correlations in the quantum hard-sphere system. The conditions cover from the normal fluid phase to the solid phases face-centered cubic (FCC) and cI16 (de Broglie wavelengths 0.2≤λB*<2, densities 0.1≤ρN*≤0.925). The focus is on the equilateral and isosceles features of the path-integral centroid and instantaneous structures. Complementary calculations of the associated pair structures are also carried out to strengthen structural identifications and facilitate closure evaluations. The three closures employed are Kirkwood superposition, Jackson–Feenberg convolution, and their average (AV3). A large quantity of new data are reported, and conclusions are drawn regarding (i) the remarkable performance of AV3 for the centroid and instantaneous correlations, (ii) the correspondences between the fluid and FCC salient features on the coexistence line, and (iii) the most conspicuous differences between FCC and cI16 at the pair and the triplet levels at moderately high densities (ρN*=0.9, 0.925). This research is expected to provide low-temperature insights useful for the future related studies of properties of real systems (e.g., helium, alkali metals, and general colloidal systems).     

Quantum chaos and magnetotransport in lateral superlattices: The transition from weak to strong modulation potential

Magnetotransport in lateral superlattices covers a variety of apparently different physics dependent on the amplitude of the modulation potential relative to the Fermi energy ranging from the weakly modulated 2DEG to the antidot lattice. Here we investigate the transition between weak modulation and antidots within the quantum-mechanical theory by studying (i) the single-particle spectrum with respect to quantum signatures of classical chaos and (ii) the change of characteristic features in the magnetoresistance starting from the Kubo formula.     

布洛赫电子的拓扑与几何

文章回顾了电子的拓扑几何理论发展的初期,大约二十多年的历史。首先介绍拓扑陈数在凝聚态物理中的两个重要应用。其一关于量子霍尔效应,绝缘条件下霍尔电导可以写成一个陈数拓扑不变量,从而解释实验结果的精确量子化。其二关于绝热泵浦,它描述布洛赫能带的绝热电流响应,与电子极化有密切联系。拓扑陈数是布里渊区上贝里曲率的积分,后者本身也有独立的物理意义。接着介绍贝里曲率对电子动力学的影响,包括反常速度和轨道磁化等概念。作者还将这个理论推广到多带情况,使其可以应用到自旋输运等现象。最后,文中展示了再量子化方法,从半经典模型来获得布洛赫电子的有效量子理论。在非相对论极限下,泡利—薛定谔方程可以看作是狄拉克电子在正能谱上的等效量子理论,其中的自旋轨道耦合即是一种几何物理效应。     

The relativistic statistical theory and Kaniadakis entropy: an approach through a molecular chaos hypothesis

We have investigated the proof of the H theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [G. Kaniadakis, Phys. Rev. E 66, 056125 (2002); G. Kaniadakis, Phys. Rev. E 72, 036108 (2005)]. As it happens in the nonrelativistic limit, the molecular chaos hypothesis is slightly extended within the Kaniadakis formalism. It is shown that the collisional equilibrium states (null entropy source term) are described by a κ power law generalization of the exponential Juttner distribution, e.g., , with θ=α(x)+β_μp^μ, where α(x) is a scalar, β_μ is a four-vector, and p^μ is the four-momentum. As a simple example, we calculate the relativistic κ power law for a dilute charged gas under the action of an electromagnetic field F^μν. All standard results are readly recovered in the particular limit κ→0.     

Random quantum states: recent developments and applications

We review the methods and use of random quantum states with particular emphasis on recent theoretical developments and applications in various fields. The guiding principle of the review is the idea that random quantum states can be understood as classical probability distributions in the Hilbert space of the associated quantum system. We show how this central concept connects questions of physical interest that cover different fields such as quantum statistical physics, quantum chaos, mesoscopic systems of both non-interacting and interacting particles, including superconducting and spin–orbit phenomena, and stochastic Schrödinger equations describing open quantum systems.     

Structure of quantum disordered wave functions: weak localization, far tails, and mesoscopic transport

We report on the comprehensive numerical study of the fluctuation and correlation properties of wave functions in three-dimensional mesoscopic diffusive conductors. Several large sets of nanoscale samples with finite metallic conductance, modeled by an Anderson model with different strengths of diagonal box disorder, have been generated in order to investigate both small and large deviations (as well as the connection between them) of the distribution function of eigenstate amplitudes from the universal prediction of random matrix theory. We find that small, weak localization-type, deviations contain both diffusive contributions (determined by the bulk and boundary conditions dependent terms) and ballistic ones which are generated by electron dynamics below the length scale set by the mean free path ℓ. By relating the extracted parameters of the functional form of nonperturbative deviations (“far tails”) to the exactly calculated transport properties of mesoscopic conductors, we compare our findings based on the full solution of the Schr?dinger equation to different approximative analytical treatments. We find that statistics in the far tail can be explained by the exp-log-cube asymptotics (convincingly refuting the log-normal alternative), but with parameters whose dependence on ℓ is linear and, therefore, expected to be dominated by ballistic effects. It is demonstrated that both small deviations and far tails depend explicitly on the sample size--the remaining puzzle then is the evolution of the far tail parameters with the size of the conductor since short-scale physics is supposedly insensitive to the sample boundaries. Received 19 August 2002 Published online 19 November 2002     

Slowly rocking symmetric, spatially periodic Hamiltonians: The role of escape and the emergence of giant transient directed transport

The nonintegrable Hamiltonian dynamics of particles placed in a symmetric, spatially periodic potential and subjected to a periodically varying field is explored. Such systems can exhibit a rich diversity of unusual transport features. In particular, depending on the setting of the initial phase of the drive, the possibility of a giant transient directed transport in a symmetric, space-periodic potential when driven with an adiabatically varying field arises. Here, we study the escape scenario and corresponding mean escape times of particles from a trapping region with the subsequent generation of a transient directed flow of an ensemble of particles. It is shown that for adiabatically slow inclination modulations the unidirectional flow proceeds over giant distances. The direction of escape and, hence, of the flow is entirely governed whether the periodic force, modulating the inclination of the potential, starts out initially positive or negative. In the phase space, this transient directed flow is associated with a long-lasting motion taking place within ballistic channels contained in the non-uniform chaotic layer. We demonstrate that for adiabatic modulations all escaping particles move ballistically into the same direction, leading to a giant directed current.