Classical Degrees Research Articles

Structure of correlated initial states that guarantee completely positive reduced dynamics

We use the Koashi-Imoto decomposition of the degrees of freedom of joint system-environment initial states to investigate the reduced dynamics. We show that a subset of joint system-environment initial states guarantees completely positive reduced dynamics, if and only if the system privately owns all quantum degrees of freedom and can locally access the classical degrees of freedom, without disturbing all joint initial states in the given subset. Furthermore, we show that the quantum mutual information for such kinds of states must be independent of the quantum degrees of freedom.

Antidamping spin-orbit torque driven by spin-flip reflection mechanism on the surface of a topological insulator: A time-dependent nonequilibrium Green function approach

Motivated by recent experiments observing spin-orbit torque (SOT) acting on the magnetization $\stackrel{P\vec}{m}$ of a ferromagnetic (F) overlayer on the surface of a three-dimensional topological insulator (TI), we investigate the origin of the SOT and the magnetization dynamics in such systems. We predict that lateral F/TI bilayers of finite length, sandwiched between two normal metal leads, will generate a large anti-damping-like SOT per very low charge current injected parallel to the interface. The large values of anti-damping-like SOT are spatially localized around the transverse edges of the F overlayer. Our analysis is based on adiabatic expansion (to first order in $\ensuremath{\partial}\stackrel{P\vec}{m}/\ensuremath{\partial}t$) of time-dependent nonequilibrium Green functions (NEGFs), describing electrons pushed out of equilibrium both by the applied bias voltage and by the slow variation of a classical degree of freedom [such as $\stackrel{P\vec}{m}(t)$]. From it we extract formulas for spin torque and charge pumping, which show that they are reciprocal effects to each other, as well as Gilbert damping in the presence of SO coupling. The NEGF-based formula for SOT naturally splits into four components, determined by their behavior (even or odd) under the time and bias voltage reversal. Their complex angular dependence is delineated and employed within Landau-Lifshitz-Gilbert simulations of magnetization dynamics in order to demonstrate capability of the predicted SOT to efficiently switch $\stackrel{P\vec}{m}$ of a perpendicularly magnetized F overlayer.

Mixed Quantum-Classical Study of Nonadiabatic Curve Crossing in Condensed Phases.

We apply the mixed quantum-classical Liouville (MQCL) equation to investigate the nonadiabatic curve crossing in condensed phases. More specifically, electron transfer rate constants of the spin-Boson model are calculated by employing a rate constant expression using the collective solvent polarization as the reaction coordinate. In the calculation, classical nuclear degrees of freedom are initially sampled at the transition state configuration, and the initial state for the electronic degree of freedom is obtained from a mixed quantum-classical Boltzmann distribution. Different contributions to the electron transfer rate from the diagonal and off-diagonal elements of the initial density matrix, and contributions from trajectories with positive and negative initial velocities are analyzed. It is shown that the off-diagonal elements of the initial density matrix play an important role in the total electron transfer rate. The MQCL results are also compared with those calculated using Ehrenfest dynamics. It is found that, although the Ehrenfest dynamics is inaccurate when the reactive flux rate expression is used directly, it can give reasonably accurate results when individual contributions from the diagonal and off-diagonal elements of the initial density matrix are calculated.

Walter Bagehot (1826–1877)

Walter Bagehot (1826–1877) Stephen Donadio WALTER BAGEHOT (1826–1877) was a Victorian political analyst and social theorist who served as editor of The Economist, which still offers its readers a column in each issue designated in his honor. Born into a banking family in Langport, Somerset, he was educated at Bristol College, where he distinguished himself in the study of classics and mathematics, as well as German and Hebrew; since his background was Unitarian, he did not pursue his university studies at Oxford or Cambridge, which were distinctly Anglican, and chose instead University College, London, where he formed a lively debating society. His health was problematic, but in 1846 he was awarded a first-class degree in classics, and he remained at the university to further his studies in moral philosophy and political economy, securing an MA degree. His subsequent training qualified him to practice law, but he decided on a career in the family bank because it would allow him more freedom to become an author. His writing career effectively began when, in Paris in late 1851, he witnessed the coup d’état by Louis Napoleon and published a reasoned defense of it in a Unitarian journal called The Inquirer. What he wrote was consciously provocative, challenging the assumptions of many of his friends and contending that situations of extreme disorder might require measures that would not be defensible in more settled circumstances. Over the next decades, Bagehot’s writings would continue to be characterized by a persistent critical independence. Not long after his return from Paris, he became a co-founder (along with his university friend Richard Holt Hutton) of the National Review, and in addition to essays in political analysis his publications in that journal would include literary assessments of authors like Shakespeare, Shelley, Dickens, Gibbon, Scott, and Macaulay. Seeking to place some of his work in The Economist, in 1857 Bagehot established a connection with its owner, James Wilson, who asked him to produce a series of articles on banking. Later that year, after a passionate courtship, he and Wilson’s eldest daughter were engaged to be married; they took up residence in the Somerset area, where Bagehot’s employment in the family bank continued and he continued to publish. Appointed to an important financial position in India, Wilson gave his son-in-law oversight over The Economist, and Bagehot went on make his friend Hutton the editor. After Wilson’s death in 1860 and Hutton’s resignation in 1861, Bagehot himself assumed the editorship, which he held until his own death in 1877. During these last years he wrote regularly on a variety of topics. His study of the workings of government entitled The English Constitution came out in 1867. Physics and Politics, which took account of work in anthropology and evolutionary theory to probe the origins of political communities, appeared in 1872, and was followed the next year by Lombard Street, a highly influential work centered on the operations of the Bank of England and the organization of the larger economy. The pages presented here are taken from “The Metaphysical Basis of Toleration,” which is included in Volume III of Bagehot’s Literary Studies, published in 1903 by Longmans, Green, and Co., London, New York, and Bombay. —SD [End Page 186] Copyright © 2016 Middlebury College Publications

The Education Background of INCOSE Systems Engineering Professional Certification Program Applicants

AbstractHelix is a multi‐year research project of the Systems Engineering Research Center (SERC) that is developing a theory of what makes those who perform systems engineering effective. To support that research, Helix is collecting data from a diverse set of systems engineers, others who perform some systems engineering activities but do not think of themselves as systems engineers, and those who work with systems engineers. Among the data sources available to Helix is the complete set of applications between 2004 and 2013 that were submitted to the INCOSE Systems Engineering Professional Certification Program. This paper reports on the education background of 2504 applicants and includes the number and types of university degrees and how the most popular degrees of applicants have changed over time. The applicants are very well educated, with almost 70% having an advanced college degree. With nearly three‐quarters of the applicants coming from the US, education from US universities dominates. At the bachelor level, classical engineering degrees dominate, but computer‐related majors are becoming increasingly popular. Two of the most popular masters degrees earned since 2000 are the Master of System Engineering and the Master of Business Administration.

Geometry and symmetry of quantum and classical-quantum variational principles

This paper presents the geometric setting of quantum variational principles and extends it to comprise the interaction between classical and quantum degrees of freedom. Euler-Poincaré reduction theory is applied to the Schrödinger, Heisenberg, and Wigner-Moyal dynamics of pure states. This construction leads to new variational principles for the description of mixed quantum states. The corresponding momentum map properties are presented as they arise from the underlying unitary symmetries. Finally, certain semidirect-product group structures are shown to produce new variational principles for Dirac’s interaction picture and the equations of hybrid classical-quantum dynamics.

An introduction to the problem of bridging quantum and classical dynamics

Simulating the exact quantum dynamics of realistic interacting systems is presently a task beyond reach but for the smallest of them, as the numerical cost for solving the time-dependent Schrodinger equation scales exponentially with the number of degrees of freedom. Mixed quantum-classical methods attempt to solve this problem by starting from a full quantum description of the system and subsequently partitioning the degrees of freedom in two subsets: the quantum subsystem and the bath. A classical limit is then taken for the bath while preserving, at least approximately, the quantum evolution of the subsystem. A key, as yet not fully resolved, theoretical question is how to do so by constructing a consistent description of the overall dynamics. An exhaustive review of this class of methods is beyond the scope of this paper and we shall limit ourselves to present, as an example, a specific approach, known as the LANDM-Map method. The method stems from an attempt at taking a rigorous limit for the classical degrees of freedom starting from a path integral formulation of the full quantum problem. The results that we discuss are not new, but our intent here is to present them as an introduction to the problem of mixed quantum classical dynamics. We shall also indicate a broad classification of the available approaches, their limitations, and some open questions in this field.

Classicalization of quantum variables and quantum–classical hybrids

The extraction of classical degrees of freedom in quantum mechanics is studied in the stochastic variational method. By using this classicalization, a hybrid model constructed from quantum and classical variables (quantum–classical hybrids) is derived. In this procedure, conservation laws such as energy are maintained, and Ehrenfest's theorem is still satisfied with modification. The criterion for the applicability of quantum–classical hybrids is also discussed.

Spectral analysis of the Earth’s topographic potential via 2D-DFT: a new data-based degree variance model to degree 90,000

Classical degree variance models (such as Kaula’s rule or the Tscherning-Rapp model) often rely on low-resolution gravity data and so are subject to extrapolation when used to describe the decay of the gravity field at short spatial scales. This paper presents a new degree variance model based on the recently published GGMplus near-global land areas 220 m resolution gravity maps (Geophys Res Lett 40(16):4279–4283, 2013). We investigate and use a 2D-DFT (discrete Fourier transform) approach to transform GGMplus gravity grids into degree variances. The method is described in detail and its approximation errors are studied using closed-loop experiments. Focus is placed on tiling, azimuth averaging, and windowing effects in the 2D-DFT method and on analytical fitting of degree variances. Approximation errors of the 2D-DFT procedure on the (spherical harmonic) degree variance are found to be at the 10–20 % level. The importance of the reference surface (sphere, ellipsoid or topography) of the gravity data for correct interpretation of degree variance spectra is highlighted. The effect of the underlying mass arrangement (spherical or ellipsoidal approximation) on the degree variances is found to be crucial at short spatial scales. A rule-of-thumb for transformation of spectra between spherical and ellipsoidal approximation is derived. Application of the 2D-DFT on GGMplus gravity maps yields a new degree variance model to degree 90,000. The model is supported by GRACE, GOCE, EGM2008 and forward-modelled gravity at 3 billion land points over all land areas within the SRTM data coverage and provides gravity signal variances at the surface of the topography. The model yields omission errors of $$\sim $$ 9 mGal for gravity ( $$\sim $$ 1.5 cm for geoid effects) at scales of 10 km, $$\sim $$ 4 mGal ( $$\sim $$ 1 mm) at 2-km scales, and $$\sim $$ 2 mGal ( $$\sim $$ 0.2 mm) at 1-km scales.

Dynamic Symmetries, Control, and Chaos with Moving Atoms in High-Quality Cavities

We consider dynamic symmetry, quantum control, and dynamic chaos of atoms moving in high-quality cavities. We review the group-theoretical approach to solve the evolution equations in the cavity quantum electrodynamics and control the quantum evolution in a micromaser device. Taking into account the photon recoil effect, we study nonlinear dynamics of the fundamental interactions between two-level atoms and a quantized cavity field. The strongly coupled atom–field system is treated as a quantum–classical hybrid with dynamically coupled quantum and classical degrees of freedom. Interaction of the purely quantum atom–field system with the classical translational atomic degree of freedom results in emergence of classical dynamic chaos from quantum electrodynamics. That chaos is shown to manifest itself in some ranges of the control parameters in the exponential sensitivity of quantum atomic variables to small variations in the initial conditions and in the appearance of dynamic atomic fractals. We discuss how to observe this effect in real experiments. Our findings establish a quantum–classical correspondence in the cavity quantum electrodynamics.

Price discrimination through communication

We study a seller’s optimal mechanism for maximizing revenue when a buyer may present evidence relevant to her value. We show that a condition very close to transparency of buyer segments is necessary and sufficient for the optimal mechanism to be deterministic—hence, akin to classic third degree price discrimination—independently of nonevidence characteristics. We also find another sufficient condition depending on both evidence and valuations, whose content is that evidence is hierarchical. When these conditions are violated, the optimal mechanism contains a mixture of second and third degree price discrimination, where the former is implemented via sale of lotteries. We interpret such randomization in terms of the probability of negotiation breakdown in a bargaining protocol whose sequential equilibrium implements the optimal mechanism.

H Tunneling Effects on Sequential Dissociation of Methane over Ni(111) and the Overall Rate of Methane Reforming

The rate constants and the activation energies for the C–H bond dissociation of CH3, CH2, and CH species adsorbed on the surface of Ni(111) are calculated in the framework of a previously developed methodology. It considers the dissociations as a result of quantum transitions between discrete vibrational energy levels of an initial C–H bond in methane and energy continuum of the dissociated final state while the system reaches the transition state along other classical degrees of freedom. These constants along with other ones calculated in our previous papers are used for calculating the overall rate using a microkinetic model of methane steam reforming (MSR) incorporating methane and water dissociation and CO formation either by C + O or by CH + O → CHO → CO + H. The analysis shows that the latter steps cannot be ignored. On this metal, where the rate of water dissociation is much larger than that of methane, the rate can be expressed explicitly. The dependencies of the MSR rate on partial pressures of methane, water, hydrogen, and carbon monoxide are studied and are compared with experiment. The isotope effect (IE) for overall rate of MSR is calculated using the rate constants of all steps including disruption of C–H, O–H, and H–H bonds. For certain applications, like MSR in a membrane reactor, rates are analyzed to show that hydrogen separation may suppress the forward reaction.

Beyond clustering: mean-field dynamics on networks with arbitrary subgraph composition.

Clustering is the propensity of nodes that share a common neighbour to be connected. It is ubiquitous in many networks but poses many modelling challenges. Clustering typically manifests itself by a higher than expected frequency of triangles, and this has led to the principle of constructing networks from such building blocks. This approach has been generalised to networks being constructed from a set of more exotic subgraphs. As long as these are fully connected, it is then possible to derive mean-field models that approximate epidemic dynamics well. However, there are virtually no results for non-fully connected subgraphs. In this paper, we provide a general and automated approach to deriving a set of ordinary differential equations, or mean-field model, that describes, to a high degree of accuracy, the expected values of system-level quantities, such as the prevalence of infection. Our approach offers a previously unattainable degree of control over the arrangement of subgraphs and network characteristics such as classical node degree, variance and clustering. The combination of these features makes it possible to generate families of networks with different subgraph compositions while keeping classical network metrics constant. Using our approach, we show that higher-order structure realised either through the introduction of loops of different sizes or by generating networks based on different subgraphs but with identical degree distribution and clustering, leads to non-negligible differences in epidemic dynamics.Electronic supplementary materialThe online version of this article (doi:10.1007/s00285-015-0884-1) contains supplementary material, which is available to authorized users.

Statistical physical theory of mode-locking laser generation with a frequency comb

A study of the Mode-locking lasing pulse formation in closed cavities is presented within a statistical mechanical framework where the onset of laser coincides with a thermodynamic phase transition driven by the optical power pumped into the system. Electromagnetic modes are represented by classical degrees of freedom of a Hamiltonian model at equilibrium in an effective ensemble corresponding to the stationary laser regime. By means of optimized Monte Carlo numerical simulations, the system properties are analyzed varying mode interaction dilution, gain profile and number of modes. Novel properties of the resulting mode-locking laser phase are presented, not observable by previous mean-field approaches. For strong dilution of the nonlinear interaction network, power condensation occurs as the whole optical intensity is taken by a few electromagnetic modes, whose number does not depend on the size of the system. For all reported cases laser thresholds, intensity spectra, and ultra-fast electromagnetic pulses are computed.

Thermal equilibrium properties of surface hopping with an implicit Langevin bath.

The ability of fewest switches surface hopping (FSSH) approach, where the classical degrees of freedom are coupled to an implicit Langevin bath, to establish and maintain an appropriate thermal equilibrium was evaluated in the context of a three site model for electron transfer. The electron transfer model consisted of three coupled diabatic states that each depends harmonically on the collective bath coordinate. This results in three states with increasing energy in the adiabatic representation. The adiabatic populations and distributions of the collective solvent coordinate were monitored during the course of 250 ns FSSH-Langevin (FSSH-L) simulations performed at a broad range of temperatures and for three different nonadiabatic coupling strengths. The agreement between the FSSH-L simulations and numerically exact results for the adiabatic population ratios and solvent coordinate distributions was generally favorable. The FSSH-L method produces a correct Boltzmann distribution of the solvent coordinate on each of the adiabats, but the integrated populations are slightly incorrect because FSSH does not rigorously obey detailed balance. The overall agreement is better at high temperatures and for high nonadiabatic coupling, which agrees with a previously reported analytical and simulation analysis [J. R. Schmidt, P. V. Parandekar, and J. C. Tully, J. Chem. Phys. 129, 044104 (2008)] on a two-level system coupled to a classical bath.

Celebrating Soft Matter's 10th anniversary: Testing the foundations of classical entropy: colloid experiments.

Defining the entropy of classical particles raises a number of paradoxes and ambiguities, some of which have been known for over a century. Several, such as Gibbs' paradox, involve the fact that classical particles are distinguishable, and in textbooks these are often 'resolved' by appeal to the quantum-mechanical indistinguishability of atoms or molecules of the same type. However, questions then remain of how to correctly define the entropy of large poly-atomic particles such as colloids in suspension, of which no two are exactly alike. By performing experiments on such colloids, one can establish that certain definitions of the classical entropy fit the data, while others in the literature do not. Specifically, the experimental facts point firmly to an 'informatic' interpretation that dates back to Gibbs: entropy is determined by the number of microstates that we as observers choose to treat as equivalent when we identify a macrostate. This approach, unlike some others, can account for the existence of colloidal crystals, and for the observed abundances of colloidal clusters of different shapes. We also address some lesser-known paradoxes whereby the physics of colloidal assemblies, which ought to be purely classical, seems to involve quantum mechanics directly. The experimental symptoms of such involvement are predicted to be 'isotope effects' in which colloids with different inertial masses, but otherwise identical sizes and properties, show different aggregation statistics. These paradoxes are caused by focussing one's attention on some classical degrees while neglecting others; when all are treated equally, all isotope effects are found to vanish.

Semipositone boundary value problems with nonlocal, nonlinear boundary conditions

We demonstrate the existence of at least one positive solution to \begin{equation} \begin{split} -y''(t)& =\lambda f(t,y(t))\text{, }t\in(0,1)\\ y(0)& =H(\varphi(y))\text{, }y(1)=0,\notag \end{split} \end{equation} where $H : \mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and $\varphi : \mathcal{C}([0,1])\rightarrow\mathbb{R}$ is a linear functional so that the boundary condition at $t=0$ may be both nonlocal and nonlinear. Since the continuous function $f : [0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ may assume negative values, our results apply to semipositone problems. The classical Leray-Schauder degree is utilized to derive the existence result, which we obtain in the case where $\lambda$ is small and which permits $f$ to be negative on its entire domain. The result is illustrated by an example.

Dynamics in the quantum/classical limit based on selective use of the quantum potential.

A classical limit of quantum dynamics can be defined by compensation of the quantum potential in the time-dependent Schrödinger equation. The quantum potential is a non-local quantity, defined in the trajectory-based form of the Schrödinger equation, due to Madelung, de Broglie, and Bohm, which formally generates the quantum-mechanical features in dynamics. Selective inclusion of the quantum potential for the degrees of freedom deemed "quantum," defines a hybrid quantum/classical dynamics, appropriate for molecular systems comprised of light and heavy nuclei. The wavefunction is associated with all of the nuclei, and the Ehrenfest, or mean-field, averaging of the force acting on the classical degrees of freedom, typical of the mixed quantum/classical methods, is avoided. The hybrid approach is used to examine evolution of light/heavy systems in the harmonic and double-well potentials, using conventional grid-based and approximate quantum-trajectory time propagation. The approximate quantum force is defined on spatial domains, which removes unphysical coupling of the wavefunction fragments corresponding to distinct classical channels or configurations. The quantum potential, associated with the quantum particle, generates forces acting on both quantum and classical particles to describe the backreaction.

Measuring Urban Agglomeration: A Refoundation of the Mean City-Population Size Index

In this paper, we put forth the view that the potential for urbanization economies increases with interaction opportunities. From that premise follow three fundamental properties that an agglomeration index should possess: (1) to increase with the concentration of population and conform to the Pigou–Dalton transfer principle; (2) to increase with the absolute size of constituent population interaction zones; and (3) to be consistent in aggregation. Limiting our attention to pairwise interactions, and invoking the space-analytic foundations of local labor market area (LLMA) delineation, we develop an index of agglomeration based on the number of interaction opportunities per capita in a geographical area. This leads to Arriaga’s mean city-population size, which is the mathematical expectation of the size of the LLMA in which a randomly chosen individual lives. The index has other important properties. It does not require an arbitrary population threshold to separate urban from non-urban areas. It is easily adapted to situations where an LLMA lies partly outside the geographical area for which agglomeration is measured. Finally, it can be satisfactorily approximated when data is truncated or aggregated into size-classes. We apply the index to the Spanish NUTS III regions, and evaluate its performance by examining its correlation with the location quotients of several knowledge intensive business services known to be highly sensitive to urbanization economies. The Arriaga index’s correlations are clearly stronger than those of either the classical degree of urbanization or the Hirshman–Herfindahl concentration index.

A quantum-classical theory with nonlinear and stochastic dynamics

The method of constrained dynamical systems on the quantum-classical phase space is utilized to develop a theory of quantum-classical hybrid systems. Effects of the classical degrees of freedom on the quantum part are modeled using an appropriate constraint, and the interaction also includes the effects of neglected degrees of freedom. Dynamical law of the theory is given in terms of nonlinear stochastic differential equations with Hamiltonian and gradient terms. The theory provides a successful dynamical description of the collapse during quantum measurement.