Top PDF The Theory of Classical Dynamics, Griffiths.pdf

The Theory of Classical Dynamics, Griffiths.pdf

The Theory of Classical Dynamics, Griffiths.pdf

Here it is assumed that the theory to be applied is that of classical dynamics, and so the representa- tion is stated in terms of mass distributions, forces, frames of reference, initial[r]

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TDDFT and quantum-classical dynamics : a universal tool describing the dynamics of matter.

TDDFT and quantum-classical dynamics : a universal tool describing the dynamics of matter.

Abstract Time-dependent density functional theory (TDDFT) is currently the most efficient approach allowing to describe electronic dynamics in complex systems, from isolated molecules to the condensed-phase. TDDFT has been employed to in- vestigate an extremely wide range of time-dependent phenomena, as spin dynamics in solids, charge and energy transport in nanoscale devices, and photoinduced exci- ton transfer in molecular aggregates. It is therefore nearly impossible to give a gen- eral account of all developments and applications of TDDFT in material science, as well as in physics and chemistry. A large variety of aspects are covered throughout these volumes, see e.g. Chapters X, Y (to be indicated). In the present chapter, we will limit our presentation to the description of TDDFT developments and applica- tions in the field of quantum molecular dynamics simulations in combination with trajectory-based approaches for the study of nonadiabatic excited-state phenomena. We will present different quantum-classical strategies used to describe the coupled dynamics of electrons and nuclei underlying nonadiabatic processes. In addition, we will give an account of the most recent applications with the aim of illustrating the nature of the problems that can be addressed with the help of these approaches. The potential, as well as the limitations, of the presented methods are discussed,
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Classical dynamics of free electromagnetic laser pulses

Classical dynamics of free electromagnetic laser pulses

Motivated by the desire to ameliorate the divergences in perturbative QED a number of generalized theories of electromagnetism have been proposed. To date, there is little experimental evidence for testing their predicted de- partures from Maxwell’s theory. However, with the in- crease in laser technology one may now be entering regimes that may discriminate between such theories. In particu- lar, Bopp-Land´e-Podolsky electrodynamics is linear in the

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Control Systems Design Based on Classical Dynamics

Control Systems Design Based on Classical Dynamics

Kalman filter theory can be used for state equation (9) and measurement equation(10), φ and Γ are constant matrixes, meet rank ( , ) φ Γ = n , rank ( , ) φ c = n , so the system is completely controllable and observable in Kalman filter theory( the proof is given the following lemma), when t s is very short and filtering time is very long , the covariance matrix,

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Quantum spin dynamics in external classical fields

Quantum spin dynamics in external classical fields

Abstract. We study the quantum and classical dynamics of spinning particles in the framework of the general-relativistic covariant Dirac theory. The exact Foldy-Wouthuysen transformation for the most general case of a fermion in ar- bitrary configurations of the gravitational, inertial and electromagnetic fields is derived. We demonstrate the complete consistency of the quantum and classical dynamics. As physical applications, we discuss prospects of probing spacetime structure and using spin e ff ects for gravitational wave detection.

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Quantum and classical dynamics of molecular scale structures

Quantum and classical dynamics of molecular scale structures

closest fit was found for 𝐸 𝐹 − 𝐸 𝐹 𝐷𝐹𝑇 = 0.65 𝑒𝑉. Figure 4.6 shows a logarithmic plot of predicted single-molecule conductances versus the number of (−𝐶𝐻 2 ) units in the alkane chain, along with a comparison with experiment. The close agreement between theory and experiment for 𝛽 𝑁 suggests that the difference between the attenuation coefficients of graphene-molecule-Au and Au-molecule- Au junctions arises from a difference in the positions of their frontier orbitals relative to 𝐸 𝐹 . Figure 4.6 also shows that the theoretical conductance values are slightly higher than measured ones for all molecular lengths, which can be attributed to the tendency of LDA to underestimate the HOMO-LUMO gap, which results in an overestimated of the conductance [14].
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Elimination of the Incompleteness of Classical Dynamics

Elimination of the Incompleteness of Classical Dynamics

Now it is clear that from all what has been made for 300 years of a “victorious march of N body paradigm” [11], only results obtained for stable “mathematical” motions are applicable to the physical mechanics of real (not abstract structure-free bodies). That means that the Poincaré recurrence theorem, as well as the Zermelo and Loschmidt paradoxes are valid only in the framework of the АММ, i.e. for structure-free mathematical objects. As for real (structured) mechanical objects/bodies, Poincaré’s theorem and Zermelo’s paradox are applicable only to the class of stable motions with a constant tube. For unstable motions of real bodies, the theorems of Picard, Poincaré, Khinchin, etc. should be replaced by limiting and ergodic theorems of the theory of stochastic processes, that means that the Laplacian (mechanical) determinism does not exist in the RMM. For example, according to the Khinchin theorem, the trajectory of a structure-free mathematical point again and again revisits almost any point of a bounded phase space. On the contrary, any configuration in the RMM, if not limiting, repeats itself only with the probability of measure zero. In other words, unlike in the АММ, the following ergodic replacement of the Khinchin theorem is applicable to the RMM:
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On the Interaction of Extended Charges in Classical Relativistic Theory

On the Interaction of Extended Charges in Classical Relativistic Theory

2) There are many principles and results of the classical theory that manifest as quantum analogs in one way or another (to name a few, corrections to the potential yield energy spectrum shift, classical trajectory corresponds to the evolution of coordinate mean values with time, etc.). Therefore one cannot ex- clude the fact that classical phenomena related to a finite size charge may affect quantum ones.

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Geometry and symmetry of quantum and classical-quantum dynamics.

Geometry and symmetry of quantum and classical-quantum dynamics.

C(z, X) = det(2X − zz) . This result comes as no surprise. The relation between the Jacobi group and Gaussian states has been known for decades in the theory of coherent states [ 70 , 71 ], under the statement that Gaussian states evolve under the action of the semidirect product Mp(R 2n ) s H(R 2n ) [ 7 ], where Mp(R 2n ) is the metaplectic group (that is, the double covering of Sp(R 2n )). Since the Lie algebra mp(R 2n ) of the metapletic group is isomorphic to that of the symplectic group (denoted by sp(R 2n )), then jac(R 2n ) ' mp(R 2n ) s h(R 2n ). In the present treatment, the symplectic group replaces the metaplectic transformations because we have identified quantum states with Wigner functions (which can account for pure as well as mixed states) rather than wavefunctions. Indeed, while the symplectic group does not possess a representation on wavefunctions, it does possess a natural action on the space of Wigner (phase-space) functions (as seen in ( 4.22 )) as in classical mechanics and one may avoid dealing with the metaplectic representation. This leads to an action of the Jacobi group, Φ : Jac(2n, R 2n ) × Den(R 2n ) → Den(R 2n ) which is given by
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THE ENERGY MOMENTUM TENSOR IN CLASSICAL FIELD THEORY

THE ENERGY MOMENTUM TENSOR IN CLASSICAL FIELD THEORY

The concept of the energy-momentum tensor in classical field the- ory has a long history, especially in Einstein’s Theory of Gravity. We consider the inhomogenous Lorentz-group as the fundamental symmetry group of all physics. A physical Lagrange action is then Lorentz invariant. Translation invariance leads to an energy ten- sor, and Lorentz invariance to an energy-momentum tensor. These tensors are Lorentz-covariant. The energy-momentum tensor is sym- metric whereas the energy tensor in general is not symmetric. Any additional symmetry of the action will be treated separately from Lorentz-invariance. In this paper we construct the energy tensor and the energy-momentum tensor for the following action
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Discussion on Inspiration Theory in Chinese Classical Aesthetics

Discussion on Inspiration Theory in Chinese Classical Aesthetics

Now it seems that the classical aesthetics’ view of ingenious nature without fabrication accord with the modern psychology’s research achievements. Mod- ern psychology thinks that brain’s excitement and inhibition adjusted to the best state, is the physiological basis of inspiration production. If the brain has been working for a long time in a highly excited state, it will lead to extreme mental fatigue and inhibit the activity of the cerebral cortex. At this time, if the creation subject gives up focusing on meditation to have a rest. This will distract the sub- ject’s attention and inhibit the original excitement center temporarily. But brain cells’ activities don’t stop completely, and the original surrounding cortex cells turn into excited state. The subconscious mind around the excited center will be stimulated. At that time, the activity of nerve cells increases greatly and a large number of potential stored information is inspired. Then it will get rid of the conventional way of thinking and develop freely and quickly, and break the fixed neural connections under usual inertial thinking mode. In the mean time, vari- ous presentations are disrupted and reassembled. The rigid thinking mode is de- stroyed and then the irregular connections and self-awareness are constructed again. At this moment, once the particularly active subconscious activities are impacted by some factors, they will rise to the ideological level just like the sud- den volcanic eruption, and stimulate the burst of inspiration. Xie Lingyun, a famous poet had described his good verse in his dream as “spring grass grows on the side of pond”, “the verse is inspired by God”. It was obtained under the situ- ation” thinking of poetry all day” while the inspiration appearing when he went to bed.
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Quantum and classical nonlinear dynamics in a microwave cavity

Quantum and classical nonlinear dynamics in a microwave cavity

the semi-classical system as it moves from being ‘below threshold’ to ‘above threshold’. First, for positive detuning (Δ > ), the threshold parametric pumping κ =  is effectively increased by the detuning to κ = √  + Δ  (the solid green/checkered red boundary in Fig- ure . At this threshold, the semi-classical system undergoes a supercritical pitchfork bi- furcation where the stable origin goes unstable, and the stable pair of fixed points emerges from the origin and grows in separation with increasing parametric pumping (the check- ered red region in the upper half-plane in Figure ). Alternatively, for all values of negative detuning (Δ < ), at the threshold parametric pumping κ =  two saddle-node bifurca- tions produce both the stable and unstable fixed point pairs (the solid green/striped blue boundary in Figure ). The origin remains stable, and the two newly created pairs of fixed points then exist for parametric pumping above threshold (κ > ) until pumping reaches the even higher value κ = √  + Δ  . Between these two values (the striped blue region in Figure ) with increasing parametric pumping, the stable pair increases in separation and the unstable pair moves to the origin. At the higher parametric pumping κ = √  + Δ  (the striped blue/checkered red boundary in Figure ) the unstable pair annihilates in a sub- critical pitchfork bifurcation at the origin, and the origin becomes unstable for all higher parametric pumping κ > √  + Δ  . The stable pair of fixed points continues to grow in separation with further increased parametric pumping (the checkered red region in the lower half-plane in Figure ).
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Spin, the Classical Theory

Spin, the Classical Theory

With the development of local gauge theories of gravitation, it became evident that intrinsic spin was an integral part of the theory. This gave spin a classical formulation that predicted the existence of a new kind of field, the torsion field. To date only omne class of experiments has been developed to detect this field, a search for a long range dipole force. In this article, the torsion equations are de-coupled from the curved space of general relativity derived from basic princi- ples using vector calculus and the theory of electromagnetism as a guide. The results are written in vector form so that they are readily available to experimentalists, paving the way for new kinds of experiments.
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From classical to contemporary ecological economics theory

From classical to contemporary ecological economics theory

Since its institutionalization, ecological economics has so far achieved clear dimensions of its activity in a scientifi c and practical sense. It explores mutual connection of ecosystems and economic activities (Proops, 1989: 76) with an emphasis on solving the problematic issues such as the use of fossil fuels, nuclear waste disposal, deforestation, etc. Proops (1989: 78) divided the aims of ecological econom- ics into two groups: those that relate to scientifi c goals and problems, and those that are focused on political and ethical issues. Ecological economics thus represents a complete approach towards an adequate allocation of resources that does not jeop- ardize the stability of the system as a whole, nor the stability of all its components. Common and Per- rings (1992) approach ecological economics from the perspective of the system theory, pointing out that a self-regulating economic system, even if envi- ronmentally sustainable, should result in a sustaina- ble level of consumption and production (Common, Perrings, 1992: 25). Also, Söderbaum emphasizes commitment in a sustainable ecological sense as one of the crucial features of ecological economics (Söderbaum, 1999: 161). In addition, Söderbaum raises the question of the conceptual framework and the value of ecological economics, emphasizing the usefulness of interaction with scientists from diff erent scientifi c disciplines.
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Symmetrizing dynamics: from classical to quantum applications

Symmetrizing dynamics: from classical to quantum applications

Structure of the Thesis The topics covered in this dissertation are at the boundary between quan- tum theory and automatic & control theory. The dissertation is divided into two parts. In the first part we reformulate the classic consensus problem as a switching dynamics leading to symmetrization with respect to the action of a suitable finite group. We show how this symmetrization framework, in addition to being a new view point for the gossip problem, directly extends the desirable features of consensus-like algorithms, such as robustness, to new control problems. More precisely, in chapter 1 we introduce the consensus problem in the operational multi-agent picture and then we focus on the gos- sip algorithm recalling the result that characterize its convergence features. In chapter 2 we begin to establish the symmetrizing framework by showing that the gossip interaction can be written as a suitable convex combination of action of the permutation group and by characterizing the consensus situation as invariance with respect the action of the group. In chapter 3 we develop a superior point of view for the symmetrizing picture that allows us to study the convergence of our class of algorithms only in terms of the group at hand and of the switching signals. Through these chapters the gossip algorithm it used as running example. Chapter 4 is devoted to the applications of our frame- work to various problem such as, the consensus for probability distribution, the generation of random state from a set and the randomized computation of the discrete Fourier transform. Through this chapter the robustness features of the algorithms are highlighted.
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Theoretical and Practical Comparison of Classical Test Theory and Item-Response Theory

Theoretical and Practical Comparison of Classical Test Theory and Item-Response Theory

Classical test theory and item response theory are widely perceived as representing two very different measurement frameworks. Few studies have empirically examined the similarities and differences in the parameters estimated using the two frameworks. The purpose of this study was to examine how item statistics (i.e. item difficulty and item discrimination) and person statistics (i.e. ability estimates) behave under the two measurement frameworks i.e. CTT and IRT. The researchers tried to compare the two models from both theoretical and practical perspectives. For this purpose, first, a theoretical comparison of the two models was carried out; then, a sample of 3000 testees taking part in the English language university entrance exam was used in order to compare the two models practically. The findings showed that person statistics from CTT were comparable with those from IRT for all three IRT models. Item difficulty indexes from CTT were comparable with those from all IRT models and especially from the one-parameter logistic (1PL) model. Item discrimination indexes from CTT were somewhat less comparable with those from IRT.
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Interpretation Neutrality in the Classical Domain of Quantum Theory

Interpretation Neutrality in the Classical Domain of Quantum Theory

Beyond its defense of these two points, much of the novelty of the present dis- cussion lies in its explicit synthesis of various elements from different parts of the literature on decoherence, the measurement problem and the quantum-classical cor- respondence, and in its attentiveness to nuances that arise when joining these vari- ous elements - from the explicit requirement that an interpretation-neutral collapse prescription be decoherence-compatible, to subtle variations in the decoherence con- ditions required for effective collapse across different interpretations, to the imple- mentation of the open-systems form of Ehrenfest’s Theorem rather than the more commonly discussed but less appropriate closed-system form. The discussion is structured as follows. Section 2 provides an overview of sources in the decoherence literature that attempt to explain the validity of classical equations of motion and highlights points on which the present discussion serves to complement these inves- tigations. Section 3 discusses several important points of terminology and method- ology, including my usage of the term “classical” and a description of the general approach to reduction adopted here. Section 4 describes a framework for recov- ering classical behavior that is based in results from decoherence theory but that remains non-commital as to the precise mechanism for collapse and the ontology of the quantum state. Section 5 shows how the interpretation-neutral account of clas- sical behavior provided by decoherence can be specially tailored to give an account of classicality in the Everett, de Broglie-Bohm and GRW interpretations, thereby resolving - if only in a speculative way - ambiguities associated with collapse in the interpretation-neutral account. Section 6 acknowledges a number of general con- cerns about decoherence-based approaches to recovering classical behavior. Section 7, the Conclusion, briefly considers some broader implications of the analysis given here, including possible extensions of this strategy to the relations between quantum and classical field theory and classical and quantum gravity. The core claims of the article are defended in Sections 3, 4 and 5.
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Classical Sociological Theory and Foundations of American Sociology

Classical Sociological Theory and Foundations of American Sociology

to verify this initial hypothesis, we must compare the social solidarity based on the division of labor with other types of solidarity and classify these types; we can do this by compar[r]

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Interpretation Neutrality in the Classical Domain of Quantum Theory

Interpretation Neutrality in the Classical Domain of Quantum Theory

In this section, I outline an interpretation-neutral, decoherence-based strategy for re- covering classical behavior from quantum theory, which relies heavily on the general approach to reduction outlined in the previous section. In a nutshell, this strategy rests on three central pillars: 1) decoherence to generate a branching structure for the quantum state, such that the state of the system of interest relative to any single branch is always quasi-classical; 2) a decoherence-compatible collapse pre- scription that selects one of these branches in accordance with the Born Rule, and whose underlying physical mechanism is left unspecified; 3) an appropriate form of Ehrenfest’s Theorem to ensure that quasi-classical, branch-relative trajectories are approximately Newtonian over relevant timescales. In Section 5, I show how concerns about the physical mechanism underpinning this decoherence-compatible collapse prescription can be addressed as a coda - albeit a necessary one - to the interpretation-neutral account given in this section, and that interpretation-specific aspects of the quantum mechanical description of classical behavior can therefore be quarantined to a relatively narrow portion of the overall analysis. In other words, we will see how the different collapse mechanisms and ontologies associated with different interpretations can be “slotted in” to this interpretation-neutral account to give a more complete, if more speculative, picture of classical behavior than the one provided by the interpretation-neutral picture alone. Thus, we will see more explicitly than previous investigations how one can go quite far in providing a quantum-mechanical account of classical behavior without taking on the specu- lative metaphysical commitments associated with some particular interpretation of quantum theory. Of course, we must also keep in mind that at most one of these interpretation-specific accounts can be correct as a description of the mechanism that nature itself employs.
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Conceptual Metaphor Theory and Classical Theory: Affinities Rather than Divergences

Conceptual Metaphor Theory and Classical Theory: Affinities Rather than Divergences

1993: 205). All of these types of evidence, except the last one, can be studied by the tools of classical linguistics with no need to invoke the cog. sci. framework. CMT delimits the literal-metaphorical distinction differently than CT, and moreo- ver Lakoff claims that this CT makes a false assumption in this respect (Lakoff 1993: 204). But there is a great deal of conceptual confusion lurking here. For CT, the literal-metaphorical distinction is defined a priori, viz. as a novel use of language which shows a contextual abnormality (most metaphorical utterances are usually trivially false or otherwise contextually inappropriate). By this definition, CT delim- its its scope of investigation. Having identified the set of metaphorical expressions in question, CT then investigates what the underlying semantic or pragmatic mecha- nism is. This investigation is, at least partly, of an empirical nature. By, contrast CMT stipulates a priori the metaphorical mechanism, i.e. the cross-domain mappings, and then empirically investigates the various realizations of this mechanism in language and other social practices. It appears that the mechanism of conceptual metaphor covers not only poetic expressions, but also much of ordinary everyday language. There is, thus, apparent disagreement among these theories that stems from their different assumptions. Lakoff basically says that CT is wrong because its assump- tion about the literal-metaphorical distinction is at odds with his empirical account of this distinction. But from the standpoint of CT we could raise the objection that CMT is wrong because it assumes a mechanism that is too broad, for it covers not only novel uses of language, but also a great portion of conventional language. 16
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