Classical Equations Of Motion Research Articles

Classical Physics and Blackbody Radiation.

We investigate the properties of the blackbody spectrum by direct numerical solution of the classical equations of motion of a one-dimensional model that contains the essential general features of the field-matter interaction. Our results, which do not rely on any statistical assumption, show that the classical blackbody spectrum exhibits remarkable properties: (i)a quasistationary state characterized by scaling properties, (ii)consistency with the Stefan-Boltzmann law, and (iii)a high-frequency cutoff. Our Letter is a preliminary step in the understanding of statistical properties of infinite-dimensional systems.

Application of a Trajectory Tracking Algorithm for Underactuated Underwater Vehicles Using Quasi-Velocities

In this work, an application of the trajectory tracking algorithm proposed in the literature for underactuated marine vehicles is presented. The main difference relies on that here the dynamics of the vehicle are expressed in terms of some quasi-velocities (QV). This fact has a double meaning. First of all, it is shown that using the QV, it is possible to control a vehicle in the absence of one variable because the works related to marine vehicles have only concerned fully actuated systems. In addition, a controller using QV provides information that gives some insight into vehicle dynamics and that is not available in classical equations of motion. The simulations done on two 3-DOF models of different underwater vehicles and using two desired trajectories show performance of the considered control strategy. A discussion of the presented control scheme and selected control approaches from recent years was also conducted, and the benefits of the proposed approach were pointed out.

Conservative and radiative dynamics in classical relativistic scattering and bound systems

As recent work continues to demonstrate, the study of relativistic scattering processes leads to valuable insights and computational tools applicable to the relativistic bound-orbit two-body problem. This is particularly relevant in the post-Minkowskian approach to the gravitational two-body problem, where the field has only recently reached a full description of certain physical observables for scattering orbits, including radiative effects, at the third post-Minkowskian (3PM) order. As an historically instructive simpler example, we consider here the analogous problem in electromagnetism in flat spacetime. We compute the changes in linear momentum of each particle and the total radiated linear momentum, in the relativistic classical scattering of two point-charges, at sixth order in the charges (analogous to 3PM order in gravity). We accomplish this here via direct iteration of the classical equations of motion, while making comparisons where possible to results from quantum scattering amplitudes, with the aim of contributing to the elucidation of conceptual issues and scalability on both sides. We also discuss further extensions to radiative quantities of recently established relations, which analytically continue certain observables from the scattering regime to the regime of bound orbits, applicable for both the electromagnetic and gravitational cases.

Non-quantum chirality in a driven Brusselator

We report the discovery of non-quantum chirality in the a periodically driven Brusselator. In contrast to standard chirality from quantum contexts, this novel type of chirality is governed by rate equations, namely by purely classical equations of motion. The Brusselator chirality was found by computing high-resolution phase diagrams depicting the number of spikes, local maxima, observed in stable periodic oscillations of the Brusselator as a function of the frequency and amplitude of the external drive. We also discuss how to experimentally observe non-quantum chirality in generic oscillators governed by nonlinear sets of rate equations.

A reciprocal-space formulation of surface hopping.

Surface hopping has seen great success in describing molecular phenomena where electronic excitations tend to be localized, but its application to materials with band-like electronic properties has remained limited. Here, we derive a formulation of fewest-switches surface hopping where both the quantum and classical equations of motion are solved entirely in terms of reciprocal-space coordinates. The resulting method is directly compatible with band structure calculations and allows for the efficient description of band-like phenomena by means of a truncation of the Brillouin zone. Using the Holstein and Peierls models as examples, we demonstrate the formal equivalence between real-space and reciprocal-space surface hopping and assess their accuracy against mean-field mixed quantum-classical dynamics and numerically exact results. Having very similar equations of motion, reciprocal-space surface hopping can be straightforwardly incorporated in existing (real-space) surface hopping implementations.

Quantification of gas exsolution and preferential diffusion for alkane solvent(s)–CO2–heavy oil systems under nonequilibrium conditions

The in-situ formation of foamy oil has been found to be a crucial mechanism accounting for the better-than-expected production performance in heavy oil reservoirs under solution gas drive. Not only have the physical laws dominating gas exsolution in foamy oil not yet been well understood, but also the different contributions of each component in a gas mixture to the generation of foamy oil associated with an extremely complicated dynamic process has not been quantified. In this study, a novel and pragmatic technique has been proposed and validated to quantify the gas exsolution and preferential diffusion for alkane solvent(s)–CO2–heavy oil systems under nonequilibrium conditions by taking gas bubble size distribution and preferential mass transfer of each gas component into account. Experimentally, constant-composition expansion (CCE) tests with various constant-pressure decline rates are utilized to describe gas exsolution behaviour of alkane solvent(s)–CO2–heavy oil systems under nonequilibrium conditions, during which not only pressure and volume are simultaneously monitored and measured, but also gas samples were respectively collected at the beginning and end of experiments to perform compositional analysis. Theoretically, a mathematical model has been formulated to quantify gas exsolution and preferential mass transfer between each gas component and the liquid phase in alkane solvent(s)–CO2–heavy oil systems under nonequilibrium conditions. More specifically, quasi-equilibrium boundary conditions, real gas equation, and Rayleigh distribution function are combined with the classical equation of motion, continuity equation, and mass transfer equation to form a novel equation matrix for quantifying gas bubble growth in foamy oil. Considering gas bubble size distribution and preferential diffusion of each component in a gas mixture, the total number of gas bubbles and individual diffusion coefficient of each gas component are determined by minimizing the discrepancy between the measured and calculated volume of alkane solvent(s)–CO2–heavy oil systems. More importantly, the dynamic composition of the gas phase and the amounts of both entrained gas and evolved gas can also be obtained simultaneously during the gas exsolution processes. Excellent agreements between the experimentally measured parameters (i.e., volume of foamy oil, composition of evolved gas, and volume of free gas) and the calculated ones have been respectively achieved. Compared with the individual diffusion coefficient for each component in a gas mixture determined under the traditional equilibrium conditions, a relatively large value has been found during mass transfer processes in a supersaturated oleic phase. Moreover, pseudo-bubblepoint pressure and gas exsolution rate are found to be two main mechanisms dominating the volume-growth rate of the evolved gas.

Lorentz-Abraham-Dirac and Landau-Lifshitz equations of motion and the solution to a relativistic electron in a counterpropagating laser beam

Beginning with a critical examination of the Lorentz-Abraham (LA) classical equation of motion for an extended charge and the closely related Lorentz-Abraham-Dirac (LAD) equation of motion for a mass-renormalized point-charge, the Landau-Lifshitz (LL) approximate solution to the LAD equation of motion is determined for an electron subject to a counterpropagating linearly or circularly polarized plane-wave pulse with an arbitrarily shaped envelope. A convenient three-vector formulation of the LL equation is used to derive closed-form expressions for the velocities and associated powers of the electron directly in terms of the time in the laboratory frame. The three-vector formulation also reveals definitive criteria for the LL solution to be an accurate approximation to the LAD equation of motion and for the LL solution to reduce to the solution of the Lorentz force equation of motion that ignores radiation reaction. Semiclassical analyses are used to obtain simple conditions for determining the regimes where the quantum effects of either Compton electron scattering by the incident photons or electron recoil produced by the emitted photons is significant. It is proven that the LL approximation becomes an inaccurate solution to the LAD equation of motion only for large enough electron velocities and plane-wave intensities that quantum recoil effects on the electron can greatly alter the classical solution. Comparisons are made with previously published analytical and numerical solutions to the LL equation of motion for the velocity of an electron in a counterpropagating plane wave.10 MoreReceived 18 August 2021Accepted 1 November 2021DOI:https://doi.org/10.1103/PhysRevAccelBeams.24.114002Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasElectromagnetismElectron beams & opticsLaser driven electron accelerationQuantum theorySingle-particle dynamicsAccelerators & BeamsGeneral PhysicsParticles & FieldsNonlinear DynamicsCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Energy and Magnetic Moment of a Quantum Charged Particle in Time-Dependent Magnetic and Electric Fields of Circular and Plane Solenoids.

We consider a quantum spinless nonrelativistic charged particle moving in the plane under the action of a time-dependent magnetic field, described by means of the linear vector potential , with two fixed values of the gauge parameter : (the circular gauge) and (the Landau gauge). While the magnetic field is the same in all the cases, the systems with different values of the gauge parameter are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are circles for and straight lines for . We derive general formulas for the time-dependent mean values of the energy and magnetic moment, as well as for their variances, for an arbitrary function . They are expressed in terms of solutions to the classical equation of motion , with . Explicit results are found in the cases of the sudden jump of magnetic field, the parametric resonance, the adiabatic evolution, and for several specific functions , when solutions can be expressed in terms of elementary or hypergeometric functions. These examples show that the evolution of the mentioned mean values can be rather different for the two gauges, if the evolution is not adiabatic. It appears that the adiabatic approximation fails when the magnetic field goes to zero. Moreover, the sudden jump approximation can fail in this case as well. The case of a slowly varying field changing its sign seems especially interesting. In all the cases, fluctuations of the magnetic moment are very strong, frequently exceeding the square of the mean value.

Thermal gauge theories with Lagrange multiplier fields

We study the Yang–Mills theory and quantum gravity at finite temperature, in the presence of Lagrange multiplier fields. These restrict the path integrals to field configurations, which obey the classical equations of motion. This has the effect of doubling the usual one-loop thermal contributions and of suppressing all radiative corrections at higher-loop order. Such theories are renormalizable at all temperatures. Some consequences of this result in quantum gravity are briefly examined.

Energy and pairing dependence of dissipation in real-time fission dynamics

This work presents a microscopic study of dissipation in fission dynamical evolutions with the time-dependent Hartree-Fock+BCS method in terms of energy dependence and pairing dependence. The friction coefficients and dissipation energies are extracted by mapping the symmetric fission process of $^{258}$Fm into a classical equation of motion. Density-constrained calculations are used to obtain the dynamical potential. The obtained friction coefficients have a strong dependence of deformations, and averagely match the coefficients adopted in statistical models. The dissipation indeed increase with increasing initial excitation energies or with decreasing pairings. It is also shown that post-scission dissipations play a significant role. The demonstrated characteristics of dissipations will be valuable for calibrations of various fission models.

Restricting loop expansions in gauge theories coupled to matter

Quantizing any model in which a Lagrange multiplier (LM) field is used to restrict field configurations to those that satisfy the classical equations of motion, leads to at most one-loop radiative corrections. This approach can be used with both the Yang–Mills (YM) and Einstein–Hilbert (EH) action; the resulting theory is both renormalizable and unitary, has a positive energy spectrum and has no negative norm states contributing to physical processes. Although this approach cannot be consistently used with scalar fields alone, scalar fields can be coupled to gauge fields so that loop effects in the gauge sector are restricted to one-loop order in a way that satisfies the usual criterion for a consistent quantum field theory. The tree-level diagrams are those of the classical theory in which the metric couples to the energy–momentum tensor.

Control of concerted back-to-back double ionization dynamics in helium.

Double ionization (DI) is a fundamental process that despite its apparent simplicity provides rich opportunities for probing and controlling the electronic motion. Even for the simplest multielectron atom, helium, new DI mechanisms are still being found. To first order in the field strength, a strong external field doubly ionizes the electrons in helium such that they are ejected into the same direction (front-to-back motion). The ejection into opposite directions (back-to-back motion) cannot be described to first order, making it a challenging target for control. Here, we address this challenge and optimize the field with the objective of back-to-back double ionization using a (1 + 1)-dimensional model. The optimization is performed using four different control procedures: (1) short-time control, (2) derivative-free optimization of basis expansions of the field, (3) the Krotov method, and (4) control of the classical equations of motion. All four procedures lead to fields with dominant back-to-back motion. All the fields obtained exploit essentially the same two-step mechanism leading to back-to-back motion: first, the electrons are displaced by the field into the same direction. Second, after the field turns off, the nuclear attraction and the electron-electron repulsion combine to generate the final motion into opposite directions for each electron. By performing quasi-classical calculations, we confirm that this mechanism is essentially classical.

A bosonic perspective on the classical mapping of fermionic quantum dynamics

We consider the application of the original Meyer-Miller (MM) Hamiltonian to mapping fermionic quantum dynamics to classical equations of motion. Non-interacting fermionic and bosonic systems share the same one-body density dynamics when evolving from the same initial many-body state. The MM classical mapping is exact for non-interacting bosons, and therefore, it yields the exact time-dependent one-body density for non-interacting fermions as well. Starting from this observation, the MM mapping is compared to different mappings specific for fermionic systems, namely, the spin mapping with and without including a Jordan-Wigner transformation and the Li-Miller mapping (LMM). For non-interacting systems, the inclusion of fermionic anti-symmetry through the Jordan-Wigner transform does not lead to any improvement in the performance of the mappings, and instead, it worsens the classical description. For an interacting impurity model and for models of excitonic energy transfer, the MM and LMM mappings perform similarly, and in some cases, the former outperforms the latter when compared to a full quantum description. The classical mappings are able to capture interference effects, both constructive and destructive, that originate from equivalent energy transfer pathways in the models.

Backreaction of Schwinger pair creation in massive QED2

Particle-antiparticle pairs can be produced by background electric fields via the Schwinger mechanism provided they are unconfined. If, as in QED in (3+1)-d these particles are massive, the particle production rate is exponentially suppressed below a threshold field strength. Above this threshold, the energy for pair creation must come from the electric field itself which ought to eventually relax to the threshold strength. Calculating this relaxation in a self-consistent manner, however, is difficult. Chu and Vachaspati addressed this problem in the context of capacitor discharge in massless QED2 [1] by utilizing bosonization in two-dimensions. When the bare fermions are massless, the dual bosonized theory is free and capacitor discharge can be analyzed exactly [1], however, special care is required in its interpretation given that the theory exhibits confinement. In this paper we reinterpret the findings of [1], where the capacitors Schwinger-discharge via electrically neutral dipolar meson-production, and generalize this to the case where the fermions have bare masses. Crucially, we note that when the initial charge of the capacitor is large compared to the charge of the fermions, Q » e, the classical equation of motion for the bosonized model accurately characterizes the dynamics of discharge. For massless QED2, we find that the discharge is suppressed below a critical plate separation that is commensurate with the length scale associated with the meson dipole moment. For massive QED2, we find in addition, a mass threshold familiar from (3+1)-d, and show the electric field relaxes to a final steady state with a magnitude proportional to the initial charge. We discuss the wider implications of our findings and identify challenges in extending this treatment to higher dimensions.

The quantum pencil revisited

A quantum description is used to find the time limit for balancing an idealized pencil-like object on end. In agreement with some prior studies, analysis for this system shows that quantum effects are not observable in any foreseeable circumstances, either by casual experimentation or by extensive laboratory preparation. A direct solution of Schrodinger’s equation avoids an a priori use of the Heisenberg uncertainty relation in the classical equations of motion, which, although very much a proper usage, may lead to misunderstanding of the results. The classical equations of motion are adequate for the pencil in all circumstances.

Quantum-classical dynamical brackets

We study the problem of constructing a general hybrid quantum-classical bracket from a partial classical limit of a full quantum bracket. Introducing a hybrid composition product, we show that such a bracket is the commutator of that product. From this we see that the hybrid bracket will obey the Jacobi identity and the Leibniz rule provided the composition product is associative. This suggests that the set of hybrid variables belonging to an associative subalgebra with the composition product will have consistent quantum-classical dynamics. This restricts the class of allowed quantum-classical interaction Hamiltonians. Furthermore, we show that pure quantum or classical variables can interact in a consistent framework, unaffected by no-go theorems in the literature or the restrictions for hybrid variables. In the proposed scheme, quantum backreaction appears as quantum-dependent terms in the classical equations of motion.

On the consistency of the Lagrange multiplier method in classical mechanics

Problems involving rolling without slipping or no sideways skidding, to name a few, introduce velocity-dependent constraints that can be efficiently treated by the method of Lagrange multipliers in the Lagrangian formulation of the classical equations of motion. In doing so, one finds, as a bonus, the constraint forces, which must be independent of the solution of the equations of motion and can only depend on the generalized coordinates and velocities, as well as time. In this paper, we establish conditions the Lagrangian and the constraints should obey in order to guarantee that the constraint forces can be obtained consistently.

From local to nonlocal: higher fidelity simulations of photon emission in intense laser pulses

State-of-the-art numerical simulations of quantum electrodynamical (QED) processes in strong laser fields rely on a semiclassical combination of classical equations of motion and QED rates, which are calculated in the locally constant field approximation. However, the latter approximation is unreliable if the amplitude of the fields, a 0, is comparable to unity. Furthermore, it cannot, by definition, capture interference effects that give rise to harmonic structure. Here we present an alternative numerical approach, which resolves these two issues by combining cycle-averaged equations of motion and QED rates calculated in the locally monochromatic approximation. We demonstrate that it significantly improves the accuracy of simulations of photon emission across the full range of photon energies and laser intensities, in plane-wave, chirped and focused background fields.

Supersymmetry breaking and stability in string vacua

We review some aspects of the dramatic consequences of supersymmetry breaking on string vacua. In particular, we focus on the issue of vacuum stability in ten-dimensional string models with broken, or without, supersymmetry, whose perturbative spectra are free of tachyons. After formulating the models at stake, we introduce their unified low-energy effective description and present a number of vacuum solutions to the classical equations of motion. In addition, we present a generalization of previous no-go results for de Sitter vacua in warped flux compactifications. Then we analyze the classical and quantum stability of these vacua, studying linearized field fluctuations and bubble nucleation. Then, we describe how the resulting instabilities can be framed in terms of brane dynamics, examining in particular brane interactions, back-reacted geometries and commenting on a brane-world string construction along the lines of a recent proposal. After providing a summary, we conclude with some perspectives on possible future developments.

Effective Toughness of Heterogeneous Materials with Rate-Dependent Fracture Energy.

We investigate the dynamic fracture of heterogeneous materials experimentally by measuring displacement fields as a rupture propagates through a periodic array of obstacles of controlled fracture energy. Our measurements demonstrate the applicability of the classical equation of motion of cracks at a discontinuity of fracture energy: the crack speed jumps at the entrance and exit of an obstacle, as predicted by the crack-tip energy balance within the brittle fracture framework. The speed jump amplitude is governed by the fracture energy contrast and by the combination of the rate dependency of the fracture energy and the inertia of the medium, which allows the crack to cross a fracture energy discontinuity at a constant energy release rate. This discontinuous dynamics and the rate dependence cause higher effective toughness, which governs the coarse-grained behavior of these cracks.