Classical Equations Of Motion Research Articles

Classical trajectories as solutions of the Schrödinger equation

A method is developed for deriving solutions to a classical equation of motion with a time-dependent mass and frequency in terms of exact solutions to the one-dimensional Schrodinger equation. The method works in two directions. Analytical solutions of the classical equation of motion may be used to find the interaction potential to exactly solve the Schrodinger equation. Two simple examples have been considered that illustrate possible applications of the method.

Phases of holographic d-wave superconductor

We study different phases in the holographic model of d-wave superconductor. These are described by solutions to the classical equations of motion found in different ansatze. Apart from the known homogeneous d-wave superconducting phase we find three new solutions. Two of them represent two distinct families of the spatially modulated solutions, which realize the charge density wave phases in the dual theory. The third one is the new homogeneous phase with nonzero anapole moment. These phases are relevant to the physics of cuprate high-Tc superconductor in pseudogap region. While the d-wave phase preserves translation, parity and time reversal symmetry, the striped phases break translations spontaneously. Parity and time-reversal are preserved when combined with discrete half-periodic shift of the wave. In anapole phase translation symmetry is preserved, but parity and time reversal are spontaneously broken. All of the considered solutions break the global U(1). Thermodynamical treatment shows that in the simplest d-wave model the anapole phase is always preferred, while the stripe phases realize the continuous transition in solution space between the normal phase and two homogeneous condensed phases.

Mixed Quantum/Classical Theory for Molecule-Molecule Inelastic Scattering: Derivations of Equations and Application to N2 + H2 System.

The mixed quantum classical theory, MQCT, for inelastic scattering of two molecules is developed, in which the internal (rotational, vibrational) motion of both collision partners is treated with quantum mechanics, and the molecule-molecule scattering (translational motion) is described by classical trajectories. The resultant MQCT formalism includes a system of coupled differential equations for quantum probability amplitudes, and the classical equations of motion in the mean-field potential. Numerical tests of this theory are carried out for several most important rotational state-to-state transitions in the N2 + H2 system, in a broad range of collision energies. Besides scattering resonances (at low collision energies) excellent agreement with full-quantum results is obtained, including the excitation thresholds, the maxima of cross sections, and even some smaller features, such as slight oscillations of energy dependencies. Most importantly, at higher energies the results of MQCT are nearly identical to the full quantum results, which makes this approach a good alternative to the full-quantum calculations that become computationally expensive at higher collision energies and for heavier collision partners. Extensions of this theory to include vibrational transitions or general asymmetric-top rotor (polyatomic) molecules are relatively straightforward.

Three-dimensional superintegrable systems in a static electromagnetic field

We consider a charged particle moving in a static electromagnetic field described by the vector potential and the electrostatic potential We study the conditions on the structure of the integrals of motion of the first and second order in momenta, in particular how they are influenced by the gauge invariance of the problem. Next, we concentrate on the three possibilities for integrability arising from the first order integrals corresponding to three nonequivalent subalgebras of the Euclidean algebra, namely and For these cases we look for additional independent integrals of first or second order in the momenta. These would make the system superintegrable (minimally or maximally). We study their quantum spectra and classical equations of motion. In some cases nonpolynomial integrals of motion occur and ensure maximal superintegrability.

Point Particle with Extrinsic Curvature as a Boundary of a Nambu–Goto String: Classical and Quantum Model

It is shown how a string living in a higher dimensional space can be approximated as a point particle with squared extrinsic curvature. We consider a generalized Howe–Tucker action for such a “rigid particle” and consider its classical equations of motion and constraints. We find that the algebra of the Dirac brackets between the dynamical variables associated with velocity and acceleration contains the spin tensor. After quantization, the corresponding operators can be represented by the Dirac matrices, projected onto the hypersurface that is orthogonal to the direction of momentum. A condition for the consistency of such a representation is that the states must satisfy the Dirac equation with a suitable effective mass. The Pauli–Lubanski vector composed with such projected Dirac matrices is equal to the Pauli–Lubanski vector composed with the usual, non projected, Dirac matrices, and its eigenvalues thus correspond to spin one half states.

Quantum transitions between classical histories

In a quantum theory of gravity spacetime behaves classically when quantum probabilities are high for histories of geometry and field that are correlated in time by the Einstein equation. Probabilities follow from the quantum state. This quantum perspective on classicality has important implications: (a) Classical histories are generally available only in limited patches of the configuration space on which the state lives. (b) In a given patch states generally predict relative probabilities for an ensemble of possible classical histories. (c) In between patches classical predictability breaks down and is replaced by quantum evolution connecting classical histories in different patches. (d) Classical predictability can break down on scales well below the Planck scale, and with no breakdown in the classical equations of motion. We support and illustrate (a)-(d) by calculating the quantum transition across the de Sitter like throat connecting asymptotically classical, inflating histories in the no-boundary quantum state. This supplies probabilities for how a classical history on one side transitions and branches into a range of classical histories on the opposite side. We also comment on the implications of (a)-(d) for the dynamics of black holes and eternal inflation.

Soliton solutions of classical equations of motions in the modified formulation of the Yang-Mills theory

Показано, что в модифицированной формулировке классической теории Янга-Миллса, приводящей к формальной теории возмущений, которая совпадает со стандартной, существуют солитонные решения классических уравнений движения.

Soliton solutions of classical equations of motions in the modified formulation of the Yang–Mills theory

We show that soliton solutions of classical field equations exist in the modified formulation of the Yang–Mills theory, which produces the same formal perturbation theory as the standard formulation.

Derivation of the Drude conductivity from quantum kinetic equations

The Drude formula of ac (frequency-dependent) electric conductivity has been established as a simple and practically useful model to understand the electromagnetic response of simple free-electron-like metals. In most textbooks of solid-state physics, the Drude formula is derived from either a classical equation of motion or the semiclassical Boltzmann transport equation. On the other hand, quantum-mechanical derivation of the Drude conductivity, which requires an appropriate treatment of phonon-assisted intraband transitions with small momentum transfer, has not been well documented except for the zero- or high-frequency case. Here, a lucid derivation of the Drude conductivity that covers the entire frequency range is presented by means of quantum kinetic equations in the density-matrix formalism. The derivation is straightforward so that advanced undergraduate students or early-year graduate students will be able to gain insight into the link between the microscopic Schrödinger equation and macroscopic transport.

Extended sigma-model in nontrivially deformed field–antifield formalism

We propose an action for the extended sigma-models in the most general setting of the kinetic term allowed in the nontrivially deformed field–antifield formalism. We show that the classical motion equations do naturally take their desired canonical form.

Methodology for Vertical-Navigation Flight-Trajectory Cost Calculation Using a Performance Database

Trajectory optimization has been identified as an important way to reduce flight costs and polluting emissions. Due to the power capacity limitations in airborne devices such as the flight management system, a fast method should be implemented to calculate the full trajectory cost. Many flight management systems use a set of lookup tables with experimental data for each flight phase, and they are called performance databases. In this paper, the trajectory flight cost is calculated using a performance database instead of using classical equations of motion. The trajectory to be calculated is composed of climb, acceleration, cruise, descent, and deceleration. The influence of the crossover altitude during climb and descent, as well as step climbs in cruise, was considered. Lagrange linear interpolations were performed within the performance database discrete values to calculate the required values. By providing a takeoff weight, the initial and final coordinates, and the desired flight plan, the trajectory model provides the top-of-climb coordinates, the top-of-descent coordinates, the fuel burned, and the flight time needed to follow the given flight plan. The accuracy of the trajectory costs calculated with the proposed method was validated with an aerodynamic model in FlightSIM®, which is software developed by Presagis®, and with the trajectory cost given by the flight management system benchmark of reference. Results showed that, for the same reference trajectories and for the same inputs, the cost computed by the method proposed in this paper is close to the costs provided by FlightSIM and by the flight management system benchmark or reference.

Enhanced Sampling of an Atomic Model with Hybrid Nonequilibrium Molecular Dynamics-Monte Carlo Simulations Guided by a Coarse-Grained Model.

Molecular dynamics (MD) trajectories based on a classical equation of motion provide a straightforward, albeit somewhat inefficient approach, to explore and sample the configurational space of a complex molecular system. While a broad range of techniques can be used to accelerate and enhance the sampling efficiency of classical simulations, only algorithms that are consistent with the Boltzmann equilibrium distribution yield a proper statistical mechanical computational framework. Here, a multiscale hybrid algorithm relying simultaneously on all-atom fine-grained (FG) and coarse-grained (CG) representations of a system is designed to improve sampling efficiency by combining the strength of nonequilibrium molecular dynamics (neMD) and Metropolis Monte Carlo (MC). This CG-guided hybrid neMD-MC algorithm comprises six steps: (1) a FG configuration of an atomic system is dynamically propagated for some period of time using equilibrium MD; (2) the resulting FG configuration is mapped onto a simplified CG model; (3) the CG model is propagated for a brief time interval to yield a new CG configuration; (4) the resulting CG configuration is used as a target to guide the evolution of the FG system; (5) the FG configuration (from step 1) is driven via a nonequilibrium MD (neMD) simulation toward the CG target; (6) the resulting FG configuration at the end of the neMD trajectory is then accepted or rejected according to a Metropolis criterion before returning to step 1. A symmetric two-ends momentum reversal prescription is used for the neMD trajectories of the FG system to guarantee that the CG-guided hybrid neMD-MC algorithm obeys microscopic detailed balance and rigorously yields the equilibrium Boltzmann distribution. The enhanced sampling achieved with the method is illustrated with a model system with hindered diffusion and explicit-solvent peptide simulations. Illustrative tests indicate that the method can yield a speedup of about 80 times for the model system and up to 21 times for polyalanine and (AAQAA)3 in water.

Toward the correspondence between Q-clouds and sphalerons

Non-linear classical equations of motion may admit degenerate solutions at fixed charges. Whereas the solutions with lower energies are classically stable, those with larger energies are unstable and are referred to as Q-clouds. We consider a theory in which a homogeneous charged condensate is classically stable and argue that Q-clouds correspond to sphalerons between the stable Q-balls and the condensate. For a model with an analytical solution, we present the Arrhenius formula for the quantum production of Q-balls from a condensate at high temperatures.

Hyperscaling violation and electroweak symmetry breaking

We consider a class of simplified models of dynamical electroweak symmetry breaking built in terms of their five-dimensional weakly-coupled gravity duals, in the spirit of bottom-up holography. The sigma-model consists of two abelian gauge bosons and one real, non-charged scalar field coupled to gravity in five dimensions. The scalar potential is a simple exponential function of the scalar field. The background metric resulting from solving the classical equations of motion exhibits hyperscaling violation, at least at asymptotically large values of the radial direction. We study the spectrum of scalar composite states of the putative dual field theory by fluctuating the sigma-model scalars and gravity, and discuss in which cases we find a parametrically light scalar state in the spectrum. We model the spontaneous breaking of the (weakly coupled) gauge symmetry to the diagonal subgroup by the choice of IR boundary conditions. We compute the mass spectrum of spin-1 states, and the precision electroweak parameter S as a function of the hyperscaling coefficient. We find a general bound on the mass of the lightest spin-1 resonance, by requiring that the indirect bounds on the precision parameters be satisfied, that implies that precision electroweak physics excludes the possibility of a techni-rho meson with mass lighter than several TeV.

Hadron formation from interaction among quarks

This paper deals with the hadronization process of quark system. A phenomenological potential is introduced to describe the interaction between a quark pair. The potential depends on the color charge of those quarks and their relative distances. Those quarks move according to classical equations of motion. Due to the color interaction, coloring quarks are separated to form color neutral clusters which are supposed to be the hadrons.

Representing the thermal state in time-dependent density functional theory.

Classical molecular dynamics (MD) provides a powerful and widely used approach to determining thermodynamic properties by integrating the classical equations of motion of a system of atoms. Time-Dependent Density Functional Theory (TDDFT) provides a powerful and increasingly useful approach to integrating the quantum equations of motion for a system of electrons. TDDFT efficiently captures the unitary evolution of a many-electron state by mapping the system into a fictitious non-interacting system. In analogy to MD, one could imagine obtaining the thermodynamic properties of an electronic system from a TDDFT simulation in which the electrons are excited from their ground state by a time-dependent potential and then allowed to evolve freely in time while statistical data are captured from periodic snapshots of the system. For a variety of systems (e.g., many metals), the electrons reach an effective state of internal equilibrium due to electron-electron interactions on a time scale that is short compared to electron-phonon equilibration. During the initial time-evolution of such systems following electronic excitation, electron-phonon interactions should be negligible, and therefore, TDDFT should successfully capture the internal thermalization of the electrons. However, it is unclear how TDDFT represents the resulting thermal state. In particular, the thermal state is usually represented in quantum statistical mechanics as a mixed state, while the occupations of the TDDFT wavefunctions are fixed by the initial state in TDDFT. We work to address this puzzle by (A) reformulating quantum statistical mechanics so that thermodynamic expectations can be obtained as an unweighted average over a set of many-body pure states and (B) constructing a family of non-interacting (single determinant) TDDFT states that approximate the required many-body states for the canonical ensemble.

Real-time quantum trajectories for classically allowed dynamics in strong laser fields

Both the physical picture of the dynamics of atoms and molecules in intense infrared fields and its theoretical description use the concept of electron trajectories. Here, we address a key question which arises in this context: Are distinctly quantum features of these trajectories, such as the complex-valued coordinates, physically relevant in the classically allowed region of phase space, and what is their origin? First, we argue that solutions of classical equations of motion can account for quantum effects. To this end, we construct an exact solution to the classical Hamilton–Jacobi equation which accounts for dynamics of the wave packet, and show that this solution is physically correct in the limit . Second, we show that imaginary components of classical trajectories are directly linked to the finite size of the initial wave packet in momentum space. This way, if the electronic wave packet produced by optical tunnelling in strong infrared fields is localised both in coordinate and momentum, its motion after tunnelling ipso facto cannot be described with purely classical trajectories – in contrast to popular models in the literature.

On the interaction of propagating intrinsic localized modes with an isolated impurity in a linear atomic chain

We study the interaction between an intrinsic localized mode and a substitutional defect in a linear chain of atoms interacting with their nearest neighbors through a translational invariant potential function with a harmonic and a quartic anharmonic terms. The classical equations of motion are solved by means of a numerical iterative procedure and moving solutions are sought. The frequencies and atomic displacements from the equilibrium positions are calculated and the interaction between the excitation and the impurity is observed through the changes of the mode profile and the frequency as the breather moves towards and through the impurity.

Matrix Mechanics of the Relativistic Point Particle and String in Clifford Space

We resolve the space-time canonical variables of the relativistic point particle into inner products of Weyl spinors with components in a Clifford algebra and find that these spinors themselves form a canonical system with generalized Poisson brackets. For N particles, the inner products of their Clifford coordinates and momenta form two NxN Hermitian matrices X and P which transform under a U(N) symmetry in the generating algebra. This is used as a starting point for defining matrix mechanics for a point particle in Clifford space. Next we consider the string. The Lorentz metric induces a metric and a scalar on the world sheet which we represent by a Jackiw-Teitelboim term in the action. The string is described by a polymomenta canonical system and we find the wave solutions to the classical equations of motion for a flat world sheet. Finally, we show that the SL(2.C) charge and space-time momentum of the quantized string satisfy the Poincare algebra.

Non-thermal fixed point in a holographic superfluid

We study the far-from-equilibrium dynamics of a (2+1)-dimensional superfluid at finite temperature and chemical potential using its holographic description in terms of a gravitational system in 3+1 dimensions. Starting from various initial conditions corresponding to ensembles of vortex defects we numerically evolve the system to long times. At intermediate times the system exhibits Kolmogorov scaling the emergence of which depends on the choice of initial conditions. We further observe a universal late-time regime in which the occupation spectrum and different length scales of the superfluid exhibit scaling behaviour. We study these scaling laws in view of superfluid turbulence and interpret the universal late-time regime as a non-thermal fixed point of the dynamical evolution. In the holographic superfluid the non-thermal fixed point can be understood as a stationary point of the classical equations of motion of the dual gravitational description.