Classical Equations Of Motion Research Articles

Imaginary past of a quantum particle moving on imaginary time

The analytical continuation of classical equations of motion to complex times suggests that a tunneling particle spends in the barrier an imaginary duration i|T|. Does this mean that it takes a finite time to tunnel, or should tunneling be seen as an instantaneous process? It is well known that examination of the adiabatic limit in a small additional AC field points towards |T| being the time it takes to traverse the barrier. However, this is only half the story. We probe the transmitted particle's history, and find that it very little of the field's past behavior, as if the transit time were close to zero. The ensuing contradiction suggests that the question is ill posed, and we explain why. Published by the American Physical Society 2024

Higher Time-Derivative Theories from Space–Time Interchanged Integrable Field Theories

We compare a relativistic and a nonrelativistic version of Ostrogradsky’s method for higher-time derivative theories extended to scalar field theories and consider as an alternative a multi-field variant. We apply the schemes to space–time rotated modified Korteweg–de Vries systems and, exploiting their integrability, to Hamiltonian systems built from space–time rotated inverse Legendre transformed higher-order charges of these systems. We derive the equal-time Poisson bracket structures of these theories, establish the integrability of the latter theories by means of the Painlevé test and construct exact analytical period benign solutions in terms of Jacobi elliptic functions to the classical equations of motion. The classical energies of these partially complex solutions are real when they respect a certain modified CPT-symmetry and complex when this symmetry is broken. The higher-order Cauchy and initial-boundary value problem are addressed analytically and numerically. Finally, we provide the explicit quantization of the simplest mKdV system, exhibiting the usual conundrum of having the choice between having to deal with either a theory that includes non-normalizable states or spectra that are unbounded from below. In our non-Hermitian system, the choice is dictated by the correct sign in the decay width.

Field redefinition invariant Lagrange multiplier formalism with gauge symmetries

It has been shown that by using a Lagrange multiplier field to ensure that the classical equations of motion are satisfied, radiative effects beyond one-loop order are eliminated. It has also been shown that through the contribution of some additional ghost fields, the effective action becomes form invariant under a redefinition of field variables, and furthermore, the usual one-loop results coincide with the quantum corrections obtained from this effective action. In this paper, we consider the consequences of a gauge invariance being present in the classical action. The resulting gauge transformations for the Lagrange multiplier field as well as for the additional ghost fields are found. These gauge transformations result in a set of Faddeev–Popov ghost fields arising in the effective action. If the gauge algebra is closed, we find the Becci–Rouet–Stora–Tyutin (BRST) transformations that leave the effective action invariant.

Activation energy analysis of mobile microorganisms using conductive nanofluid flows: Mitigating toxic algal blooms in biotechnology applications

Motile microorganisms play a central role in ecosystems in both natural habitats and laboratory settings. Their contribution to nutrient cycling is crucial and their impact on human health can be significant. Furthermore, they are promising in various biotechnological applications. Ongoing studies of microbial motility represent a dynamic and evolving field of research with profound implications for our understanding of biology, ecology, and environmental science. The thermal properties of the base material are extremely advanced and enable a wide range of industrial, technical and process applications due to their thermal radiation, variable heat transfer properties and activation energy. Researchers are still working hard to find renewable energy sources that are both economical and environmentally friendly. In this regard, the past decade has seen growing interest in the potential of nanoparticles as renewable energy sources. The current study aimed to learn more about the rheological properties of thixotropic nanofluids on the surface of Riga surrounded by rotating bacteria. For this, theoretical investigations have been performed on 3D magneto-hydrodynamic micropolar-based Casson nanofluid on the surface of Riga surrounded by rotating bacteria. The influence of thermal radiation, activation energy and heat generation are considered in the present scenario. The rheology of Brownian motion, micro rotation and thermophoresis also accounted for. Classical equations of motion in the form of PDEs are transformed into ODEs by applying similarity variables. The converted ODEs are then solved with the help of a shooting algorithm using Bvp4c MATLAB software. From an engineering point of view, the impression of physical flow parameters dimensionless profiles is demonstrated graphically and in the form of tables. The outcomes show that enhancing the porosity parameter significantly reduces the velocity profiles while the opposite behaviour is noted for the viscoelastic parameter. Moreover, temperature and concentration fields are boosted by thermophoresis and concentration exponents. Present results are validated with the existing ones. The current study has the potential to improve the stability achieved by the bioconvection of nanomaterials.

Universal Definition of the Nonconformal Trace Anomaly.

We show that there exists a generalized, universal notion of the trace anomaly for theories which are not conformally invariant at the classical level. The definition is suitable for any regularization scheme and clearly states to what extent the classical equations of motion should be used, thus resolving existing controversies surrounding previous proposals. Additionally, we exhibit the link between our definition of the anomaly and the functional Jacobian arising from a Weyl transformation.

Molecular Simulation Study of Surface-Enhanced Raman Scattering of Liquid Water.

We developed in our previous study [J. Chem. Phys., 2021, 155, 174118, J. Phys. Chem. A, 2022, 126, 4762] a classical electronic and molecular dynamics simulation method to describe the optical response of metal material in solution based on an atomistic model by incorporating the classical equation of motion for free electrons under an applied electric field. To show further usefulness of the method, in the present study, we apply it to surface-enhanced Raman scattering of liquid water to examine the signal enhancement of the solution system caused by plasmon resonance effects of a silver nanoparticle.

Molecular dynamics simulation of 100 keV argon ion radiation effects in metals and comparison with stopping and range of ions in matter code

Simulation is essential to analyse radiation-induced effects because energetic ions have short transit times within materials (Ar ions of 1 keV and 100 keV have transit times of 0.29 picoseconds and 29 femtoseconds respectively, through a 25 nm Al film). The binary collision approximation (BCA) is extensively used in simulations to study damage caused by energetic ions due to its computational efficiency. Stopping and Range of Ions in Matter (SRIM), a widely used Monte Carlo software using BCA, provides accurate ion ranges consistent with experiments. Molecular dynamics simulations are powerful computational method, where classical equations of motion are solved to track atomic movements from radiation damage, but computational resources required are heavier for this method. In our current effort, the energy transfer from 100 keV Ar+ to metal targets is analysed as a function of target properties, including melting temperature and atomic density, using these two methodologies. A detailed investigation of 100 keV Ar+ ion impact on metal targets is done using the classical Molecular Dynamics simulation program ‘MDRange’, which uses the Recoil Interaction Analysis (RIA) approximation, where the interaction between the ion and its closest neighbours is considered via a two-body potential. The correlation of energy transferred to the metal with the target properties is examined considering the physical approximations of each technique.

Visualizing Ultrasound Sources Using Signal Time Reversal in the Particle Dynamics Model

A method is proposed for solving the inverse problem of reconstructing acoustic wave sources from field measurements on some surface using wavefront reversal in the particle dynamics method. In this method, the studied medium is represented as a set of interacting particles (material points or solid bodies), for which classical equations of motion are written. The paper considers the representation of a medium as a set of particles in a body-centered cubic crystal lattice. The case of a linear dependence of the force of attraction of particles on distance is considered. The advantage of this approach is the ability to take into account wave propagation in arbitrarily inhomogeneous media using a single numerical model. The possibility of visualizing two spherical acoustic wave sources in water behind an obstacle has been demonstrated numerically and experimentally, despite the presence of transverse waves in the considered model of a solid body; their influence is negligible in this case. The method was tested experimentally on a soundproof screen with an aperture simulating a sound-emitting object of complex shape. A wave from a point source of short pulses passes through the aperture. Using a receiving acoustic sensor mounted on a two-dimensional scanner, the spatiotemporal distribution of sound vibrations on the water surface was measured. By processing the datausing wavefront reversal in the particle model, the image of the aperture in the soundproof screen was reconstructed.

Mass spectrum in a six-dimensional SU(n) gauge theory on a magnetized torus

We examine six-dimensional SU(n) gauge theories compactified on a two-dimensional torus with a constant magnetic flux background to obtain a comprehensive low-energy mass spectrum. We introduce general background configurations including the magnetic flux and continuous Wilson line phases, consistent with classical equations of motion. Under the standard gauge fixing procedure, the complete mass spectrum in low-energy effective theory for the SU(n) case is newly presented without imposing restrictions on the gauge fixing parameter. Our analysis confirms the inevitable existence of tachyonic modes, which neither depend on the background configurations of Wilson line phases nor are affected by the gauge fixing parameter. Masses for some low-energy modes exhibit dependence on the gauge fixing parameter, and these modes are identified as would-be Goldstone bosons that are absorbed by massive four-dimensional vector fields. We discuss the phenomenological implications associated with stabilization or condensation of the tachyonic states. Various mass spectra and symmetry-breaking patterns are expected with flux backgrounds in the SU(n) case. They are helpful for constructing phenomenologically viable models beyond the standard model, such as gauge-Higgs unification and grand unified theories.

Revealing quantum effects in bosonic Josephson junctions: a multi-configuration atomic coherent state approach

The mean-field approach to two-site Bose–Hubbard systems is well-established and leads to non-linear classical equations of motion for population imbalance and phase difference. It can, for example, be based on the representation of the solution of the time-dependent Schrödinger equation either by a single Glauber state or by a single atomic (SU(2)) coherent state [S. Wimberger et al., Phys. Rev. A 103, 023326 (2021)]. We demonstrate that quantum effects beyond the mean-field approximation are easily uncovered if, instead, a multiconfiguration ansatz with a few time-dependent SU(2) basis functions is used in the variational principle. For the case of plasma oscillations, the use of just two basis states, whose time-dependent parameters are determined variationally, already gives a good qualitative agreement of the phase space dynamics with numerically exact quantum solutions. In order to correctly account for more non-trivial effects, like macroscopic quantum self-trapping, moderately more basis states are needed. For the onset of spontaneous symmetry breaking, however, a multiplicity of 2 gives a significant improvement already. In any case, the number of variational trajectories needed for good agreement with the full quantum results is orders of magnitude smaller than that in the semi-classical case, which is based on multiple mean-field trajectories.

Gravity from thermodynamics: Optimal transport and negative effective dimensions

We prove an equivalence between the classical equations of motion governing vacuum gravity compactifications (and more general warped-product spacetimes) and a concavity property of entropy under time evolution. This is obtained by linking the theory of optimal transport to the Raychaudhuri equation in the internal space, where the warp factor introduces effective notions of curvature and (negative) internal dimension. When the Reduced Energy Condition is satisfied, concavity can be characterized in terms of the cosmological constant \LambdaΛ; as a consequence, the masses of the spin-two Kaluza-Klein fields obey bounds in terms of \LambdaΛ alone. We show that some Cheeger bounds on the KK spectrum hold even without assuming synthetic Ricci lower bounds, in the large class of infinitesimally Hilbertian metric measure spaces, which includes D-brane and O-plane singularities. As an application, we show how some approximate string theory solutions in the literature achieve scale separation, and we construct a new explicit parametrically scale-separated AdS solution of M-theory supported by Casimir energy.

Simple Physical Model for the Estimation of Irreversible Dissociation Rates for Bimolecular Complexes.

In this article, we propose a simple method of estimating dissociation rates of bimolecular van der Waals complexes ("wells"), rooted in rigid body dynamics, requiring as input parameters only the bimolecular binding energy, together with the intermolecular equilibrium distance and moments of inertia of the complex. The classical equations of motion are solved for the intermolecular and rotational degrees of freedom in a coordinate system considering only the relative motion of the two molecules, thus bypassing the question of whether the energy of the complex is statistically distributed. Well-escaping trajectories are modeled from these equations, and the escape rate as a function of relative velocity and angular momentum is fitted to an empirical function, which is then integrated over a probability distribution of said quantities. By necessity, this approach makes crude assumptions on the shape of the potential well and neglects the impact of energy quantization, and, more crucially, the coupling between the degrees of freedom included in the equations of motion with those that are not. We quantify the error caused by the first assumption by comparing our model potential with a quantum chemical potential energy surface (PES) and show that while the model does make several compromises and may not be accurate for all classes of bimolecular complexes, it is able to produce physically consistent dissociation rate coefficients within typical atmospheric chemistry confidence intervals for triplet state alkoxyl radical complexes, for which the detailed balance approach has been shown to fail.

Two-Dimensional Terahertz Spectroscopy of Nonlinear Phononics in the Topological Insulator MnBi_{2}Te_{4}.

The interaction of a single-cycle terahertz electric field with the topological insulator MnBi_{2}Te_{4} triggers strongly anharmonic lattice dynamics, promoting fully coherent energy transfer between the otherwise noninteracting Raman-active E_{g} and infrared (IR)-active E_{u} phononic modes. Two-dimensional terahertz spectroscopy combined with modeling based on the classical equations of motion and symmetry analysis reveals the multistage process underlying the excitation of the Raman-active E_{g} phonon. In this nonlinear combined photophononic process, the terahertz electric field first prepares a coherent IR-active E_{u} phononic state and subsequently interacts with this state to efficiently excite the E_{g} phonon.

Quasiclassical model of inhomogeneous cosmology

Fluctuation terms and higher moments of a quantum state imply corrections to the classical equations of motion that may have implications in early-Universe cosmology, for instance in the state-dependent form of effective potentials. In addition, space-time properties are relevant in cosmology, in particular when combined with quantum corrections required to maintain general covariance in a consistent way. Here, an extension of previous investigations of static quasiclassical space-time models to dynamical ones is presented, describing the evolution of one-dimensional space as in the classical Lemaitre–Tolman–Bondi models. The corresponding spatial metric has two independent components, both of which are in general subject to quantum fluctuations. The main result is that individual moments from both components are indeed required for general covariance to be maintained at a semiclassical level, while quantum correlations between the components are less relevant.

External gauge field-coupled quantum dynamics: Gauge choices, Heisenberg algebra representations and gauge invariance in general, and the Landau problem in particular

Even though its classical equations of motion are then left invariant, when an action is redefined by an additive total derivative or divergence term (in time, in the case of a mechanical system) such a transformation induces non-trivial consequences for the system’s canonical phase space formulation. This is even more true and then in more subtle ways for the canonically quantized dynamics, with, in particular, an induced transformation in the unitary configuration space representation of the Heisenberg algebra being used for the quantum system. When coupled to a background gauge field, such considerations become crucial for a proper understanding of the consequences for the system’s quantum dynamics of gauge transformations of that classical external background gauge field, while under such transformations, the system’s degrees of freedom, abstract quantum states and quantum dynamics are certainly strictly invariant. After a detailed analysis of these different points in a general context, these are then illustrated specifically in the case of the quantum Landau problem with its classical external background magnetic vector potential for which the most general possible parametrized gauge choice is implemented herein. The latter discussion aims as well to clarify some perplexing statements in the literature regarding the status of gauge choices to be made for the magnetic vector potential for that quantum system. The role of the global spacetime symmetries of the Landau problem and their gauge invariant Noether charges is then also emphasized.

Color-factor symmetry of the amplitudes of Yang-Mills and biadjoint scalar theory using perturbiner methods

Color-factor symmetry is a property of tree-level gauge-theory amplitudes containing at least one gluon. BCJ relations among color-ordered amplitudes follow directly from this symmetry. Color-factor symmetry is also a feature of biadjoint scalar theory amplitudes as well as of their equations of motion. In this paper, we present a new proof of color-factor symmetry using a recursive method derived from the perturbiner expansion of the classical equations of motion.

Simulation of classical axion electrodynamics using COMSOL multiphysics

The axion is a hypothetical particle motivated to address the strong CP problem, and is one of the appealing dark matter candidates. Numerous experimental searches for dark matter axions have been proposed relying on their coupling with photons. The classical equations of motion for the axion-photon coupling are well known but need to be fully computed for complex experimental setups. The partial differential equations of axion electrodynamics can be numerically solved using finite element methods. In this work, we simulate axion electrodynamics using COMSOL Multiphyics, a commercially available simulation software, for various experimental schemes, including the dish antenna haloscope, cavity haloscope, dielectric haloscope, and axion-photon regeneration. We show that the numerical results are in good agreement with the analytical solutions.

Nonlinear ion-stopping calculations for a classical free-electron gas at high projectile energies

In this work, we solved the classical equations of motion and Poisson equation self-consistently, equivalent to the nonlinear Vlasov-Poisson equation, for a projectile moving in a static free-electron gas to calculate the full noncentral self-consistent electron-ion potential, and thus the ion stopping power. We investigated the origin of the Barkas effect, namely, the first nonlinear effect for projectiles at high velocities responsible for the difference between the energy-loss results for positively and negatively charged ions traversing the same target. This effect is strongly enhanced by the multipolar part of the electron-ion potential as first suggested by Lindhard [J. Lindhard, Nucl. Instr. and Meth. 132, 1438 (1976)]. Moreover, this effect is partially related to the nonconservation of the angular momentum in electron-ion collisions. These nonlinear calculations are applied to understanding the stopping of protons and antiprotons in Al at high projectile energies.

Collision of two interacting electrons on a mesoscopic beam splitter: Exact solution in the classical limit

Experiments on collisions of isolated electrons guided along the edges in quantum Hall setups can mimic mixing of photons with the important distinction that electrons are charged fermions. In the so-called electronic Hong-Ou-Mandel (HOM) setup uncorrelated pairs of electrons are injected towards a beamsplitter. If the two electron wave packets were identical, Fermi statistics would force the electrons to scatter to different detectors, yet this quantum antibunching may be confounded by Coulomb repulsion. Here we model an electronic HOM experiment using a quadratic 2D saddle point potential for the beamsplitter and unscreened Coulomb interaction between the two injected electrons subjected to a strong out-of-plane magnetic field. We show that classical equations of motion for the drift dynamics of electrons' guiding centers take on the form of Hamilton equations for canonically conjugated variables subject to the saddle point potential and the Coulomb potential where the dynamics of the center-of-mass coordinate and the relative coordinate separate. We use these equations to determine collision outcomes in terms of a few experimentally tuneable parameters: the initial energies of the uncorrelated electrons, relative time delay of injection and the shape of the saddle point potential. A universal phase diagram of deterministic bunching and antibunching scattering outcomes is presented with a single energy scale characterizing the increase of the effective barrier height due to interaction of coincident electrons. We suggest clear-cut experimental strategies to detect the predicted effects and give analytical estimates of conditions when the classical dynamics is expected to dominate over quantum effects.

Observation of Bohm trajectories and quantum potentials of classical waves

In 1952 David Bohm proposed an interpretation of quantum mechanics, in which the evolution of states results from trajectories governed by classical equations of motion but with an additional potential determined by the wave function. There exist only a few experiments that test this concept and they employed weak measurement of non-classical light. In contrast, we reconstruct the Bohm trajectories in a classical hydrodynamic system of surface gravity water waves, by a direct measurement of the wave packet. Our system is governed by a wave equation that is analogous to the Schrödinger equation which enables us to transfer the Bohm formalism to classical waves. In contrast to a quantum system, we can measure simultaneously their amplitude and phase. In our experiments, we employ three characteristic types of surface gravity water wave packets: two and three Gaussian temporal slits and temporal Airy wave packets. The Bohm trajectories and their energy flows follow the valleys and bounce off the hills in the corresponding quantum potential landscapes.