Function Of Momentum Research Articles

Klein–Gordon equation for electrically charged particles with new energy-momentum operator

We address the Klein–Gordon equation for a spinless charged particle in the presence of an electromagnetic (EM) field, and focus on its known shortcoming, related to the existence of solutions with a negative probability density. We disclose a principal way to overcome this shortcoming, using our recent results obtained in the analysis of quantum phase effects for charges and dipoles, which prove the need to abandon the customary definition of the momentum operator for a charged particle in an EM field through its canonical momentum, and to adopt the more general definition of this operator through the sum of mechanical and EM momenta for the system ‘charged particle in an EM field’. We show that the application of the new energy-momentum operator to the Klein–Gordon equation actually eliminates solutions with negative probability density. Some implications of the obtained results are discussed.

Recombination mechanism for D0-meson production and [formula omitted] production asymmetry in the LHCb p+20Ne fixed-target experiment

We discuss production of neutral D mesons in proton-proton collisions at the LHC (fixed target mode) in the framework of the BJM recombination model. We present rapidity and transverse momentum distributions of D mesons and compare the recombination contribution to the dominant gluon-gluon fusion mechanism. Both the direct production, as dictated by the matrix element, and fragmentation of the associated c or c¯ are included. The latter mechanism generates D mesons with smaller rapidities than those produced directly. We calculate the D0+D‾0 meson distributions relevant for fixed target p+4He collisions at s = 86.6 GeV as well as for p+20Ne collisions at s = 69 GeV. The recombination component is consistent with the LHCb data and in addition results in production asymmetry. The asymmetries in D0−D0‾ production as a function of rapidity and transverse momentum are shown and the cancellation of terms for direct production and associated c/c¯ fragmentation is discussed.

New Challenges for Classical and Quantum Probability

The discovery that any classical random variable with all moments gives rise to a full quantum theory (that in the Gaussian and Poisson cases coincides with the usual one) implies that a quantum–type formalism will enter into practically all applications of classical probability and statistics. The new challenge consists in finding the classical interpretation, for different types of classical contexts, of typical quantum notions such as entanglement, normal order, equilibrium states, etc. As an example, every classical symmetric random variable has a canonically associated conjugate momentum. In usual quantum mechanics (associated with Gaussian or Poisson classical random variables), the interpretation of the momentum operator was already clear to Heisenberg. How should we interpret the conjugate momentum operator associated with classical random variables outside the Gauss–Poisson class? The Introduction is intended to place in historical perspective the recent developments that are the main object of the present exposition.

Measurement of global polarization of Λ hyperons in few-GeV heavy-ion collisions

The global polarization of Λ hyperons along the total orbital angular momentum of a relativistic heavy-ion collision is presented based on the high statistics data samples collected in Au+Au collisions at sNN=2.4 GeV and Ag+Ag at 2.55 GeV with the High-Acceptance Di-Electron Spectrometer (HADES) at GSI, Darmstadt. This is the first measurement below the strangeness production threshold in nucleon-nucleon collisions. Results are reported as a function of the collision centrality as well as a function of the hyperon's transverse momentum (pT) and rapidity (yCM) for the range of centrality 0–40%. We observe a strong centrality dependence of the polarization with an increasing signal towards peripheral collisions. For mid-central (20 – 40%) collisions the polarization magnitudes are 〈PΛ〉(%)=6.8±1.3(stat.)±2.1(syst.) for Au+Au and 〈PΛ〉(%)=6.2±0.4(stat.)±0.6(syst.) for Ag+Ag, which are the largest values observed so far. This observation thus provides a continuation of the increasing trend previously observed by STAR and contrasts expectations from recent theoretical calculations predicting a maximum in the region of collision energies about 3 GeV. The observed polarization is of a similar magnitude as predicted by 3D-fluid-dynamics and the UrQMD plus thermal vorticity model and significantly above results from the AMPT model.

The complex Maxwell stress tensor theorem: The imaginary stress tensor and the reactive strength of orbital momentum. A novel scenery underlying electromagnetic optical forces

We uncover the existence of a universal phenomenon concerning the electromagnetic optical force exerted by light or other electromagnetic waves on a distribution of charges and currents in general, and of particles in particular. This conveys the appearence of underlying reactive quantities that hinder radiation pressure and currently observed time-averaged forces. This constitutes a novel paradigm of the mechanical efficiency of light on matter, and completes the landscape of the optical, and generally electromagnetic, force in photonics and classical electrodynamics; widening our understanding in the design of both illumination and particles in optical manipulation without the need of increasing the illuminating power, and thus lowering dissipation and heating. We show that this may be accomplished through the minimization of what we establish as the reactive strength of orbital (or canonical) momentum, which plays against the optical force a role analogous to that of the reactive power versus the radiation efficiency of an antenna. This long time overlooked quantity, important for current progress of optical manipulation, and that stems from the complex Maxwell theorem of conservation of complex momentum that we put forward, as well as its alternating flow associated to the imaginary part of the complex Maxwell stress tensor, conform the imaginary Lorentz force that we introduce in this work, and that like the reactive strength of orbital momentum, is antagonistic to the well-known time-averaged force; thus making this reactive Lorentz force indirectly observable near wavelengths at which the time-averaged force is lowered. The Minkowski and Abraham momenta are also addressed.

Modelling the Alfvén eigenmode induced fast-ion flow measured by an imaging neutral particle analyzer

An imaging neutral particle analyzer (INPA) provides energy and radially resolved measurements of the confined fast-ion population ranging from the high-field side to the edge on the midplane of the DIII-D tokamak. In recent experiments, it was used to diagnose fast-ion flow in the INPA-interrogated phase-space driven by multiple, marginally unstable Alfvén eigenmodes (AEs). The key features of this measured fast-ion flow are: (I) a fast-ion flow from q min and the injection energy (81 keV) towards lower energies and plasma periphery.(II) A flow from the same location towards higher energies and the plasma core, (III) a phase-space ‘hole’ at the injected energy and plasma core and (IV) a pile-up at the plasma core at lower energies (∼60 keV). Ad hoc energetic particle diffusivity modelling of TRANSP significantly deviates from the observation. Comparably, a reduced modelling, i.e. a combination of NOVA-K and ASCOT5 code with the measured mode structure and amplitude, generally reproduce some key features of the observed phase-space flow, but largely failed to interpret fast ion depletion near the plasma axis. At last, self-consistent, first-principle multi-phase hybrid simulations that include realistic neutral beam injection and collisions are able to reproduce most features of the time-resolved phase-space flow. During consecutive hybrid phases, an RSAE consistent with the experiment grows and saturates, redistributing the injected fast ions. The resulting synthetic INPA images are in good agreement with the measurement near the injection energy. The simulations track the fast-ion redistribution within the INPA range, confirming that the measured fast-ion flow follows streamlines defined by the intersection of phase-space surfaces of constant magnetic moment μ and constant E′ = nE + ωP ϕ , where n and ω are the instability toroidal mode number and frequency, and E and P ϕ the ion energy and toroidal canonical momentum. Nonperturbative effects are required to reproduce the depletion of fast ions near the magnetic axis at the injection energy.

Analysis of nanofluid flow subject to velocity slip and Joule heating over a nonlinear stretching Riga plate with varying thickness

Nanofluid flow towards a nonlinear stretched Riga plat of varying thickness is considered in this study. Riga plate is an electromagnetic sheet containing electrodes at fixed positions alternatively. This pattern causes an electromagnetic effect in nanoliquid flow. The flow develops due to the nonlinear stretchable Riga plate. Mathematical modelling and analysis are carried out in the presence of Joule heating, slip effect and heat generation. Brownian motion and thermophoresis aspects for nanoliquid are incorporated using Buongiorno's model. Resulting nonlinear partial differential equations are transformed into nonlinear ordinary differential equations using suitable similarity transformations. Homotopy analysis method is used for series solutions. Convergent of series solutions are analyzed for momentum, energy and concentration functions. Drag coefficient and heat transfer rate are modelled and presented. The findings of analysis show that the modified Hartman number enhances the velocity profile. Opposite behavior is seen for velocity and temperature against variable thickness parameter. Heat transfer coefficient declines for larger Brownian motion and thermophoresis variables. The comparison of current study with published data is presented.

Revealing the order parameter dynamics of 1T-TiSe_2 following optical excitation

The formation of a charge density wave state is characterized by an order parameter. The way it is established provides unique information on both the role that correlation plays in driving the charge density wave formation and the mechanism behind its formation. Here we use time and angle resolved photoelectron spectroscopy to optically perturb the charge-density phase in 1T-TiSe_2 and follow the recovery of its order parameter as a function of energy, momentum and excitation density. Our results reveal that two distinct orders contribute to the gap formation, a CDW order and pseudogap-like order, manifested by an overall robustness to optical excitation. A detailed analysis of the magnitude of the the gap as a function of excitation density and delay time reveals the excitonic long-range nature of the CDW gap and the short-range Jahn–Teller character of the pseudogap order. In contrast to the gap, the intensity of the folded Se_{4p}* band can only give access to the excitonic order. These results provide new information into the the long standing debate on the origin of the gap in TiSe_2 and place it in the same context of other quantum materials where a pseudogap phase appears to be a precursor of long-range order.

Elliptic flow of charged particles at midrapidity relative to the spectator plane in Pb–Pb and Xe–Xe collisions

Measurements of the elliptic flow coefficient relative to the collision plane defined by the spectator neutrons v2{ΨSP} in collisions of Pb ions at center-of-mass energy per nucleon–nucleon pair sNN=2.76 TeV and Xe ions at sNN=5.44 TeV are reported. The results are presented for charged particles produced at midrapidity as a function of centrality and transverse momentum for the 5–70% and 0.2–6 GeV/c ranges, respectively. The ratio between v2{ΨSP} and the elliptic flow coefficient relative to the participant plane v2{4}, estimated using four-particle correlations, deviates by up to 20% from unity depending on centrality. This observation differs strongly from the magnitude of the corresponding eccentricity ratios predicted by the TRENTo and the elliptic power models of initial state fluctuations that are tuned to describe the participant plane anisotropies. The differences can be interpreted as a decorrelation of the neutron spectator plane and the reaction plane because of fragmentation of the remnants from the colliding nuclei, which points to an incompleteness of current models describing the initial state fluctuations. A significant transverse momentum dependence of the ratio v2{ΨSP}/v2{4} is observed in all but the most central collisions, which may help to understand whether momentum anisotropies at low and intermediate transverse momentum have a common origin in initial state fluctuations. The ratios of v2{ΨSP} and v2{4} to the corresponding initial state eccentricities for Xe–Xe and Pb–Pb collisions at similar initial entropy density show a difference of (7.0±0.9)% with an additional variation of +1.8% when including RHIC data in the TRENTo parameter extraction. These observations provide new experimental constraints for viscous effects in the hydrodynamic modeling of the expanding quark–gluon plasma produced in heavy-ion collisions at the LHC.

Physics aware analytics for accurate state prediction of dynamical systems

It has been successfully demonstrated that synchronisation of physical prior, like conservation laws with a conventional neural network significantly decreases the amount of training necessary to learn the dynamics of non-linear physical systems. Recent research shows how parameterisation of Lagrangian and Hamiltonian using conventional Neural Network’s weights and biases are achieved, and then executing the Euler-Lagrangian and Hamilton’s equation of motion for prediction of future state vectors proved to be more efficient than conventional Neural Networks in predicting non-linear dynamical systems. In classical mechanics, the Lagrangian formalism predicts the trajectory by solving the second-order Euler-Lagrangian equation following all the symmetries of the system while, the Hamiltonian mechanics explicitly uses the Poisson bracket of canonical position and canonical momentum to arrive at two independent first-order equations of motion. For a physical system that follows energy conservation, we obtain the momentum taking the first derivative of the Lagrangian which supplements the energy conservation via Hamiltonian Mechanics. The dynamics of the physical system converges when the generalised coordinates transform to suitable action angle variables using Poisson bracket formalism conserving the energy. In this work, we have demonstrated the prediction by integrating Hamiltonian and Lagrangian neural network leading to faster convergence with the actual dynamics from less number of training epochs. Hence, integration of HNN(Hamiltonian Neural Network) and LNN(Lagrangian Neural Network) outperforms their autonomous implementation. We have tested our proposed model in three conservative non-linear dynamical systems and found promising results.

Limit cycles and chaos in the hybrid atom-optomechanics system

We consider atoms in two different periodic potentials induced by different lasers, one of which is coupled to a mechanical membrane via radiation pressure force. The atoms are intrinsically two-level systems that can absorb or emit photons, but the dynamics of their position and momentum are treated classically. On the other hand, the membrane, the cavity field, and the intrinsic two-level atoms are treated quantum mechanically. We show that the mean excitation of the three systems can be stable, periodically oscillating, or in a chaotic state depending on the strength of the coupling between them. We define regular, limit cycle, and chaotic phases, and present a phase diagram where the three phases can be achieved by manipulating the field-membrane and field-atom coupling strengths. We also computed other observable quantities that can reflect the system’s phase such as position, momentum, and correlation functions. Our proposal offers a new way to generate and tune the limit cycle and chaotic phases in a well-established atom-optomechanics system.

Variational learning of Euler–Lagrange dynamics from data

The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this, we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.

Gasdynamical flows in star forming regions

A review of gas-dynamical flows in astrophysics is given with an emphasis on accretion flows and outflows. Analytical estimates of the dependence of the velocity and density on the radius during spherical accretion are considered, as well as the rate of mass inflow and the corresponding luminosity of the accretion disk are also estimated. Some features of gas inflow and outflow near the star are mentioned, like mass-loading flows where source functions of mass and momentum are presented and analyzed. The velocities and radii of the ionization fronts of hot stars are considered. The Sedov-Taylor analytical formula on the dependence of the radius of the fireball on time during a strong explosion is given, where the explosion energy and density of the medium act as parameters. Examples of the use of dimensional theory in astrophysics, in particular, in accretion flows used above are considered. An example of a numerical solution of the interaction of an interstellar cloud with the solar wind is also given, followed by an estimate of the inflow of the cloud’s gas environment into the Earth’s atmosphere, which affects the terrestrial ozone content.

Nonholonomic Virtual Constraints for Control of Powered Prostheses Across Walking Speeds.

This paper presents a method to design a nonholonomic virtual constraint (NHVC) controller that produces multiple distinct stance-phase trajectories for corresponding walking speeds. NHVCs encode velocity-dependent joint trajectories via momenta conjugate to the unactuated degree(s)-of-freedom of the system. We recently introduced a method for designing NHVCs that allow for stable bipedal robotic walking across variable terrain slopes. This work extends the notion of NHVCs for application to variable-cadence powered prostheses. Using the segmental conjugate momentum for the prosthesis, an optimization problem is used to design a single stance-phase NHVC for three distinct walking speed trajectories (slow, normal, and fast). This stance-phase controller is implemented with a holonomic swing phase controller on a powered knee-ankle prosthesis, and experiments are conducted with an able-bodied user walking in steady and non-steady velocity conditions. The control scheme is capable of representing 1) multiple, task-dependent reference trajectories, and 2) walking gait variance due to both temporal and kinematic changes in user motion.

Supercanonical coordinates for orthosymplectic evolution operators

A time-dependent self-adjoint even Hamiltonian is defined by a linear combination of generators of the semidirect sum osp(m′/n′)⊕sne(n′+1), of the orthosymplectic plus the even Heisenberg algebra by computing the supercommutator of odd binary forms Π, given as linear combinations of odd bilinear generators of the odd Heisenberg algebra no(m′+1) elements times ne(n′+1) elements, establishing a relationship between entangled boson systems and entangled fermion systems. This approach leads to the concept of intertwining, defined through the resulting quadratic Hamiltonians of bosons and, separately, of fermions with coefficients given in terms of the same coefficients of Π. Intertwining is invariant under transformations of Π, which leave certain binary forms of the coefficients of Π in the Hamiltonian unchanged. Alternatively, the coefficients can be interpreted as simultaneous time-dependent (super-) control parameters for both spin-statistics. Time-dependent inhomogeneous linear supercanonical transformations of wave vectors leave invariant the Heisenberg superalgebra n(m′/n′) and belong to the semidirect product Osp(m′/n′) ⋉ Ne(n′ + 1) of the orthosymplectic supergroup with the even Heisenberg group. The unitary time evolution operator is constructed using the adjoint map in canonical coordinates determined by the supercanonical transformation. The method is a generalization of an Inönu–Wigner contraction procedure and a Wei–Norman method for superalgebras with a selection of subalgebras associated with the root space decomposition of the Lie superalgebra. Analogously, this is a separation of variables method for quantum mechanical problems in systems with bosons and fermions. The standard Floquet theory leads to new results concerning stability for locally periodic coefficients. The lowest dimensional cases are explicitly computed. The intertwining of boson and fermions systems and the Hamiltonians considered here are of interest in quantum control theory for systems including fermions and bosons, in quantum optics, and quantum computation.

Anisotropic Z n-graded classical r-matrix, deformed A n Toda- and Gaudin-type models, and separation of variables

We consider a problem of separation of variables for Lax-integrable Hamiltonian systems governed by gl(n) ⨂ gl(n)-valued classical r-matrices r(u, v). We find a new class of classical non-skew-symmetric non-dynamical gl(n) ⨂ gl(n)-valued r-matrices rJ(u, v) for which the “magic recipe” of Sklyanin [Prog. Theor. Phys. Suppl. 118, 35 (1995)] in the theory of variable separation is applicable, i.e., for which standard separating functions A(u) and B(u) of Gekhtman [Commun. Math. Phys. 167, 593 (1995)] and Scott [“Classical functional Bethe ansatz for SL(N): Separation of variables for the magnetic chain,” arXiv:hep-th 940303] produce a complete set of canonical coordinates satisfying the equations of separation. We illustrate the corresponding separation of variable theory by the example of the anisotropically deformed An Toda models proposed in the work of Skrypnyk [J. Phys. A: Math. Theor. 38, 9665–9680 (2005)] and governed by the r-matrices rJ(u, v) and by the generalized Gaudin models [T. Skrypnyk, Phys. Lett. A 334(5–6), 390 (2005)] governed by the same classical r-matrices. The n = 2 and n = 3 cases are considered in detail.

First Measurement of the Z→μ^{+}μ^{-} Angular Coefficients in the Forward Region of pp Collisions at sqrt[s]=13 TeV.

The first study of the angular distribution of μ^{+}μ^{-} pairs produced in the forward rapidity region via the Drell-Yan reaction pp→γ^{*}/Z+X→ℓ^{+}ℓ^{-}+X is presented, using data collected with the LHCb detector at a center-of-mass energy of 13TeV, corresponding to an integrated luminosity of 5.1 fb^{-1}. The coefficients of the five leading terms in the angular distribution are determined as a function of the dimuon transverse momentum and rapidity. The results are compared to various theoretical predictions of the Z-boson production mechanism and can also be used to probe transverse-momentum-dependent parton distributions within the proton.

Evidence for Nonlinear Gluon Effects in QCD and Their Mass Number Dependence at STAR.

The STAR Collaboration reports measurements of back-to-back azimuthal correlations of di-π^{0}s produced at forward pseudorapidities (2.6<η<4.0) in p+p, p+Al, and p+Au collisions at a center-of-mass energy of 200GeV. We observe a clear suppression of the correlated yields of back-to-back π^{0} pairs in p+Al and p+Au collisions compared to the p+p data. The observed suppression of back-to-back pairs as a function of transverse momentum suggests nonlinear gluon dynamics arising at high parton densities. The larger suppression found in p+Au relative to p+Al collisions exhibits a dependence of the saturation scale Q_{s}^{2} on the mass number A. A linear scaling of the suppression with A^{1/3} is observed with a slope of -0.09±0.01.

Particle production at large p T in Xe+Xe collisions with jet quenching using the higher twist approach* *Supported by the Guangdong Major Project of Basic and Applied Basic Research (2020B0301030008) and the Natural Science Foundation of China with Project (11935007, 11805167)

The production of , η, and ϕ in the most central (0%–5%) Xe+Xe collisions at = 5.44 TeV is investigated in the framework of the perturbative QCD (pQCD) improved parton model at an accuracy of next-to-leading order (NLO). The jet quenching effect is effectively incorporated by medium-modified fragmentation functions via the higher-twist approach. Predictions of the nuclear modification factors of , η, and ϕ as functions of the transverse momentum are made with the jet transport parameter , which is extracted from the available experimental data of charged hadrons provided by ALICE and CMS. The particle ratios as functions of in Xe+Xe collisions at = 5.44 TeV as well as in 0%–5% Pb + Pb collisions at = 5.02 TeV are also presented. The numerical simulations of the scaled ratios of charged hadron production in the Xe+Xe 5.44 TeV system over those in the Pb+Pb 5.02 TeV system give a good description of the CMS data, and the scaled ratios of , η, and ϕ production coincide with the curve of charged hadron production.

Ponderomotive forces, Stokes drift, and momentum in acoustic and electromagnetic waves

This work illuminates fundamental connections between three wave-induced phenomena: (i) the ponderomotive force acting on the medium particles, (ii) the Stokes drift of free medium particles, and (iii) the canonical wave momentum. The results are applicable to sound waves in fluids or gases and to electromagnetic waves in media with free electrons.