Canonical Perturbation Theory Research Articles

Condition monitoring of rotating machines under time-varying conditions based on adaptive canonical variate analysis

Abstract Condition monitoring signals obtained from rotating machines often demonstrate a highly non-stationary and transient nature due to internal natural deterioration characteristics of their constituent components and external time-varying operational conditions. Traditional multivariate statistical monitoring approaches are based on the assumption that the underlying processes are linear and static and are apt to interpret the normal changes in operating conditions as faults, which would result in high false positive rates. On the other hand, the development of robust diagnostic techniques for the detection of incipient faults remains a challenge for researchers, given the difficulty of finding an appropriate trade-off between a low false positive ratio and early detection of emerging faults. To address these issues, this paper proposes a novel adaptive fault detection approach based on the canonical residuals (CR) induced by the combination of canonical variate analysis (CVA) and matrix perturbation theory for the monitoring of dynamic processes where variations in operating conditions are incurred. The canonical residuals are calculated based upon the distinctions between past and future measurements and are able to effectively detect emerging faults while still maintaining a low false positive rate. The effectiveness of the developed diagnostic model for the detection of abnormalities in industrial processes was demonstrated for slow involving faults in case studies of two operational industrial high-pressure pumps. In comparison with the variable-based and canonical correlation-based statistical monitoring approaches, the proposed canonical residuals-adaptive canonical variate analysis (CR-ACVA) fault detection method has demonstrated its superiorities by the detailed performance comparisons.

Dynamics of Dirac observables in canonical cosmological perturbation theory

The relational formalism based on geometrical clocks and Dirac observables in linearized canonical cosmological perturbation theory is used to introduce an efficient method to find evolution equations for gauge invariant variables. Our method generalizes an existing technique by Pons et al (2010 J. Phys.: Conf. Ser. 222 012018); (2009 Phys. Rev. D 80 084015) to relate the evolution of gauge invariant observables with the one of gauge variant quantities, and is applied as a demonstration for the longitudinal and spatially flat gauges. Gauge invariant evolution equations for the Bardeen potential and the Mukhanov–Sasaki variable are derived in the extended ADM phase space. Our method establishes a full agreement at the dynamical level between the canonical and conventional cosmological perturbation theory at the linear order using Dirac observables.

Uphill acceleration in a spatially modulated electrostatic field particle accelerator

Spatially modulated electrostatic fields can be designed to efficiently accelerate particles by exploring the relationships between the amplitude, the phase velocity, the shape of the potential, and the initial velocity of the particle. The acceleration process occurs when the value of the velocity excursions of the particle surpasses the phase velocity of the carrier, as a resonant mechanism. The ponderomotive approximation based on the Lagrangian average is usually applied in this kind of system in non-accelerating regimes. The mean dynamics of the particle is well described by this approximation far from resonance. However, the approximation fails to predict some interesting features of the model near resonance, such as the uphill acceleration phenomenon. A canonical perturbation theory is more accurate in these conditions. In this work, we compare the results from the Lagrangian average and from a canonical perturbation theory, focusing in regions where the results of these two approaches differ from each other.

Gauge invariant variables for cosmological perturbation theory using geometrical clocks

Using the extended ADM-phase space formulation in the canonical framework we analyze the relationship between various gauge choices made in cosmological perturbation theory and the choice of geometrical clocks in the relational formalism. We show that various gauge invariant variables obtained in the conventional analysis of cosmological perturbation theory correspond to Dirac observables tied to a specific choice of geometrical clocks. As examples, we show that the Bardeen potentials and the Mukhanov–Sasaki variable emerge naturally in our analysis as observables when gauge fixing conditions are determined via clocks in the Hamiltonian framework. Similarly other gauge invariant variables for various gauges can be systematically obtained. We demonstrate this by analyzing five common gauge choices: longitudinal, spatially flat, uniform field, synchronous and comoving gauge. For all these, we apply the observable map in the context of the relational formalism and obtain the corresponding Dirac observables associated with these choices of clocks. At the linear order, our analysis generalizes the existing results in canonical cosmological perturbation theory twofold. On the one hand we can include also gauges that can only be analyzed in the context of the extended ADM-phase space and furthermore, we obtain a set of natural gauge invariant variables, namely the Dirac observables, for each considered choice of gauge conditions. Our analysis provides insights on which clocks should be used to extract the relevant natural physical observables both at the classical and quantum level. We also discuss how to generalize our analysis in a straightforward way to higher orders in the perturbation theory to understand gauge conditions and the construction of gauge invariant quantities beyond linear order.

Absolute IR vibrational band intensities of hexafluoroacetone: Comparison of experiment and anharmonic ab initio calculation using the second-order operator canonical Van Vleck perturbation theory

Hexafluoroacetone (HFA, OC(CF3)2) is widely utilized in organic synthesis, chemistry of heat resistant polymers and biochemistry. In this work, infrared (IR) spectra of gaseous HFA were recorded in the range 2000–340 cm−1 with a resolution of 1 cm−1. For the first time, experimental absolute IR intensities of fourteen fundamental bands of HFA were measured and compared with intensities predicted by anharmonic ab initio calculations. Theoretical molecular structures, semi-diagonal quartic potential energy surfaces (PES), and cubic surfaces of dipole moment (DM) components of HFA were calculated using MP2/cc-pVTZ ab initio quantum-mechanical model. In addition, harmonic frequencies were also calculated at MP2/cc-pVQZ level of theory and used for refining the PES. The numerical-analytic implementation of the second-order operator canonical Van Vleck perturbation theory (CVPT2) was employed for predicting anharmonic IR spectra of HFA. The deviation from observations of predicted IR intensities for ten fundamentals (ν1, ν2, ν8, ν13–ν21) was less than 25% on average. The significant difference between ab initio harmonic intensities calculated with MP2/cc-pVTZ and MP2/cc-pVQZ models for some normal modes indicates that an inaccuracy of the quantum-mechanical model is probably the major source of errors in theoretical intensities. An interpretation of the IR experimental spectra and associated literature data for HFA is accomplished; the ν4, ν7, ν10 and ν15 fundamentals were reassigned. Three predicted Fermi resonances were identified in experimental spectra, ν3/ν5 + ν8, ν17/ν10 + ν19 and ν20/ν9 + ν23. An empirical fit of harmonic frequencies within anharmonic model assisted in a reliable identification of a number of overtones and binary combination tones. The fitted harmonic frequencies were compared with MP2/cc-pVQZ and a simplified complete basis set (CBS) extrapolation. It was found that MP2/cc-pVQZ frequencies were more reliable. The efficiency and convenience of the theoretical model employed and associated techniques for interpretation of complex spectra of HFA are demonstrated.

Gauge invariant canonical cosmological perturbation theory with geometrical clocks in extended phase-space — A review and applications

The theory of cosmological perturbations is a well-elaborated field and has been successfully applied, e.g. to model the structure formation in our universe and the prediction of the power spectrum of the cosmic microwave background. To deal with the diffeomorphism invariance of general relativity, one generally introduces combinations of the metric and matter perturbations, which are gauge invariant up to the considered order in the perturbations. For linear cosmological perturbations, one works with the so-called Bardeen potentials widely used in this context. However, there exists no common procedure to construct gauge invariant quantities also for higher-order perturbations. Usually, one has to find new gauge invariant quantities independently for each order in perturbation theory. With the relational formalism introduced by Rovelli and further developed by Dittrich and Thiemann, it is in principle possible to calculate manifestly gauge invariant quantities, that is quantities that are gauge invariant up to arbitrary order once one has chosen a set of so-called reference fields, often also called clock fields. This article contains a review of the relational formalism and its application to canonical general relativity following the work of Garcia, Pons, Sundermeyer and Salisbury. As the starting point for our application of this formalism to cosmological perturbation theory, we also review the Hamiltonian formulation of the linearized theory for perturbations around FLRW spacetimes. The main aim of our work will be to identify clock fields in the context of the relational formalism that can be used to reconstruct quantities like the Bardeen potential as well as the Mukhanov–Sasaki variable. This requires a careful analysis of the canonical formulation in the extended ADM-phase-space where lapse and shift are treated as dynamical variables. The actual construction of such observables and further investigations thereof will be carried out in our companion paper.

Comparing the accuracy of perturbative and variational calculations for predicting fundamental vibrational frequencies of dihalomethanes.

Three dihalogenated methane derivatives (CH2F2, CH2FCl, and CH2Cl2) were used as model systems to compare and assess the accuracy of two different approaches for predicting observed fundamental frequencies: canonical operator Van Vleck vibrational perturbation theory (CVPT) and vibrational configuration interaction (VCI). For convenience and consistency, both methods employ the Watson Hamiltonian in rectilinear normal coordinates, expanding the potential energy surface (PES) as a Taylor series about equilibrium and constructing the wavefunction from a harmonic oscillator product basis. At the highest levels of theory considered here, fourth-order CVPT and VCI in a harmonic oscillator basis with up to 10 quanta of vibrational excitation in conjunction with a 4-mode representation sextic force field (SFF-4MR) computed at MP2/cc-pVTZ with replacement CCSD(T)/aug-cc-pVQZ harmonic force constants, the agreement between computed fundamentals is closer to 0.3 cm-1 on average, with a maximum difference of 1.7 cm-1. The major remaining accuracy-limiting factors are the accuracy of the underlying electronic structure model, followed by the incompleteness of the PES expansion. Nonetheless, computed and experimental fundamentals agree to within 5 cm-1, with an average difference of 2 cm-1, confirming the utility and accuracy of both theoretical models. One exception to this rule is the formally IR-inactive but weakly allowed through Coriolis-coupling H-C-H out-of-plane twisting mode of dichloromethane, whose spectrum we therefore revisit and reassign. We also investigate convergence with respect to order of CVPT, VCI excitation level, and order of PES expansion, concluding that premature truncation substantially decreases accuracy, although VCI(6)/SFF-4MR results are still of acceptable accuracy, and some error cancellation is observed with CVPT2 using a quartic force field.

Anharmonic vibrational analysis of s-trans and s-cis conformers of acryloyl fluoride using numerical-analytic Van Vleck operator perturbation theory

A new gas-phase infrared (IR) spectrum of acryloyl fluoride (ACRF, CH2CHCFO) with a resolution of 0.1cm−1 in the range 4000–450cm−1 was measured. Theoretical ab initio molecular structures, full quartic potential energy surfaces (PES), and cubic surfaces of dipole moments and polarizability tensor components (electro-optical properties, EOP) of the s-trans and s-cis conformers of the ACRF were calculated by the second-order Møller-Plesset electronic perturbation theory with a correlation consistent Dunning triple-ζ basis set. The numerical-analytic implementation of the second-order operator canonical Van Vleck perturbation theory was employed for predicting anharmonic IR and Raman scattering (RS) spectra of ACRF. To improve the anharmonic predictions, harmonic frequencies were replaced by their counterparts evaluated with the higher-level CCSD(T)/cc-pVTZ model, to form a “hybrid” PES. The original operator representation of the Hamiltonian is analytically reduced to a quasi-diagonal form, integrated in the harmonic oscillator basis and diagonalized to account for strong resonance couplings. Double canonical transformations of EOP expansions enabled prediction of integral intensities of both fundamental and multi-quanta transitions in IR/RS spectra. Enhanced band shape analysis reinforced the assignments. A thorough interpretation of the new IR experimental spectra and existing matrix-isolation literature data for the mixture of two conformers of ACRF was accomplished, and a number of assignments clarified.

Higher-order formulas of amplitude-dependent tune shift caused by a sextupole magnetic field distribution

Nowadays, designs for ring-based light sources use multibend lattices for achieving a very small emittance of around 100 pmrad. In this type of storage ring, the chromaticity correcting sextupoles generally have greater strengths than those used in typical third-generation light sources. Therefore, controlling lattice nonlinearity such as amplitude-dependent tune shift (ADTS) is important for enabling stable operations and smooth beam commissioning. As the strength of the sextupoles increases, their higher-order terms contribute significantly to ADTS, rendering well-known lowest-order formulas inadequate for describing tune variations at large horizontal amplitudes. In response, we have derived explicit expressions of ADTS up to the fourth order in sextupole strength based on the canonical perturbation theory, assuming that the amplitude of a vertical betatron oscillation is smaller compared with the horizontal one. The new formulas express the horizontal and vertical betatron tune variations as functions of the action variables: ${J}_{x}$ and ${J}_{y}$ up to $O({J}_{x}^{2})$ and $O({J}_{y})$. The derived formulas were applied to a five-bend achromat lattice designed for the SPring-8 upgrade. By comparing the calculated results with the tracking simulations, we found that (1) the formulas accurately express ADTS around a horizontal amplitude of $\ensuremath{\sim}10\text{ }\text{ }\mathrm{mm}$ and (2) the nonlinear terms of the fourth order in sextupole strength govern the behaviors of circulating electrons at large horizontal amplitudes. In this paper, we present explicit expressions of fourth-order formulas of ADTS and provide some examples to illustrate their effectiveness.

Application of first-order canonical perturbation method with dissipative Hori-like kernel

Lie-Hori canonical perturbation theory provides asymptotic solutions for conservative Hamiltonian systems. This restriction prevents the canonical method from being applied directly to dissipative mechanical systems. There are, however, two main alternatives to overcome this difficulty, enabling the application of canonical perturbation methods. The first one consists in constructing a time-dependent Hamiltonian, through a generating function, related to the energy dissipation pattern of the system. The second embeds the original phase space into a double dimensional one where the dynamics of the system can be formulated in a Hamiltonian way. In this paper, a modified Lie-Hori method that avoid the disadvantages of the former approaches is proposed. Namely, it is not necessary to find out a time-dependent generating function, nor doubling the number of the canonical variables of the original problem. The new algorithm provides first order analytical solutions for a certain set of dissipative non-linear dynamical systems. It is based on a suitable modification of the Hori kernel in the double-dimensional embedding phase space, allowing the inclusion of the dissipative (or generalized) forces. By means of this redefined auxiliary system, the path-integrals of the method can be performed in a domain of the phase space with the same dimensionality as the original problem.

Geostationary secular dynamics revisited: application to high area-to-mass ratio objects

The long-term dynamics of the geostationary Earth orbits (GEO) is revisited through the application of canonical perturbation theory. We consider a Hamiltonian model accounting for all major perturbations: geopotential at order and degree two, lunisolar perturbations with a realistic model for the Sun and Moon orbits, and solar radiation pressure. The long-term dynamics of the GEO region has been studied both numerically and analytically, in view of the relevance of such studies to the issue of space debris or to the disposal of GEO satellites. Past studies focused on the orbital evolution of objects around a nominal solution, hereafter called the forced equilibrium solution, which shows a particularly strong dependence on the area-to-mass ratio. Here, we i) give theoretical estimates for the long-term behavior of such orbits, and ii) we examine the nature of the forced equilibrium itself. In the lowest approximation, the forced equilibrium implies motion with a constant non-zero average `forced eccentricity', as well as a constant non-zero average inclination, otherwise known in satellite dynamics as the inclination of the invariant `Laplace plane'. Using a higher order normal form, we demonstrate that this equilibrium actually represents not a point in phase space, but a trajectory taking place on a lower-dimensional torus. We give analytical expressions for this special trajectory, and we compare our results to those found by numerical orbit propagation. We finally discuss the use of proper elements, i.e., approximate integrals of motion for the GEO orbits.

Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures.

We consider exact and asymptotic solutions of the stationary cubic nonlinear Schrödinger equation on metric graphs. We focus on some basic example graphs. The asymptotic solutions are obtained using the canonical perturbation formalism developed in our earlier paper [S. Gnutzmann and D. Waltner, Phys. Rev. E 93, 032204 (2016)2470-004510.1103/PhysRevE.93.032204]. For closed example graphs (interval, ring, star graph, tadpole graph), we calculate spectral curves and show how the description of spectra reduces to known characteristic functions of linear quantum graphs in the low-intensity limit. Analogously for open examples, we show how nonlinear scattering of stationary waves arises and how it reduces to known linear scattering amplitudes at low intensities. In the short-wavelength asymptotics we discuss how genuine nonlinear effects may be described using the leading order of canonical perturbation theory: bifurcation of spectral curves (and the corresponding solutions) in closed graphs and multistability in open graphs.

Kato expansion in quantum canonical perturbation theory

This work establishes a connection between canonical perturbation series in quantum mechanics and a Kato expansion for the resolvent of the Liouville superoperator. Our approach leads to an explicit expression for a generator of a block-diagonalizing Dyson’s ordered exponential in arbitrary perturbation order. Unitary intertwining of perturbed and unperturbed averaging superprojectors allows for a description of ambiguities in the generator and block-diagonalized Hamiltonian. We compare the efficiency of the corresponding computational algorithm with the efficiencies of the Van Vleck and Magnus methods for high perturbative orders.

On Tomonaga’s theory of split-anode magnetrons

This article is meant to formulate the equations of motion of an electron in a cavity magnetron using action-angle variables. This means following the electron's path on its way from a cylindrical cathode moving toward a co-axial cylindrical anode in presence of a uniform magnetic field parallel to the common axis. After analyzing the situation without coupling to an external oscillatory electric field, we employ methods of canonical perturbation theory to find the resonance condition between the frequencies of the free theory w_r, w_phi and the applied perturbing oscillatory frequency w. A long-time averaging process will then eliminate the periodic terms in the equation for the now time-dependent action-angle variables. The terms that are no longer periodic will cause secular changes so that the canonical action-angle variables (J, delta) change in a way that the path of the electron will deform gradually so that it can reach the anode. How the ensemble of the initially randomly distributed electrons forms spokes and how their energy is conveyed to the cavity-field oscillation is the main focus of this article. Some remarks concerning the importance of results in QED and the invention of radar theory and application conclude the article.

Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory.

In this paper we present a general framework for solving the stationary nonlinear Schrödinger equation (NLSE) on a network of one-dimensional wires modeled by a metric graph with suitable matching conditions at the vertices. A formal solution is given that expresses the wave function and its derivative at one end of an edge (wire) nonlinearly in terms of the values at the other end. For the cubic NLSE this nonlinear transfer operation can be expressed explicitly in terms of Jacobi elliptic functions. Its application reduces the problem of solving the corresponding set of coupled ordinary nonlinear differential equations to a finite set of nonlinear algebraic equations. For sufficiently small amplitudes we use canonical perturbation theory, which makes it possible to extract the leading nonlinear corrections over large distances.

Haag's theorem in renormalisable quantum field theories

We review a package of no-go results in axiomatic quantum field theory with Haag's theorem at its centre. Since the concept of operator-valued distributions in this framework comes very close to what we believe canonical quantum fields are about, these results are of consequence to quantum field theory: they suggest the seeming absurdity that this highly victorious theory is incapable of describing interactions. We single out unitarity of the interaction picture's intertwiner as the most salient provision of Haag's theorem and critique canonical perturbation theory to argue that renormalisation bypasses Haag's theorem by violating this very assumption.

Quasi-degenerate perturbation theory using matrix product states.

In this work, we generalize the recently proposed matrix product state perturbation theory (MPSPT) for calculating energies of excited states using quasi-degenerate (QD) perturbation theory. Our formulation uses the Kirtman-Certain-Hirschfelder canonical Van Vleck perturbation theory, which gives Hermitian effective Hamiltonians at each order, and also allows one to make use of Wigner's 2n + 1 rule. Further, our formulation satisfies Granovsky's requirement of model space invariance which is important for obtaining smooth potential energy curves. Thus, when we use MPSPT with the Dyall Hamiltonian, we obtain a model space invariant version of quasi-degenerate n-electron valence state perturbation theory (NEVPT), a property that the usual formulation of QD-NEVPT2 based on a multipartitioning technique lacked. We use our method on the benchmark problems of bond breaking of LiF which shows ionic to covalent curve crossing and the twist around the double bond of ethylene where significant valence-Rydberg mixing occurs in the excited states. In accordance with our previous work, we find that multi-reference linearized coupled cluster theory is more accurate than other multi-reference theories of similar cost.

Conservative perturbation theory for nonconservative systems.

In this paper, we show how to use canonical perturbation theory for dissipative dynamical systems capable of showing limit-cycle oscillations. Thus, our work surmounts the hitherto perceived barrier for canonical perturbation theory that it can be applied only to a class of conservative systems, viz., Hamiltonian systems. In the process, we also find Hamiltonian structure for an important subset of Liénard system-a paradigmatic system for modeling isolated and asymptotic oscillatory state. We discuss the possibility of extending our method to encompass an even wider range of nonconservative systems.

Canonical density matrix perturbation theory.

Density matrix perturbation theory [Niklasson and Challacombe, Phys. Rev. Lett. 92, 193001 (2004)] is generalized to canonical (NVT) free-energy ensembles in tight-binding, Hartree-Fock, or Kohn-Sham density-functional theory. The canonical density matrix perturbation theory can be used to calculate temperature-dependent response properties from the coupled perturbed self-consistent field equations as in density-functional perturbation theory. The method is well suited to take advantage of sparse matrix algebra to achieve linear scaling complexity in the computational cost as a function of system size for sufficiently large nonmetallic materials and metals at high temperatures.

Anharmonic Vibrational Analysis of the Infrared and Raman Gas-Phase Spectra of s-trans- and s-gauche-1,3-Butadiene.

A quantum-mechanical (hybrid MP2/cc-pVTZ and CCSD(T)/cc-pVTZ) full quartic potential energy surface (PES) in rectilinear normal coordinates and the second-order operator canonical Van Vleck perturbation theory (CVPT2) are employed to predict the anharmonic vibrational spectra of s-trans- and s-gauche-butadiene (BDE). These predictions are used to interpret their infrared and Raman scattering spectra. New high-temperature Raman spectra in the gas phase are presented in support of assignments for the gauche conformer. The CVPT2 solution is based on a PES and electro-optical properties (EOP; dipole moment and polarizability) expanded in Taylor series. Higher terms than those routinely available from Gaussian09 software were calculated by numerical differentiation of quadratic force fields and EOP using the MP2/cc-pVTZ model. The integer coefficients of the polyad quantum numbers were derived for both conformers of BDE. Replacement of harmonic frequencies by their counterparts from the CCSD(T)/cc-pVTZ model significantly improved the agreement with experimental data for s-trans-BDE (root-mean-square deviation ≈ 5.5 cm(-1)). The accuracy in predicting the rather well-studied spectrum of fundamentals of s-trans-BDE assures good predictions of the spectrum of s-gauche-BDE. A nearly complete assignment of fundamentals was obtained for the gauche conformer. Many nonfundamental transitions of the BDE conformers were interpreted as well. The predictions of multiple Fermi resonances in the complex CH-stretching region correlate well with experiment. It is shown that solving a vibrational anharmonic problem through a numerical-analytic implementation of CVPT2 is a straightforward and computationally advantageous approach for medium-size molecules in comparison with the standard second-order vibrational perturbation theory (VPT2) based on analytic expressions.