Canonical Perturbation Research Articles

A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

Validity ranges of Lie canonical perturbation theory (LCPT) are investigated in terms of non-blow-up regions. We investigate how the validity ranges depend on the perturbation order in two systems, one of which is a simple Hamiltonian system with one degree of freedom and the other is a HCN molecule. Our analysis of the former system indicates that non-blow-up regions become reduced in size as the perturbation order increases. In case of LCPT by Dragt and Finn and that by Deprit, the non-blow-up regions enclose the region inside the separatrix of the Hamiltonian, but it may not be the case for LCPT by Hori. We also analyze how well the actions constructed by these LCPTs approximate the true action of the Hamiltonian in the non-blow-up regions and find that the conventional truncated LCPT does not work over the whole region inside the separatrix, whereas LCPT by Dragt and Finn without truncation does. Our analysis of the latter system indicates that non-blow-up regions do not necessarily cover the whole regions inside the HCN well. We propose a new perturbation method to improve non-blow-up regions and validity ranges inside them. Our method is free from blowing up and retains the same normal form as the conventional LCPT. We demonstrate our method in the two systems and show that the actions constructed by our method have larger validity ranges than those by the conventional and our previous methods proposed in Teramoto and Komatsuzaki (J Chem Phys 129:094302, 2008; Phys Rev E 78:017202, 2008).

올리브 오일을 첨가한 어육 소시지의 최적화 연구

Abstract The purpose of this study was to determine the optimal mixing ratio of Alaska Pollack (Theragra chalcogramma) and olive oil in the preparation of sausage. The experiment was designed according to the central composite design for estimating the response surface, which demonstrated 10 experimental points including 2 replicates for Alaska Pollack and olive oil. The physical, mechanical and sensory properties of test materials were measured. A canonical form and perturbation plot showed the influence of each ingredient on the final product mixture. Measurement results of the physical and mechanical properties showed a significant increase or decrease in the following properties: dough sweetness (p<0.05); sausage L (p<0.05), a (p<0.001), and b (p<0.01); hardness (p<0.01), chewiness (p<0.05), and gumminess (p<0.01). Also, the sensory measurements showed a significant improvement in color (p<0.05), flavor (p<0.01), taste (p<0.001), tenderness (p<0.05), chewiness (p<0.01), mositness (p<0.05), and overall quality (p<0.01). As a result, the optimum formulation by numerical and graphical methods was calculated as Alaska Pollack 35.74 g and olive oil 7 g. Key words: Alaska Pollack (Theragra chalcogramma), olive oil, fish sausage, response surface methodology (RSM), optimization

Eccentricity estimates in hierarchical triple systems

AbstractPerturbation theory in the three body problem has greatly advanced our ability to understand and model a variety of systems ranging from artificial satellites to stars and from extrasolar planets to asteroid-Jupiter interactions. In a series of papers, we developed an analytical technique for estimating the orbital eccentricity of the inner binary in hierarchical triple systems. The method combined the secular theory with calculations of short period terms. The derivation of the short term component was based on an expansion of the rate of change of the Runge-Lenz vector by using first order perturbation theory, while canonical perturbation theory was used to investigate the secular evolution of the system. In the present work we extend the calculation to the orbit of the outer binary. At the same time, we provide an improved version for some previous results. A post-Newtonian correction is included in our model. Our analytical estimates are compared with numerical and analytical results on the subject and applications to stellar triples and extrasolar planets are discussed.

Eccentric motion of spinning compact binaries

The equations of motion for spinning compact binaries on eccentric orbits are treated perturbatively in powers of a fractional mass-difference ordering parameter. The solution is valid through first order in the mass-difference parameter. A canonical point transformation removes the leading order terms of the spin-orbit Hamiltonian which induce a wiggling precession of the orbital angular momentum around the conserved total angular momentum, a precession which disappears in the case of equal masses or one single spin. Action-angle variables are applied which make a canonical perturbation theory easily treatable.

삼채가루 첨가 식빵의 제조조건 최적화 및 저장성 연구

Abstract The purpose of this study was to determine the optimal mixing ratio of Allium hookeri powder and butter in the preparation of bread. The experiment was designed according to the central composite design for estimating the response surface, which demonstrated 10 experimental points, including 2 replicates for Allium hookeri powder and butter. Further, the mechanical and sensory properties of test materials were measured. A canonical form and perturbation plot conveyed the influence of each ingredient on the final product mixture. Overall, the measurement results of the mechanical properties showed a significant increase or decrease in dough pH, sweetness, volume, weight, height, specific volume, a & b-value of crumb and, springiness (p<0.05). Moreover, the sensory measurements demonstrated a significant improvement in color, appearance, texture, moistness and, overall quality. As a result, the optimum formulation from the numerical and graphical methods was calculated as 22.65 g of Allium hookeri powder and 47.77 g butter. After optimization, DPPH free radical scavenging activity, total phenolic content, and total plate counts over 10 days were evaluated. In sum, the results revealed the antioxidant and antibiotic actions of bread with Allium hookeri powder.Key words: Allium hookeri powder, bread, sensory, response surface design(RSM), optimization

Effective Hamiltonians for fastly driven tight-binding chains

We consider a single particle tunnelling in a tight-binding model with nearest-neighbour couplings, in the presence of a periodic high-frequency force. An effective Hamiltonian for the particle is derived using an averaging method resembling classical canonical perturbation theory. Three cases are considered: uniform lattice with periodic and open boundary conditions, and lattice with a parabolic potential. We find that in the latter case, interplay of the potential and driving leads to appearence of the effective next-nearest neighbour couplings. In the uniform case with periodic boundary conditions the second- and third-order corrections to the averaged Hamiltonian are completely absent, while in the case with open boundary conditions they have a very simple form, found before in some particular cases by S.Longhi [Phys. Rev. B 77, 195326 (2008)]. These general results may found applications in designing effective Hamiltonian models in experiments with ultracold atoms in optical lattices, e.g. for simulating solid-state phenomena.

Polyad quantum numbers and multiple resonances in anharmonic vibrational studies of polyatomic molecules

In the theory of anharmonic vibrations of a polyatomic molecule, mixing the zero-order vibrational states due to cubic, quartic and higher-order terms in the potential energy expansion leads to the appearance of more-or-less isolated blocks of states (also called polyads), connected through multiple resonances. Such polyads of states can be characterized by a common secondary integer quantum number. This polyad quantum number is defined as a linear combination of the zero-order vibrational quantum numbers, attributed to normal modes, multiplied by non-negative integer polyad coefficients, which are subject to definition for any particular molecule. According to Kellman's method [J. Chem. Phys. 93, 6630 (1990)], the corresponding formalism can be conveniently described using vector algebra. In the present work, a systematic consideration of polyad quantum numbers is given in the framework of the canonical Van Vleck perturbation theory (CVPT) and its numerical-analytic operator implementation for reducing the Hamiltonian to the quasi-diagonal form, earlier developed by the authors. It is shown that CVPT provides a convenient method for the systematic identification of essential resonances and the definition of a polyad quantum number. The method presented is generally suitable for molecules of significant size and complexity, as illustrated by several examples of molecules up to six atoms. The polyad quantum number technique is very useful for assembling comprehensive basis sets for the matrix representation of the Hamiltonian after removal of all non-resonance terms by CVPT. In addition, the classification of anharmonic energy levels according to their polyad quantum numbers provides an additional means for the interpretation of observed vibrational spectra.

Asymptotic Stability of Breathers in Some Hamiltonian Networks of Weakly Coupled Oscillators

We consider a Hamiltonian chain of weakly coupled anharmonic oscillators. It is well known that if the coupling is weak enough then the system admits families of periodic solutions exponentially localized in space (breathers). In this paper we prove asymptotic stability in energy space of such solutions. The proof is based on two steps: first we use canonical perturbation theory to put the system in a suitable normal form in a neighborhood of the breather, second we use dispersion in order to prove asymptotic stability. The main limitation of the result rests in the fact that the nonlinear part of the on site potential is required to have a zero of order 8 at the origin. From a technical point of view the theory differs from that developed for Hamiltonian PDEs due to the fact that the breather is not a relative equilibrium of the system.

Does the orbit-averaged theory require a scale separation between periodic orbit size and perturbation correlation length?

Using the canonical perturbation theory, we show that the orbit-averaged theory only requires a time-scale separation between equilibrium and perturbed motions and verifies the widely accepted notion that orbit averaging effects greatly reduce the microturbulent transport of energetic particles in a tokamak. Therefore, a recent claim [Hauff and Jenko, Phys. Rev. Lett. 102, 075004 (2009); Jenko et al., ibid. 107, 239502 (2011)] stating that the orbit-averaged theory requires a scale separation between equilibrium orbit size and perturbation correlation length is erroneous.

Long-term evolution of Galileo operational orbits by canonical perturbation theory

Galileo operational orbits are slightly affected by the 3 to 5 tesseral resonance, an effect that can be much more important in the case of disposal orbits. Proceeding by canonical perturbation theory we show that the part of the long-term Hamiltonian corresponding to the non-centralities of the Earth's gravitational potential can be replaced by an intermediary that shows the pendulum dynamics of the 3 to 5 tesseral resonance problem. Inclusion of lunisolar perturbations requires a semi-analytical integration, which is compared with the corresponding results from the well-established Draper Semi-analytical Satellite Theory.

Heating of ions by high frequency electromagnetic waves in magnetized plasmas

The heating of ions by high frequency electrostatic waves in magnetically confined plasmas has been a paradigm for studying nonlinear wave-particle interactions. The frequency of the waves is assumed to be much higher than the ion cyclotron frequency and the waves are taken to propagate across the magnetic field. In fusion type plasmas, electrostatic waves, like the lower hybrid wave, cannot access the core of the plasma. That is a domain for high harmonic fast waves or electron cyclotron waves—these are primarily electromagnetic waves. Previous studies on heating of ions by two or more electrostatic waves are extended to two electromagnetic waves that propagate directly across the confining magnetic field. While the ratio of the frequency of each wave to the ion cyclotron frequency is large, the frequency difference is assumed to be near the ion cyclotron frequency. The nonlinear wave-particle interaction is studied analytically using a two time-scale canonical perturbation theory. The theory elucidates the effects of various parameters on the gain in energy by the ions—parameters such as the amplitudes and polarizations of the waves, the ratio of the wave frequencies to the cyclotron frequency, the difference in the frequency of the two waves, and the wave numbers associated with the waves. For example, the ratio of the phase velocity of the envelope formed by the two waves to the phase velocity of the carrier wave is important for energization of ions. For a positive ratio, the energy range is much larger than for a negative ratio. So waves like the lower hybrid waves will impart very little energy to ions. The theoretical results are found to be in good agreement with numerical simulations of the exact dynamical equations. The analytical results are used to construct mapping equations, simplifying the derivation of the motion of ions, which are, subsequently, used to follow the evolution of an ion distribution function. The heating of ions can then be properly quantified in terms of the wave parameters and can be conveniently used to find ideal conditions needed to heat ions by high frequency electromagnetic waves.

A multiscale approach to a synthetic aperture radar in dispersive random media

We consider synthetic aperture radar (SAR) image formation in the situation when the propagation medium is random and dispersive. The propagation model is the Klein–Gordon equation with a random index of refraction and a random dispersive term. We show via a multiscale analysis how medium heterogeneities and dispersion affect the image. In fact, in a situation with a strong source chirp signal, the main effect of the medium heterogeneities is to introduce random phase distortions in the SAR data. We carry out a novel scaling analysis that gives a precise characterization of this canonical phase perturbation and how it affects image resolution and stability. The main effect of the phase perturbation is to reduce the azimuthal resolution and the signal-to-noise ratio and we quantify this performance degradation.

토마토 분말 첨가 시폰 케이크의 제조조건 최적화 및 품질 특성

The purpose of this study was to determine the optimal mixing condition of tomato powder and sugar for producing chiffon cake. The experiment was designed according to the central composite design of response surface, which yielded ten experimental points, including two replicates. Physiochemical and sensory properties were measured, and theses values applied to mechanical models. A canonical form and perturbation plot showed the influence of each ingredient on the final product mixture. The results of the physiochemical analysis of each sample showed significant differences in sweetness (P

The Moduli Space of Germs of Generic Families of Analytic Diffeomorphisms Unfolding a Parabolic Fixed Point

In this paper we describe the moduli space of germs of generic families of analytic diffeomorphisms which unfold a parabolic fixed point of codimension 1. In [MRR] (and also [R]), it was shown that the Ecalle-Voronin modulus can be unfolded to give a complete modulus for such germs. The modulus is defined on a ramified sector in the canonical perturbation parameter $\eps$. As in the case of the Ecalle-Voronin modulus, the modulus is defined up to a linear scaling depending only on $\eps$. Here, we characterize the moduli space for such unfoldings by finding the compatibility conditions on the modulus which are necessary and sufficient for realization as the modulus of an unfolding. The compatibility condition is obtained by considering the region of sectorial overlap in $\eps$-space. This lies in the Glutsyuk sector where the two fixed points are hyperbolic and connected by the orbits of the diffeomorphism. In this region we have two representatives of the modulus which describe the same dynamics. We identify the necessary compatibility condition between these two representatives by comparing them both with their common Glutsyuk modulus. The compatibility condition implies the existence of a linear scaling for which the modulus is 1/2-summable in $\eps$, whose direction of non-summability coincides with the direction of real multipliers at the fixed points. Conversely, we show that the compatibility condition (which implies the summability property) is sufficient to realize the modulus as coming from an analytic unfolding, thus giving a complete description of the space of moduli.

Hamiltonian formulation of nonequilibrium quantum dynamics: Geometric structure of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy

Time-resolved measurement techniques are opening a window on nonequilibrium quantum phenomena that is radically different from the traditional picture in the frequency domain. The simulation and interpretation of nonequilibrium dynamics are conspicuous challenges for theory. This paper presents an approach to quantum many-body dynamics that is based on a Hamiltonian formulation of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations of motion for reduced density matrices. These equations have an underlying symplectic structure, and we write them in the form of the classical Hamilton equations for canonically conjugate variables. Applying canonical perturbation theory or the Krylov-Bogoliubov averaging method to the resulting equations yields a systematic approximation scheme. The possibility of using memory-dependent functional approximations to close the Hamilton equations at a given level of the hierarchy is discussed. The geometric structure of the equations gives rise to reduced geometric phases that are observable even for noncyclic evolutions of the many-body state. The approach is applied to a finite Hubbard chain which undergoes a quench in on-site interaction energy $U$. Canonical perturbation theory, carried out to second order, fully captures the nontrivial real-time dynamics of the model, including resonance phenomena and the coupling of fast and slow variables.

Directed transport in a classical lattice with a high-frequency driving

We analyze the dynamics of a classical particle in a spatially periodic potential under the influence of a periodic in time uniform force. It was shown by S. Flach and coworkers [Phys. Rev. Lett. 84, 2358 (2000)] that despite zero average force, directed transport is possible in the system. Asymptotic description of this phenomenon for the case of slow driving was developed by X. Leoncini and coworkers [Phys. Rev. E 79, 026213 (2009)]. Here we consider the case of fast driving using the canonical perturbation theory. An asymptotic formula is derived for the average drift velocity as a function of the system parameters and the driving law. We show that directed transport arises in an effective Hamiltonian that does not possess chaotic dynamics, thereby clarifying the relation between chaos and transport in the system. Sufficient conditions for transport are derived.

Fermi acceleration in time-dependent rectangular billiards due to multiple passages through resonances

We consider a slowly rotating rectangular billiard with moving boundaries and use canonical perturbation theory to describe the dynamics of a billiard particle. In the process of slow evolution, certain resonance conditions can be satisfied. Correspondingly, phenomena of scattering on a resonance and capture into a resonance happen in the system. These phenomena lead to destruction of adiabatic invariance and to unlimited acceleration of the particle.

On the Error Reduction of a Simple Symplectic Integrator

Abstract An analysis of truncation errors of the simplest symplectic integrator is revisited. We performed an analytical error estimate of this integrator within the framework of a canonical perturbation method, taking the two-body system as an example. Our study was motivated by the desire to remove any confusion in the derivation of this issue in one standard literature along this line of research. We repeated the derivation, and examined how a secular numerical error arises in symplectic integration. We showed that it is possible to substantially eliminate a secular numerical error by choosing appropriate initial conditions, as previous studies have shown. On the other hand, depending on boundary parameter values, such as the eccentricity in the two-body system, we could see that it is sometimes impossible to reduce the secular numerical error by changing the starting conditions. We also numerically demonstrated this reduction, or lack of reduction of numerical errors by taking a restricted three-body system as an example. We confirmed that the so-called “iterative start” of symplectic integrators works in certain systems.

Numerical-Analytic Implementation of the Higher-Order Canonical Van Vleck Perturbation Theory for the Interpretation of Medium-Sized Molecule Vibrational Spectra

Anharmonic vibrational states of semirigid polyatomic molecules are often studied using the second-order vibrational perturbation theory (VPT2). For efficient higher-order analysis, an approach based on the canonical Van Vleck perturbation theory (CVPT), the Watson Hamiltonian and operators of creation and annihilation of vibrational quanta is employed. This method allows analysis of the convergence of perturbation theory and solves a number of theoretical problems of VPT2, e.g., yields anharmonic constants y(ijk), z(ijkl), and allows the reliable evaluation of vibrational IR and Raman anharmonic intensities in the presence of resonances. Darling-Dennison and higher-order resonance coupling coefficients can be reliably evaluated as well. The method is illustrated on classic molecules: water and formaldehyde. A number of theoretical conclusions results, including the necessity of using sextic force field in the fourth order (CVPT4) and the nearly vanishing CVPT4 contributions for bending and wagging modes. The coefficients of perturbative Dunham-type Hamiltonians in high-orders of CVPT are found to conform to the rules of equality at different orders as earlier proven analytically for diatomic molecules. The method can serve as a good substitution of the more traditional VPT2.

Interaction of charged particles with localized electrostatic waves in a magnetized plasma

Charged particle interaction with localized wave packets in a magnetic field is formulated using the canonical perturbation theory and the Lie transform theory. An electrostatic wave packet characterized by a wide range of group and phase velocities as well as spatial extent along and across the magnetic field is considered. The averaged changes in the momentum along the magnetic field, the angular momentum, and the guiding center position for an ensemble of particles due to their interaction with the wave packet are determined analytically. Both resonant and ponderomotive effects are included. For the case of a Gaussian wave packet, closed-form expressions include the dependency of the ensemble averaged particle momenta and guiding center position variations on wave packet parameters and particle initial conditions. These expressions elucidate the physics of the interaction which is markedly different from the well known case of particle interaction with plane waves and are relevant to a variety of applications ranging from space and astrophysical plasmas to laboratory and fusion plasmas, as well as accelerators and microwave devices.