Unperturbed Propagator Research Articles

Coupling between turbulence and solar-like oscillations: A combined Lagrangian PDF/SPH approach

Context. The development of space-borne missions such as CoRoT and Kepler now provides us with numerous and precise asteroseismic measurements that allow us to put better constraints on our theoretical knowledge of the physics of stellar interiors. In order to utilise the full potential of these measurements, however, we need a better theoretical understanding of the coupling between stellar oscillations and turbulent convection. Aims. The aim of this series of papers is to build a new formalism specifically tailored to study the impact of turbulence on the global modes of oscillation in solar-like stars. In building this formalism, we circumvent some fundamental limitations inherent to the more traditional approaches, in particular the need for separate equations for turbulence and oscillations, and the reduction of the turbulent cascade to a unique length and timescale. In this first paper we derive a linear wave equation that directly and consistently contains the turbulence as an input to the model, and therefore naturally contains the information on the coupling between the turbulence and the modes through the stochasticity of the equations. Methods. We use a Lagrangian stochastic model of turbulence based on probability density function methods to describe the evolution of the properties of individual fluid particles through stochastic differential equations. We then transcribe these stochastic differential equations from a Lagrangian frame to a Eulerian frame more adapted to the analysis of stellar oscillations. We combine this method with smoothed particle hydrodynamics, where all the mean fields appearing in the Lagrangian stochastic model are estimated directly from the set of fluid particles themselves, through the use of a weighting kernel function allowing to filter the particles present in any given vicinity. The resulting stochastic differential equations on Eulerian variables are then linearised. As a first step the gas is considered to follow a polytropic relation, and the turbulence is assumed anelastic. Results. We obtain a stochastic linear wave equation governing the time evolution of the relevant wave variables, while at the same time containing the effect of turbulence. The wave equation generalises the classical, unperturbed propagation of acoustic waves in a stratified medium (which reduces to the exact deterministic wave equation in the absence of turbulence) to a form that, by construction, accounts for the impact of turbulence on the mode in a consistent way. The effect of turbulence consists of a non-homogeneous forcing term, responsible for the stochastic driving of the mode, and a stochastic perturbation to the homogeneous part of the wave equation, responsible for both the damping of the mode and the modal surface effects. Conclusions. The stochastic wave equation obtained here represents our baseline framework to properly infer properties of turbulence-oscillation coupling, and can therefore be used to constrain the properties of the turbulence itself with the help of asteroseismic observations. This will be the subject of the rest of the papers in this series.

About the validity of complex mass scheme for the Δ resonance and higher energy region approaches

It is a well known fact that when a resonance (R) is produced and decays, its unperturbed propagator will be singular at when it appears in an s-type Feynman pole graph. Nevertheless this is not a problem, since it must be dressed by the successive re-interactions of the resonance with itself to all orders through the non-pole T-matrix, which we include in this work for the self-energy one-loop pion–nucleon contributions. Recently, we have shown that within the one-loop approximation for the absorptive part it is possible to justify the use of the complex-mass description over the unperturbed propagator, adopted in previous calculations to treat pion–nucleon elastic and radiative scattering, and photo and weak pion production in the resonance region. Here we discuss the limits of the validity of this approximation, since in actual calculations we must go over this range. We show that, when the resonance energy is increased above , this approach becomes invalid and an alternative approximation should be considered. In addition, we discuss the usual energy dependent width approximation, present in several actual isobar models used to analyze the last neutrino scattering experiments, and show that also it could be unappropriate.

Coherent stochastic resonance in a sextic double-well potential

We study the response of a stochastic system described by the one-dimensional Fokker–Planck equation with a sextic double-well potential U 6( x)=(− c 4 x 4+ c 6 x 6) to a weak sinusoidal periodic signal. Two types of boundary conditions have been investigated: (i) absorbing boundary conditions, and (ii) natural boundary conditions. The unperturbed propagators are calculated by the eigenfunction-expansion method. We find that for case (i), the mean survival time shows the coherent stochastic resonance, and for case (ii), the signal-to-noise ratio exhibits the stochastic resonance. The results are compared with those of the bistable quartic double-well potential U 4(x)=(− 1 2 x 2+ 1 4 x 4) .

Perturbation method with triangular propagators and anharmonicities of intermediate strength

We propose a new, very flexible version of the Rayleigh–Schrodinger perturbation method which admits a lower triangular matrix in place of the usual diagonal unperturbed propagator. The technique and its enhanced efficiency are illustrated on rational anharmonicities V(1)(x)=β×polynomial(x)/polynomial(x). They are shown tractable, in the intermediate coupling regime, as \(\mathcal{O}(\beta {\text{ - }}\beta ^{{\text{(0)}}} )\) perturbations of exact states at non-vanishing β(0)≠0. In this sense our method bridges the gap between the current weak- and strong-coupling expansions.

A perturbation model to predict thermal sensitivity of transverse waves on thin quartz plates

The development of new acoustic wave devices has received increasing interest for various applications, particularly for sensors, but also in the telecommunication domain. The need to increase operating frequencies together with the efficiency of such devices pushes new principles to produce high-quality acoustic wave resonators and filters. Previous work has been done to experimentally measure the thermal sensitivity of transverse waves trapped by metal strip gratings deposited on thin quartz plates. Temperature compensation was found for AT-cut devices in an extended range of frequency, with the turnover temperature varying from 30 to 0 °C. Moreover, thermal compensation was found for higher-order modes in the Z cut, although the first mode exhibited a very high sensitivity to temperature. A theoretical model has been developed to explain these results. It is based on the Sinha–Tiersten perturbation approach combined with a model of unperturbed propagation, taking piezoelectricity into account. Comparison between the theoretical results provided by this model and experimental data is presented. A very good agreement is found for Z and AT-cut devices, even if second-order effects appear to be predominant in the latter case.

Nonequilibrium perturbation theory for complex scalar fields

Real-time perturbation theory is formulated for complex scalar fields away from thermal equilibrium in such a way that dissipative effects arising from the absorptive parts of loop diagrams are approximately resummed into the unperturbed propagators. Low order calculations of physical quantities then involve quasiparticle occupation numbers which evolve with the changing state of the field system, in contrast to standard perturbation theory, where these occupation numbers are frozen at their initial values. The evolution equation of the occupation numbers can be cast approximately in the form of a Boltzmann equation. Particular attention is given to the effects of a non-zero chemical potential, and it is found that the thermal masses and decay widths of quasiparticle modes are different for particles and antiparticles.

Double well model V( r) = ar2 + br4 + cr6 with a < 0 and perturbation method with triangular propagators

We propose a new version of the Rayleigh-Schrödinger perturbation prescription suitable for tridiagonal zero-order Hamiltonians. It picks up an unusual model space and its unperturbed propagators remain easily invertible lower triangular matrices as a consequence. Its efficiency is illustrated on sextic potentials with double-well shape.

Nonlinearized Perturbation Theories

A brief review is presented of the two recent perturbation algorithms. Their common idea lies in a not quite usual treatment of linear Schrodinger equations via nonlinear mathematical means. The first approach (let us call it a quasi-exact perturbation theory, QEPT) tries to get the very zero-order approximants already "almost exact", at a cost of leaving the higher-order computations more complicated. Technically, it constructs and employs solutions of certain auxiliary nonlinear systems of algebraic equations for the suitable zero-order couplings and energies. The second approach (a fixed-point perturbation theory, FPPT) pays more atten- tion to the higher-order corrections. Its purpose lies in an improvement of construction of unperturbed propagators or, alternatively, of the closely related (so-called eective) finite-dimensional auxiliary Hamiltonians. On a technical level, it employs a factor- ization interpreted via certain nonlinear mappings and, finally, approximates some matrix elements by fixed points of these mappings. In a broad context of the "generalized Rayleigh-Schrodinger" perturbation strat- egy, both the prescriptions need just more summations over "intermediate states". QEPT defines its nondiagonal unperturbed propagators in terms of infinite continued fractions. FPPT introduces a further simplification via another finite system of non- linear algebraic equations for fixed points. Thus, both the subsequent QE and FP steps of construction share the same mathematics.

Diagonalization of propagators in thermo field dynamics for relativistic quantum fields

Two-point functions for interacting quantum fields in statistical systems can be diagonalized by matrix transformations. It is shown, that within the framework of time-dependent Thermo Field Dynamics this diagonalization can be understood as a thermal Bogoliubov transformation to non-interacting statistical quasi-particles. The condition for their unperturbed propagation relates these states to the thermodynamic properties of the system: It requires global equilibrium for stationary situations, or specifies the time evolution according to a kinetic equation.

Feynman rules for nonequilibrium field theory

Within the closed-time-path formalism, a set of real- and imaginary-time propagators is obtained, which are suitable for the perturbative evaluation of Green functions in nonequilibrium states of a scalar field theory. A generalized renormalization procedure, described in earlier works, allows for the partial resummation of absorptive parts of loop diagrams, so that finite quasiparticle lifetimes are incorporated in the unperturbed propagators. In this way, low-order calculations are made to reflect the time evolution of the nonequilibrium state, which is not correctly described by standard perturbation theory.

Effect of bending on zero dispersion operation of single-mode optical fibers

Perturbation theory for bent optical fibers predicts that the unperturbed (straight) propagation constant (or effective index) will be corrected by a small term that is proportional to (R(b))(-2), where R(b) is the radius of the bend. We examine how this small correction affects the chromatic dispersion of the fundamental mode, and in particular we examine the changes that occur when the fiber operates at the zero dispersion wavelength lambda(o). For a bend radius of 1 cm, lambda(o) may shift by up to 20 nm.

Perturbative description of dissipation in nonequilibrium field theory.

Dissipative effects, such as the relaxation of quasiparticle occupation numbers, arise from absorptive parts of Green's functions, which typically appear first at the second order of perturbation theory. Within the closed-time-path formalism, it is shown, using a generalized renormalization technique, that these absorptive parts may be approximately resummed so as to appear in unperturbed propagators. In this way, it becomes possible to study, in low-order perturbation theory, the evolution in time of a field theory which is driven away from thermal equilibrium by the presence in its Hamiltonian of explicitly time-dependent parameters. Particular attention is given to a scalar field with time-dependent mass, which is relevant to the dynamics of phase transitions in the very early Universe. Under favorable conditions, the analysis leads to a kinetic equation of the Boltzmann type, and an approximate numerical solution of this equation is presented for illustrative purposes.

Ionization by an intense slowly varying laser field: Gauge-invariant approximations.

A method to calculate amplitudes for ionization in a strong laser field is developed that is constrained in such a way that gauge invariance is necessarily satisfied when approximations are introduced. A variational approximation procedure is adopted here, in the context of a simplified model of a charged particle bound in a short-range potential. The physical situation considered in an illustration of the method is one in which the external field is slowly varying relative to the time required for the particle to penetrate the potential barrier. The approximation derived here represents an extension of results obtained previously for problems of this type in that the effect of the laser field on the propagation of the particle during the ionization process is accounted for, in a manner consistent with the assumed slowly varying nature of the field. This is accomplished, specifically, through the replacement of the unperturbed propagator in the integral equation defining the continuum (momentum-space) wave function of the particle in the potential by a generalized Airy function. The relation between the approximation derived here and some results obtained previously is briefly discussed.

Padé oscillators and a new formulation of perturbation theory

Being guided by the problem of bound states in potentials close to their Padé approximants, a new Rayleigh–Schrödinger-type perturbation theory is developed. The unperturbed system is understood: here in a broader sense: its solutions are not needed, but merely the related nondiagonal unperturbed propagator R. In particular, all the chain models H0ψ=ES0ψ (H0, S0=band matrices) with arbitrary perturbations are then perturbatively solvable, with R constructed in terms of auxiliary matrix continued fraction fn. Alternatively, a ‘‘generalized unperturbed spectrum’’ f̂n may be required as an input: The algebraically constructed asymptotics of the fn’s play this role in our Padé examples. Due to S≠I, the‘‘Sturmians’’ may also be constructed. In the test evaluations of the binding energies and/or couplings, the simultaneous upper and lower bounds of high precision are shown to be numerically obtainable.

Pure bending‐induced birefringence in optical waveguides: A simple analytical expression

AbstractThe local field in the bent optical waveguide and the birefringence caused by the pure bent geometry effect are investigated. It is found that, this birefringence is independent of the local field in the bent guide and only dependent on the unperturbed propagation constant and the radius of the guide bend.

Modified Rayleigh-Schrödinger perturbation theory.

A new version of perturbation theory is described. It extends the admissible class of Hamiltonians ${H}_{0}$ to nondiagonal matrices via use of the nondiagonal unperturbed propagators. Numerically, its efficiency is demonstrated on the standard anharmonic-oscillator example where it works as an effective resummation technique.

Momentum density of electrons in liquid aluminum

The electron momentum density in liquid aluminum is calculated. The electron correlation effects are included in this calculation by following the scheme of Daniel and Vosko. The unperturbed electron propagators are modified to include the effects of the ion potentials. Numerical results show a substantial change from the case of a homogeneous electron gas of corresponding density, and also from a calculation including the ion potentials but ignoring correlations.

On the Critical Dynamics in Bose Systems and the Time-Dependent Ginzburg-Landau Model

An extended expression of the dynamic self-energy of one-particle Green function is obtained up to order 1/m in m-component Bose systems, by means of a complex unperturbed propagator. As applications of it, dynamic critical exponents of the TDGL model as well - as of a Bose system are evaluated to order 1/m, for both short-range and long-range inter­ actions. These confirm the previous results of the TDGL model and of the Bose system for short-range interactions. An effective interaction V.rr(p, i(J).) is obtained to order 1/m, which is useful for discussion on anomalies of transport coefficients near the A.-transition.

The Ginzburg criterion-rationalized

Following a critique of the various arguments leading to Ginzburg's criterion for the width of the critical region the author shows that if the unperturbed propagator of fluctuations is expressed in terms of the full susceptibility (mass renormalization) then the width of the critical region follows systematically. The procedure gives, automatically, the width of the region about the real transition temperature for which both the Landau and the Ginzburg-Landau theories are valid. Exactly the same logic applies above and below the transition temperature.

Green's function equations of motion for a many-fermion system: II. Inhomogeneous system at finite temperature

The equations of motion for many-time causal Green's functions are extended to an inhomogeneous many-fermion system at finite temperature. The boundary condition that the perturbation vanishes in the remote past and distant future (adiabatic hypothesis) is used to determine the unperturbed propagator. The temperature enters the theory only as a parameter. Thus there is no need for analytic continuations in the complex temperature-time plane. The theory is used to derive thermal Hartree-Fock theory and Wick's theorem at finite temperature. A linked cluster perturbation expansion at finite temperature is obtained by iterating the equations of motion, without unlinked disconnected diagrams even appearing. After integration over frequency, the present theory gives the perturbation theory rules in terms of global propagators that Baym and Sessler obtained from the imaginary-time theory.