Terms Of Initial Conditions Research Articles

Interplay of exciton or electron transfer and relaxation: Time-convolution generalized master equation approach

Generalized Master Equation theory of the time-convolution type for exciton or electron interacting with harmonic phonons is suggested yielding the full single-carrier density matrix. A new form of the projection superoperator used makes the initial condition term zero for a broad class of initial conditions with a partially formed polaron cloud around carrier. Specification to a particle in a symmetric dimer and comparison with standard Haken-Strobl-Reineker model is performed. The theory is shown to yield that, surprisingly, in the resulting memory functions, apart of information about initial polaron cloud survives even after a short redressing time-interval following each individual (and not only initial) carrier hop.

Soliton solution for coupled periodically twisted birefringent optical fibers

Recently there has been interest in all-optical switching devices that may be implemented in optical communications or optical computing. I derive soliton solutions for the birefringent fiber filter coupler with the assumption of an ellipticity angle of 35°. In particular, I calculate the soliton switching behavior and illustrate it with some simple examples. I describe the conditions for soliton switching in terms of initial conditions and optical fiber parameters.

Chemical energy release and dynamics of transitional, reactive shear flows

Results are reported from numerical studies of a compressible, subsonic reactive mixing layer, on the the effects of chemical-reaction exothermicity on the shear-layer development, and the dependence of these effects on initial conditions. The model solves the unsteady, conservation equations for mass, momentum, energy, and species concentrations. The convective transport equations are solved using the flux-corrected transport (FCT) algorithm and appropriate inflow and outflow boundary conditions. A one-step, irreversible, Arrhenius chemical reaction rate, and realistic (species- and temperature-dependent) modeling of diffusive transport are coupled with the convective transport using time-step splitting. The system studied consists of nonpremixed coflowing streams, where both the fuel (faster stream, hydrogen) and the oxidizer (slower stream, oxygen) are diluted in nitrogen. To facilitate the analysis of the results the flow is organized by low-level, single-frequency velocity perturbation at the inflow. The simulations show that energy release has the effect of reducing the shear layer growth and the amount of chemical product formed−relative to the corresponding cases for which exothermicity is not accounted for, in qualitative agreement with results from previous investigations. The relative mixing-layer growth reduction becomes more pronounced for larger energy release and lower Re, and is significant in terms of both, Reynolds stress, ρu′v′, and the velocity-fluctuation correlation u′v′. In spite of the relatively fast flows studied, for the regimes considered, the results on the initial mixing-layer growth are significantly sensitive to diffusive transport effects−more so in terms of Reynolds stress, than in terms of product formation. With larger energy release here associated to larger free-stream reactant molar fractions c0, the amount of chemical product in absolute terms is found to increase with energy release−but slower than c0, so that the product formation becomes effectively less efficient. The results of the present work highlight the difficulties involved in making general statements about the effects of exothermicity on the mixing-layer growth, indicating that a careful conceptualization of these properties in terms of initial conditions and other characteristic parameters of the reactive systems under study is required.

Modeling impact on a one-link flexible robotic arm

A finite-element model for a single-link flexible robotic arm including the effects of beam damping, hub inertia and both Coulomb and viscous hub friction is derived. The initial conditions required to represent impact loading are determined, and the motion of the arm under impact loading is simulated. Simulation results are compared to experimental data. From the experimental results it is concluded that this model provides an accurate representation of the physical process. It also provides a vehicle for investigating the effects of parameters of the process model, in particular, frictional effects and load profiles. Impact loading has been effectively represented in terms of initial conditions and can be applied to a range of finite-element models.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Integral equation approach to joint state and parameter estimation in continuous-time MIMO systems

This paper shows how the problem of simultaneous estimation of the continuous-time (CT) model parameters and the state of a multi-input multi-output (MIMO) system can be decoupled into a set of multi-input single-output (MISO) subproblems. It then shows how the MISO problem can be solved following an integral equation approach with a common denominator (CD) model. The key observation is that the initial condition terms arising out of integration in the data vector, heretofore considered to be an unnecessary burden on the estimator, form the initial state of a canonical realisation. State estimation is then computationally simple. The method is suitable for real-time implementation. It is also general and permits realisation of the measures of signals and their integrals by spectral methods employing orthogonal functions (OF), numerical methods from signal samples etc. An implementation of the algorithm based on block pulse functions (BPF) has been made. A discussion on related aspects of real-time implementation, accuracy and convergence of estimates, possible variants of the scheme etc. is also presented, along with a comparison with other existing methods and scope for further improvement.

Higher-order effects in non-Markovian theory of electronic relaxation in interacting electron-phonon systems

A finite coupling generalised master equation theory of the electronic relaxation among electronic eigen-states is presented. When converted into a closed time-convolution form (TCGME) for just diagonal elements of the density matrix, the theory is appreciably non-Markovian with the memory kernel summed partially up to the infinite order in the electron-phonon coupling. On soluble examples, it is shown that: (a), higher-order effects may influence even the lowest-order relaxation rates; (b), weakness of the finite coupling is not sufficient for reducing the theory to the standard lowest-order form; (c), the Markov approximation to the TCGME is in general misleading; (d), the initial condition term may appreciably influence the long-time development of the relaxation; (e), the theory is able to yield quasi-coherent effects (quantum beats) due to the electron coupling to the bath; (f), very effective tunnelling through barriers between equivalent sites appears with non-zero tunnelling currents starting from the second order in the electron-phonon coupling. Limitations from above (below) for the time (frequency) are found for the theory to reduce to the standard lowest-order form.

GME and the Electron or Exciton Initial Condition Term Due to Lattice Relaxation Dimer

AbstractInstantaneous creation of an electron or exciton in a dimer interacting with initially statistically independent unrelaxed lattice phonons leads to an initial condition term in the time‐convolution generalized master equations (GME) for the site occupation probabilities which may be undamped with increasing time till the coherence decay time provided the finite exciton‐lifetime effects are negligible. For the Peier form of the projector diagonalizing the full density matrix in the small polaron representation with the canonical form of the reference phonon density matrix and temperature equal to that of the initially unrelaxed equilibrium harmonic phonons, this term is calculated to the second order in the relevant hopping integral and its magnitude is estimated.

Theory of geminate recombination on a lattice

Central to the photogeneration process in molecular systems is the competition between the relaxation of an excited state (to the ground state) and the separation into an ion pair via charge transfer steps. We have developed the first comprehensive model of this process which depends on molecular parameters and denumerates the competition between the various rates. The model is discrete diffusion on a lattice in a combined Coulomb and external field. The model is solved exactly and is computationally straightforward. The analytic method we have used is quite general and can easily be extended to include a broad class of problems involving large numbers of (correlated) ‘‘special sites.’’ The main feature of the computation, needed to solve these problems, is the evaluation of the lattice Green’s functions in the presence of the electric field. Our solution has a larger range of applicability than the (continuum limit) Onsager theory and is more versatile in terms of initial conditions and modeling of transient phenomena. We will elaborate this approach and discuss our results for the quantum efficiency as a function of external field, temperature, and molecular concentration, η(E,T,c). We show that η(E,T,c) exhibits a diversity of behavior dependent on the relative magnitudes of inter- and intramolecular transition rates. Onsager-like behavior for η is contained in the infinite sink limit and more generally, η can change appreciably as a function of the molecular parameters even for a fixed initial separation r0. In this theory the r0 is geometry controlled, and the initial yield φ0 can be determined by the same competition of rates as control the long time yield. The short time yield is controlled by a few discrete hops and therefore follows an exponential decay as indicated by recent picosecond experiments. In general, within this framework, one can study the influence on η of such factors as dimensionality, lattice structure, disorder, tunneling transition rates, intramolecular rates, and intrinsic energy level differences.

Initial Condition Term and Method of Artificial Temperature for Generalized Master Equations

Initial Condition Term and Method of Artificial Temperature for Generalized Master Equations

Solutions of the generalized rate law for vibrational relaxation of a pure diatomic gas

The generalized rate law for the relaxation of the vibrational energy of a pure diatomic gas, AB, derived earlier, is solved analytically for a variety of initial conditions corresponding to shock tube, laser-excited fluorescence, and chemical activation experiments. The resulting expressions can be used to easily predict whether a given system will relax according to a V–V or a T–V mechanism or both. The initial conditions and the molecular anharmonicity are shown to be as important, if not more important, for this purpose than the ratio of T–V and V–V rate constants. Behind shock waves the energy relaxes exponentially with a T–V time constant. The initial distribution remains Boltzmann. In laser or chemical activation experiments the energy does not relax exponentially, leading to phenomenological time "constants" [Formula: see text] or [Formula: see text] which are not constant in time and prevent direct comparisons with shock tube data. It is only after an incubation period during which the vibrational energy is redistributed via V–V processes that the energy then exchanges with translational energy and decays. Prescriptions are given to extract T–V and V–V rate constants from such data. The initial degree of laser excitation, a, and the time regime probed, t/τ, must be known for this purpose. However, when direct overtone excitation is used, a careful choice of α can lead to extraction of the T–V constant directly. Even though the vibrational energy itself does not relax exponentially, it is shown that the mean energy, [Formula: see text], and the mean squared energy, [Formula: see text], relax in such a way that the quantity [Formula: see text] does decrease exponentially with a time constant very closely related to the V–V rate constant for 2AB(ν = 1) → AB(ν = 2). A short survey of various laser and chemical excitations in the literature is presented and analyzed in terms of initial conditions. In general, the larger the degree of excitation and the higher the quantum numbers of the excited levels, the more V–V character does the energy relaxation have.

Ignition time in the stretched-flow field

Ignition times in a typical stretched-flow field counterflow system are calculated for both initially unmixed and premixed gases. To simplify the calculation, ideal gases with equal velocities are assumed, and asymptotic analysis for the limit of large activation energy is developed in order to reduce the six parameters which control the field. The solution of the present unsteady problem is uniquely determined in terms of initial conditions. Mixing and heat transfer should increase as the flow velocity increases, and therefore the ignition time is expected to decrease. However, in both cases of unmixed and premixed gases, the ignition time increases as the stretch rate, which is proportional to the flow velocity, increases. This indicates that a long induction time is required to reach a reaction intense enough for ignition. At a certain large flow stretch rate, the ignition time becomes infinite and ignition can no longer take place. Previously, such an ignitable limit was discussed as a function of flame temperature variation with flow velocity, but in this analysis it is derived for ignition time variation.

Generalized master equations under delocalized initial conditions

The initial condition term that must be appended to the generalized master equation (GME) when the density matrix is not initially diagonal in the representation chosen is studied and explicit expressions are obtained for several cases. The term is shown to vanish for initial occupation of a Bloch state of arbitrary wave vector if the system is a crystal and the representation is that of site states, despite the violation of the initial diagonality condition. It is pointed out how one is to use the expressions for the initial term in transport calculations.

Unconstrained Variational Statements for Initial and Boundary-Value Problems

Abstract : A procedure is developed for generating variational statements suitable for obtaining approximate solutions to boundary-initial value problems. The essence of the procedure is to introduce all boundary and initial conditions into the variational statement as natural boundary conditions. This is accomplished through the use of Lagrange multipliers in which all initial condition terms as well as boundary terms are determined analytically. The result is a variational statement in which completely unconstrained trial functions may be assumed as a basis for an approximate solution. Several applications are given, including the response of a beam subject to a moving concentrated mass loading.

The Ideal Resonance Problem: A comparison of the solutions expressed in terms of mean elements and in terms of initial conditions

In an earlier publication (Jupp, 1972), a solution of the Ideal Resonance Problem is exhibited explicitly in terms of the ‘mean’ elements; to second order in the small parameter in the case of libration, and to first order in the case of deep circulation. Both representations possess a singularity when the ‘mean’ modulus of the Jacobi elliptic functions is unity; this corresponds to the separatrix of the phase plane of the dynamical system.

Transition and Mixing in the Shear Layer Produced by Tangential Injection in Supersonic Flow

The interaction between a viscous mixing layer induced by tangential injection, and an external supersonic flow field is considered experimentally and analytically. Both subsonic and supersonic injection are investigated. The experiments were performed at freestream Mach numbers of 2.85 and 4.19 using air as the injectant. The principal observations are in the form of spark schlieren photographs, interferograms, and wall pressure distributions. The experiments were arranged to cross Lin’s neutral stability boundary for parallel streams. Transition occurred in all cases, but an increase in stability was noted with either a decrease in the injectant Mach number or an increase in the Mach number of the external flow. Both of these results follow the trends predicted by the stability theory. For the supersonic injection cases, it was found that simple inviscid theory is sufficient to predict the overall interaction pattern between streams, when the ratio of initial boundary layer thickness to the injection slot height is small. However, when the injection is subsonic, the injectant initial conditions in terms of either pressure or Mach number at the slot exit are determined by the downstream viscous-inviscid interaction with the external supersonic flow. A simple one-dimensional theory is applied to this problem to enable prediction of the initial conditions.

Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite

Gravity-gradient perturbations of the attitude motion of a tumbling tri-axial satellite are investigated. The satellite center of mass is considered to be in an elliptical orbit about a spherical planet and to be tumbling at a frequency much greater than orbital rate. In determining the unperturbed (free) motion of the satellite, a canonical form for the solution of the torque-free motion of a rigid body is obtained. By casting the gravity-gradient perturbing torque in terms of a perturbing Hamiltonian, the long-term changes in the rotational motion are derived. In particular, far from resonance, there are no long-period changes in the magnitude of the rotational angular momentum and rotational energy, and the rotational angular momentum vector precesses abound the orbital angular momentum vector. At resonance, a low-order commensurability exists between the polhode frequency and tumbling frequency. Near resonance, there may be small long-period fluctuations in the rotational energy and angular momentum magnitude. Moreover, the precession of the rotational angular momentum vector about the orbital angular momentum vector now contains substantial long-period contributions superimposed on the non-resonant precession rate. By averaging certain long-period elliptic functions, the mean value near resonance for the precession of the rotational angular momentum vector is obtained in terms of initial conditions.

Effects of Finite Lattice Heat Capacity on Spin-Lattice Relaxation

The spin-lattice approach to equilibrium is studied for situations in which the littice heat capacity is not infinite. It is assumed that the spins and the lattice each have a well-defined temperature, and that a bath either has been removed or is only very weakly coupled to the lattice. It is shown that asymptotically the magnetization relaxes exponentially with a relaxation time which can be very much different from ${T}_{1}$ (the spin-lattice relaxation time for an infinite lattice heat capacity), not only in numerical value but in its functional dependence upon temperature, magnetic field, and spin concentration. A general formula yielding the final common temperature in terms of initial conditions is given. Some examples are presented in which the effect of the finite lattice is to lock the magnetization in at its initial value for times 1 to 2 orders of magnitude longer than the ultimate relaxation time.

Theory of the Orbit of an Artificial Satellite with Use of Spheroidal Coordinates.

view Abstract Citations (12) References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Theory of the Orbit of an Artificial Satellite with Use of Spheroidal Coordinates. Vinti, John P. Abstract The author has developed the orbital theory for an artificial satellite, using a gravitational potential, (Vinti, 1959, J. Research Nat. Bur. Standards 63B 105) in spheroidal coordinates, p, ~, 4), which leads to quadrature solutions. This procedure accounts for all of the second harmonic (of coefficient J2) and for most of the fourth, but omits entirely the local anomalies, drag, lunar-solar perturbations, and the odd harmonics when the origin is chosen at the earth's center of mass. A previous report (Vinti, 1960, Bull. Am. Phys. Soc. 5, 8) outlined a solution for the secular motions of CL, and ~. The complete orbital solution now requires evaluation of the integrals in the kinetic equations and their subsequent solution for p and ,i; all other quantities follow readily from these. Only the p integrals cause trouble; if .1... P4 are the zeros of a certain quartic F(p) that they all involve, one must first find Pi+P2, P3+P4, PiP2, and p~p~, where p' and P2 are those zeros of F(p) which would reduce to the perigee and apogee radii if J2 vanished. With this fact as a start, a method of successive approximations leads to power series in J2 for these sums and products, the present treatment stopping with J22 Expressions of order J22 follow readily for the p integrals and the ~ integrals, the former involving two angles closely related to v and E, the latter involving two other angles closely related to CL, + ~ and 4) - ~. Here v and E are the true and eccentric anomalies, CL, is the argument of perigee, and 4) and ~ are the right ascensions of the satellite and the node. The constants that appear are all expressible in terms of initial conditions. A method of successive approximations, started by neglecting terms involving J2, then leads to the solution for p and ~ as functions of time, through terms of order J22 Calculation of the polar coordinates is then easy. This work was supported by the U. S. Air Force, through the Office of Scientific Research of the Air Research and Development Command. Publication: The Astronomical Journal Pub Date: August 1960 DOI: 10.1086/108267 Bibcode: 1960AJ.....65..353V full text sources ADS |