Stationary solutions

The stationary solution of (A3.3.61). when /R(x) is very small, satisfies... 

For long times, the smaller eigenvalue will dominate (A3.4.141T yielding the stationary solution... 

The typical dependence of the stable stationary solutions to (4) on the control parameter of the model Xe is presented in Fig. 1. These results have been obtained as numerical solutions of (4) with equal to... 

This condition is thus the necessary and sufficient condition for the existence of a stable stationary solution (oscillation) of the differential equation (6-127). 

For partially ordered media the stationary solution (A8.10) is an eigenfunction of the integral operator in (A8.10), belonging to the eigen-... 

The classification of possible regimes of flow are proposed. It is based on a non-dimensional parameter accounting for the ratio of the micro-channel length to the capillary height. It is shown that in the generic case the governing system of equations, which describes capillary flow, has three stationary solutions two stable and one (intermediate) unstable. 

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

Static measurements (stationary solution). After a coulometric pulse of specific magnitude, the resulting pH step is measured. Repeating the experiment with different pulses allows the construction of the titration curve. 

Dynamic measurements (stationary solution). The pH change during one continuous pulse (up to 150 pC) is registered. Owing to the small volume involved, diffusion times have only a limited effect and the recording gives a fair approximation of the titration curve. 

With the stationary solution ipfE, one can use asymptotic boundary conditions to extract the scattering matrix. However, for the total reaction... 

To obtain the attachment reaction efficiency in the quasi-free state, we denote the specific rates of attachment and detachment in the quasi-free state by kf and kf respectively and modify the scavenging equation (10.10a) by adding a term kfn on the right-hand side, where is the existence probability of the electron in the attached state. From the stationary solution, one gets kf/kf = (kfk ikfkf), or in terms of equilibrium constants, K(qf) = Kr.Kr, where k, and k2 are the rates of overall attachment and detachment reactions, respectively. Furthermore, if one considers the attachment reaction as a scavenging process, then one gets (see Eq. 10.11) = k f fe/(ktf + kft) = fe,f/(l + Ku) and consequently k2 = kfKJ(l + KJ. 

The Fokker-Planck equation is a partial differential equation. In most cases, its time-dependent solution is not known analytically. Also, if the Fokker-Planck equation has more than one state variable, exact stationary solutions are... 

The theory of shape selection has been examined by many investigators concerned with solidification from the melt, and its status has recently been reviewed by Caroli and Muller-Krumbauer [63], The problem is to find stable, quasi-stationary solutions to the diffusion equation where a propagating branch maintains a constant shape and velocity. If the interface is assumed to have a uniform concentration, a family of such solutions exists, but there is no unique solution owing to the lack of a characteristic length. The solutions fix the peclet number. 

The general solution is a superposition of stationary solutions with constant coefficients Cj. In this sense, a knowledge of the stationary states provides the complete solution to the problem. The probability that measuring the energy of the system would yield EJ follows immediately as cj 2. 

To understand the physical implications of these equations it is customary to consider the non-relativistic limit of their stationary solutions, assuming... 

Nevertheless, very-long-lived quasi-stationary-state solutions of Schrodinger s equation can be found for each of the chemical structures shown in (5.6a)-(5.6d). These are virtually stationary on the time scale of chemical experiments, and are therefore in better correspondence with laboratory samples than are the true stationary eigenstates of H.21 Each quasi-stationary solution corresponds (to an excellent approximation) to a distinct minimum on the Born-Oppenheimer potential-energy surface. In turn, each quasi-stationary solution can be used to construct an alternative model unperturbed Hamiltonian //(0) and perturbative interaction L("U),... 

Both the initial- and the final-state wavefunctions are stationary solutions of their respective Hamiltonians. A transition between these states must be effected by a perturbation, an interaction that is not accounted for in these Hamiltonians. In our case this is the electronic interaction between the reactant and the electrode. We assume that this interaction is so small that the transition probability can be calculated from first-order perturbation theory. This limits our treatment to nonadiabatic reactions, which is a severe restriction. At present there is no satisfactory, fully quantum-mechanical theory for adiabatic electrochemical electron-transfer reactions. 

Columns two and three of Table E8.3 list the components of x that are the stationary solutions of the problem. Note that the solutions with u > 0 are minima, those for u < 0 are maxima, and u = 0 is a saddle point. This is because maximizing/is equivalent to minimizing -/, and the KTC for the problem in Equation (a) with / replaced by —/are the equations shown in (c) with u allowed to be negative. In Fig-... 

These equations, obtained by a simple superposition of the external force term and the relaxation term, give a quantitative agreement with experiment in the case of liquids. The stationary solution of Eq. (6) allows the calculation of %. 

For large Reynolds numbers, the right-hand side of this expression will be large, thereby forcing the scalar dissipation rate to attain a stationary solution quickly. Thus, for a fully developed scalar spectrum, the scalar mixing rate is related to the turbulent frequency by... 

A CV voltammogram can be recorded under either a dynamic or a steady state depending on the electrode design and solution convection mode. In a stationary solution with a conventional disk electrode, if the scan rate is sufficiently high to ensure a non-steady state, the current will respond differently to the forward and backward potential scan. Figure 63 shows a typical CV for a reversible reduction.1... 

In papers , unsteady-state regime arising upon propagation of the stationary fundamental mode from linear to nonlinear section of a single-mode step-index waveguide was studied via numerical modeling. It was shown that the stationary solution to the paraxial nonlinear wave equation (2.9) at some distance from the end of a nonlinear waveguide has the form of a transversely stable distribution ( nonlinear mode ) dependent on the field intensity, with a width smaller than that of the initial linear distribution. 

As it is well known, stationary solutions to Eq.(3.2) occur at the extrema of the Hamiltonian for a given power. The solutions that correspond to global or local minimum of H for a family of solitons are stable. The representation of the output nonlinear waveguide as a nonlinear dynamical system by the Hamiltonian allows to predict, to some extent, the dynamics of the total field behind the waveguide junction. 

An obvious question arises now are they stationary solutions with finite mass and energy of the diffusion Eq. (8) ... 

Assume that to the left of the combustor outlet boundary, rr = 0, there exists a stationary solution of the Euler equations p = po, P = Po, u = uq, where po, po, and Mo are the constant pressure, density, and velocity. Flow velocity has a single nonzero component, mq, along the x axis. The flow is assumed subsonic, i.e., M = uq/cq < 1, where cq is the speed of sound. We consider the solution of the nonstationary Euler equations and linearize the problem in the vicinity of the stationary solution by assuming that... 

Even though the bifurcation behavior exhibits a Z-shaped curve, it is more complicated due to the existence of the HB. For example, upon ignition, the system is expected to oscillate because no locally stable stationary solutions are found (an oscillatory ignition). Time-dependent simulations confirm the existence of self-sustained oscillations [7, 12]. The envelope of the oscillations (amplitude of H2 mole fraction) is shown in circles (a so-called continuation in periodic orbits). 

An important outcome of these simulations is the location of HB points (largely ignored in previous work), which is important for the development of extinction theory. In particular, the turning point E lies on a locally unstable stationary solution branch and does not coincide with the actual extinction, as previously thought. The actual extinction point is the termination point of oscillations. Thus, local stability analysis is essential to properly analyze flame stability and develop extinction theory. 

Figure 26.3 The mole fraction of H2 just above the surface (a) and the wall heat flux (6) as functions of the dimensionless time, 2at, for 10% H2-air mixture at a surface temperature of 1100 K. Self-sustained oscillations and stationary solutions are represented by solid and dashed curves, respectively. The pressure is 4 atm and the strain is a = 200 s ...

The transition probabilities W% C C) cannot be arbitrary but must guarantee that the equilibrium state P C) is a stationary solution of the master equation (5). The simplest way to impose such a condition is to model the microscopic dynamics as ergodic and reversible for a fixed value of X ... 

In equilibrium x (t) = x is constant in time. In this case, the stationary solution of the master equation is the equihbrium solution... 

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