Quasi-stationary solution

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. 

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),... 

The system of equations with initial and boundary conditions formulated above allows us to find the velocity distributions and pressure drop for the filled part of the mold. In order to incorporate effects related to the movement of the stream front and the fountain effect, it is possible to use the velocity distribution obtained285 for isothermal flow of a Newtonian liquid in a semi-infinite plane channel, when the flow is initiated by a piston moving along the channel with velocity uo (it is evident that uo equals the average velocity of the liquid in the channel). An approximate quasi-stationary solution can be found. Introduction of the function v /, transforms the momentum balance equation into a biharmonic equation. Then, after some approximations, the following solution for the function jt was obtained 285... 

In order to analyze the degree of accuracy of Eqs. (3.73) and (3.74), the current-potential curves calculated with rigorous (3.66) (solid lines), approximate quasi-stationary (3.73) (dotted lines), and stationary (3.74) (dashed lines) equations have been plotted in Fig. 3.9 for different values of the electrode radius and two values of k°. From this figure, it can be observed that a decrease of the electrode size facilitates the fulfillment of the Eq. (3.73) for a given value of k° such that the approximate quasi-stationary solution can be used instead of the rigorous one with an error smaller than 5 % for rs < 50 pm if k° = 10-3 cm s-1 and t = 1 s. Equation (3.74) is valid for any value of k° if rs < 3 pm. 

Since the lifetime of singlets is very short, it has a quasi-stationary solution... 

Modeling EM solitary waves in a plasma is quite a challenging problem due to the intrinsic nonlinearity of these objects. Most of the theories have been developed for one-dimensional quasi-stationary EM energy distributions, which represent the asymptotic equilibrium states that are achieved by the radiation-plasma system after long interaction times. The analytical modeling of the phase of formation of an EM soliton, which we qualitatively described in the previous section, is still an open problem. What are usually called solitons are asymptotic quasi-stationary solutions of the Maxwell equations that is, the amplitude of the associated vector potential is either an harmonic function of time (for example, for linear polarization) or it is a constant (circular polarization). Let s briefly review the theory of one-dimensional RES. 

Fig.10 Theoretical file hickness curves for 750 250 N load and 5 s shaft speed, g =6.2 10, g =1.58 10, (a) quasi-stationary solution, (b) dynamic solution, (c) defiati...

In this section pure birth-death processes will be discussed. Although such models have been dealt with extensively in the literature (see, for instance [4.1-8]), there still exist some seldom considered problems referring to the relation between the exact stationary or quasi-stationary solution of the master equation and the deterministic approach. This relation will be treated in Sect. 4.3.1 including an appUcation of the results obtained to observations on animal populations [4.15,17]. Further, the generalization of the stochastic standard model by including multistep birth or death processes will be investigated in Sect. 4.3.2. 

This solution describes a population which has completely died out. The solution is stable because from nothing comes nothing . Besides this trivial solution, however, there exists a quasi-stationary solution for which the eventual extinction occurs extremely slowly. 

In the conventional approximation the quasi-stationary solution is determined by neglecting (n - l)/n in (4.66) and by inserting the approximate factors... 

This conventional quasi-stationary distribution (4.68) or (4.70), however, has completely the wrong form for small n due to neglecting the factors (n - l)/n. On the other hand, it will be seen that the probability of complete extinction in the quasi-stationary solution depends decisively on the exact form for small values of n. 

The quasi-stationary solution to (4.63) is therefore determined in a better approximation than through (4.67) by using... 

The equations (4.84, 85) and their solutions are based on the approximation (4.83) which is valid for a uni-modal probability distribution only. Contrary to this assumption, however, any initially uni-modal distribution can be expected to develop first into the doubled-peaked quasi-stationary solution Pqs (n) shown in Fig. 4.10 and then to go over finally to the exact stationary solution, that of an extinct population. Consequently deviations of the true time paths of (n), and o t) from those described by (4.84, 85) are to be expected. A calculation of the development of a model population with time with the exact master equation (4.63) and with the parameters A = 0.5, n = 0.2, and bi = 0.01 confirms this expectation (Figs. 4.11, 12). The Fig. 4.11 shows the exactly calculated change with time of a distribution which starts as normal distribution but soon develops into the form of the bimodal quasi-stationary distribution Pqs(n). In Fig. 4.12 and for the same model parameters the exact paths of the mean value (n)(and the variance a t) are compared with the paths obtained by solving the approximate equations (4.84, 85). 

For r = 0 the quasi-stationary solution Pqs (n) is assumed to have been established according to (4.72) with ro(0) = 0 and Jii (0) = 1. Because the probability transfer from the living to the extinct state occurs slowly it is further assumed that the form of the quasi-stationary distribution is conserved during this process, while the weight of each living or non-extinct state p(n t) diminishes in proportion to This implies the assumption that ... 

The quasi-stationary solution to (4.100) should first be determined corresponding to the solution (4.72) for the single-step case. It is easily verified, however, that the condition of detailed balance (4.40) is not fulfilled in the general multi-step case and so, in fact, the quasi-stationary solution to the present problem cannot be derived in the form (4.39). 

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