Quasi-stationary approximation

Following Batchelor, we set up the equation in the quasi-stationary approximation, neglecting the displacement current and the density of free charges. We employ c = 1 and the Heaviside system (without 47r), ip = scalar potential, A = vector potential, div A = 0, J = current, div J = 0 the specific resistance of the fluid is r. 

C being the self-capacitance of the junction, U (current bias term included —U() = (ft/2e)(/c cos) for a single junction. Further we concentrate on overdamped junctions where C -C G2H/(2eIc) and neglect the capacitance term. The normal conductor part we write following [11] in quasi-stationary approximation... 

This quasi-stationary approximation is only valid if the typical time r of the motion along the saddle-point trajectory is long in comparison with h/eV, that is, eVr h. To check the validity of this, we precede the results with simple qualitative estimations. 

The resulting rate can be estimated as logT 4>q(G/Gq)x If o < 1, this reduces to log T 4>o(G/Go)Ith/If- In the opposite limit, the estimation for the rate reads log r 4>o(G/Gq ) l< (It J h), F being a dimensionless function 1. It is important to note that these expressions match the quantum tunneling rate log Jr Uqt/K (G/Gq)< provided eVr h. Therefore the quasi-stationary approximation is valid when the quantum tunneling rate is negligible and the third factor mentioned in the introduction is not relevant. For equilibrium systems, the situation corresponds to the well-known crossover between thermally activated and quantum processes at k Tr h [9]. 

According to Eqs. (259) and (260), the effective characteristics of a medium in the quasi-stationary approximation differ from the static case only in the replacement of the conductivity a (dc conductivity) by the complex conductivity a ... 

We assume further that the concentrations of vapor and impurity molecules are small in comparison with the concentration of a neutral buffer gas and use the diffusion equation to describe the mass transfer to the particles. The deposition problem is considered within the quasi-stationary approximation. For simplicity we assume the gas-particle system to be isothermal. The conditions on the particle surface and at the infinity distance from the particle can be written as... 

Bruijn (B24) employs the quasi-stationary approximation, as discussed in Bankoif (B8), to the growth of vapor bubbles in superheated binary liquid mixtures. As noted previously, this neglect of the convective term in the diffusion equation is justified only when Aj< l, which is usually not the case in atmospheric boiling. On the other hand, this technique would be applicable to isothermal gas bubble growth in three-component systems, where two of the components are dissolved gases. 

If drop sizes are small enough, and the difference in densities of the internal and external liquid is sufficiently small, the flow of the liquid can be considered as slow, and the motion of drops - as inertialess. This is the case when we separate emulsions of the w/o (water-oil) type. Then equations (11.32) are reduced to the equations of inertialess motion of drops in the quasi-stationary approximation... 

The dynamics of mass-exchange between a mono-dispersive ensemble of hydrate inhibitor drops and a hydrocarbon gas has been considered in Section 21.1 within the framework of the above-made assumptions in a quasi-stationary approximation. In dimensionless variables ... 

Let us restrict ourselves to the case of fast coagulation, assuming that each collision of bubbles results in their coalescence. The study of mutual approach of bubbles in a laminar flow is based on the analysis of trajectories of their relative motion. The equations of non-inertial motion of a bubble of radius a relative to a bubble of radius h in the quasi-stationary approximation are ... 

At quasi-stationary approximation in relation to [R...H...03 and taking into consideration the material balance, we obtain for k... 

At quasi-stationary approximation [14, 28, 104] the rates of products formation will be presented as Eqs. (44)-(46) ... 

Within the model we have assumed that the boundary curvature can be neglected. To check this assumption, let us analyze the ratio of the following fluxes jcurv (caused by the curvature gradient) and jchem (caused by the concentration gradient) in a quasi-stationary approximation. In the general case, we have cD 9 (a Sip) cD ao, dp... 

When the characteristic time of the pressure change is larger than the hydrodynamic relaxation time (or the oscillation frequency is small / (t, ) ) the quasi-stationary approximation holds... 

With the quasi-stationary approximation, the net rate v of the overall relation is equal to the difference between forward and reverse rates of each step with the suitable weighing of the stoichiometric number of that step ... 

The process of the magnetic field influence on a developed turbulence was examined by [8],and demonstrated the possibility of using the quasi-stationary approximation for the solution of the second type problem and suggested to use quasi-linear approximations to solve the problem at Rcm = 20. One of the second type problem results were reported in [9], the modeling of a diminishing MHD turbulence by LES and DNS methods and demonstrated that the magnetic field at the initial time started to decay under the influence of the total kinetic energy. This effect is consistent with Joule dissipation. A similar picture of the decay was not reported by the authors because their main objective was the evaluation... 

We want to note that exactly this form of the equation was proposed in classical Bodenstein and Christiansen works. In addition, the same equation can be derived by analysis of a multi-step kinetic scheme of the chain mechanism for thermal phosgene synthesis in a quasi-stationary approximation (see Chap. 2). 

To a very good approximation, the quasi-stationary approximation for the difiiision profiles can be applied. From the mass balance at the precipitate / matrix inter ce the classical Zener equation for the grow rate is derived. For vanadium, die equation is... 

The time-scale separation can be invoked once more to make a quasi-stationary approximation consisting in the assumption that P y x t) instantaneously evolves to the stationary distribution, P (j x t), given the current value of x. Therefore, from all the above considerations, the master equation that governs the dynamics of... 

Under the assumptions that nx and y are constant, and that + hex — nj, with nr constant, the system state is fully determined by the ( A . a) values. Let P nEx,nA, t) be the probability of having hex molecules Ex and ha molecules A at time t. To study the system stochastic dynamics one could write the master equation for P riEx,nA, t) and analyze it. However, the analysis can be simplified if we previously make a quasi-stationary approximation. Is it usually acknowledged that the reaction in (5.7) is much faster than those in (5.8) and (5.9). Since the reaction in (5.7) modifies the value of a but not of ha, we can directly apply the formalism developed in the Sect. 5.1. 

Notice that Eq. (5.17) is the same as Eq. (4.10). Hence, the quasi-stationary approximation, allowed us to reduce the system of chemical reactions in (5.1) to a birth-death process in which the effective production and degradation rates are ... 

The quasi-stationary approximation studied in the previous section allowed us to break the system of chemical reactions in (5.7)-(5.9) into a couple of subsystems that can be analyzed separately ... 

Under the assumption that the binding and unbinding of repressor and polymerase from the promoter are much faster processes than the synthesis and degradation of RNA, one can make a quasi-stationary approximation similar to the one we made in the previous section. According to this approximation we can split the system into fast and slow subsystems. The fast subsystem comprises the binding and unbinding of repressor and polymerase molecules to the promoter, while the slow subsystem accounts for the synthesis and degradation of RNA. Moreover, from the slow subsystem perspective, the fast subsystem reaches stationarity instantaneously. 

Being promoter binding and unbinding by a polymerase the fastest processes of gene expression, we can make a quasi-stationary approximation similar the ones we have done in the previous sections. As a result we get a reduced system schematically represented in Fig. 8.1. The chemical reactions governing the dynamics of this system are then as follows ... 

As a result of the above, we have to deal with a 3-dimensional system, rather than with a 4-dimensional one. Nonetheless, this system is still too complex to be analytically studied. One way to simpUly the system is to suppose that the unbound-to-bound transitions are much faster than those between the open and closed states, and use the quasi-stationary approximation introduced in Chap. 5. Let us define the probabilities that the channel is open and closed (regardless of their being bound or unbound by the regulatory molecule) as follows ... 

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