Stationarity condition

To obtain the stationarity conditions, the Lagrangian is formed and differentiated, with the details of this procedure described by McMillan (1970) and by Beveridge and Schecter (1970). The development begins by converting the inequality constraints to equality constraints through the addition of slack variables, zj,i= 1. . ..1, such that constraints (18.4) become 

Note that V L = 0 gives Eqs. (18.21), which are the definitions of the slack variables and need not be expressed in the KKT conditions. Note also that L = 2A.jZ, = 0, and, using Eqs. (18.21), Eqs. (18.26) result. These are the so-called complementary slackness equations. For constraint i, either the residual of the constraint is zero, g, = 0, or the Kuhn-Tucker multiplier is zero, X., = 0, or both are zero that is, when the constraint is inactive (gj 0), the Kuhn-Tucker multiplier is zero, and when the Kuhn-Tucker multiplier is greater than zero, the constraint must be active (g, = 0). Stated differently, there is slackness in either the constraint or the Kuhn-Tucker multiplier. Finally, it is noted that V c x is the Jacobian matrix of the equality constraints, J x, and V g i is the Jacobian matrix of the inequality constraints, K[x). 

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V L is equal to the constrained derivatives for the problem, which should be zero at the solution to the problem. Also, these stationarity conditions very neatly provide the necessaiy conditions for optimality of an equality-constrained problem. 

The present work aims to derive fully microscopic expressions for the nucleation rate J and to apply the results to realistic estimates of nucleation rates in alloys. We suppose that the state with a critical embryo obeys the local stationarity conditions (9) dFjdci — p, but is unstable, i.e. corresponds to the saddle point cf of the function ft c, = F c, — lN in the ci-space. At small 8a = c — cf we have... 

Also, all factors should satisfy a stationarity condition... 

Balance of Forces It is convenient to define the L function L(x,Xy) =f(x) + g(xfX + h(xfv, along with "weights or multipliers X and v for the constraints. The stationarity condition (balance of forces acting on the ball) is then given by... 

Thus, for the case of a turbulent boundary layer, although the stationarity condition is fairly well satisfied, the homogeneity condition is probably satisfied only for z/8 0.2. 

In terms of these conditions, a fc-particle hierarchy of approximations can be defined, with Hartree-Fock as the one-particle approximation for closed-shell states. Unfortunately, the stationarity conditions do not determine the fully, and for their constmction additional information is required, which essentially guarantees -representability. Nevertheless, the fe-particle hierarchy based on the irreducible stationarity conditions opens a promising way for the solution of the -electron problem. 

Before we come to the irreducible stationarity conditions for the energy—our main concern—let us have a short look at the traditional stationarity conditions. 

While the CSE expresses in terms of y j and y, +2> he BC,t only needs The BC,t are obviously simpler than the CSE. On the other hand, the BCj can be derived order by order from the CSE, while the converse is not true. Only if one goes up to fe = n, do the two sets of stationarity conditions become equivalent. 

We have already discussed the relations between the four stationarity conditions. In view of their separability, the two irreducible conditions are the right choice in the spirit of a many-body theory in terms of connected diagrams. 

Any of the four conditions has an infinity of solutions. Actually, the energy is stationary for any eigenstate of the Hamiltonian, so one has to specify in which state one is interested. This will usually be done at the iteration start. Moreover, the stationarity conditions do not discriminate between pure states and ensemble states. The stationarity conditions are even independent of the particle statistics. One must hence explicitly take care that one describes an n-fermion state. The hope that by means of the CSE or one of the other sets of conditions the n-representability problem is automatically circumvented has, unfortunately, been premature. 

As we shall see, the stationarity conditions determine essentially the non-diagonal elements of y and the k, while the diagonal elements are determined by the specification of the considered state and the n-representability. 

The stationarity condition (170) does not give any information on the diagonal elements yP, not even on the yP with Cp = Zq. We must get this information from another source. Fortunately, for an n-electron state, vanishing of >.2 is only compatible with Lt = 0 for > 2 and with idempotency of y. 

If the stationarity conditions (211)-(213) are satisfied, the energy expressions are simplified to... 

There are also some unexpected problems, related to the fact that the stationarity conditions do not discriminate between ground and excited states, between pure states and ensemble states, and not even between fermions and bosons. The IBQ give only information about the nondiagonal elements of y and the Xk, whereas for the diagonal elements other sources of information must be used. These elements are essentially determined by the requirement of w-representability. This can be imposed exactly to the leading order of perturbation theory. Some information on the diagonal elements is obtained from the lCSE,t, though in a very expensive and hence not recommended way. The best way to take care of -representability is probably via a unitary Fock-space transformation of the reference function, because this transformation preserves the -representability. 

Now assume that some constraint c(x) = 0 is added to the problem. It can be shown that the stationarity condition now takes the form... 

However as mentioned in a paper by Christiansen and Kramers (13) and shown experimentally, notably by Hinshelwood (14) and Semenoff (15), cases are known in which the number of intermediates is increased by one revolution of the closed sequence. In that case it is impossible to fulfil the stationarity conditions, which means that the reaction goes on with ever increasing speed, i.e., we get an explosion. Only by adding inhibitors which remove one or more of the intermediates can the reaction be turned into an ordinary smooth reaction. 

It is a gross error to apply stationary state theory to non-stationary polymerizations, of which many exist. In a great majority of cases, the stationarity conditions are only fulfilled after a certain non-stationary phase in which the concentration of active centres increases. When this occurs in a kinetically pure medium, i.e. in the absence of inhibitors or other intervening compounds, it usually signifies slow initiation and is called the pre-effect. The general shape and meaning of the pre-effect is represented graphically in Fig. 5. For very small conversions ([M] [M]0) we can write [57]... 

Radical polymerizations are almost always considered as kinetically stationary. However, the stationarity conditions are not always fulfilled. Living polymerizations with rapid initiation are stationary, but the character of the medium should not significantly change during polymerization in order to prevent shifts in the equilibria between ion pairs and free ions. All other polymerizations are non-stationary even, to some extent, living polymerizations with slow initiation. It is usually very difficult to define initiation and termination rates so as to permit exact kinetic analysis. When the concentration of active centres cannot be directly determined, indirect methods must be applied, and sometimes even just a trial search for best agreement with experiment. 

The stationarity conditions for the other internal variable fluxes are deduced in the same way. For example, the condition Jj. = 0 is identical to the condition... 

A similar-looking condition. R, — 0, holds for any immobile species in a true steady-state operation. For example, the surface species on a fixed bed of catalyst are immobile in those reactor models that neglect surface diffusion, and a mass balance then requires that R, vanish at steady state. When desired, one can include both R = 0 and R w 0 as possibilities by invoking a stationarity condition as in Example 2.4. 

This example illustrates the use of quasi-equilibria and stationarity conditions to reduce a reaction scheme to a shorter form with fewer independent parameters. The original ten parameters of the five-reaction scheme are combined into four independent ones (Tca, 77, 77i772, and 77 477 5) in the final rate expression. 

The stationarity condition of g with respect to 6 says that ... 

Taking the first y-derivatives of the As, the stationarity condition for the energy is ... 

But instead of solving the n-coupled non-linear equations given in (12) by minimizing as it merely represents satisfaction of the variance stationarity condition as opposed to the absolute minimization of V, direct minimization of V by the SAM appears to be a perfectly feasible proposition [28]. Having thus made the advocated methodologies transparent, we now turn to some simple minded applications of the techniques to the... 

We have hardly observed numerical instabilities on the MP2-R12/A level, but rather often on the higher level of approximations. These numerical instabilities can be avoided if one no longer cares to determine the full matrix c h from the stationarity condition (but rather only its eigenvalues) for eigenvectors (extremal pairs) obtained from other information. 

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