Other Variational Principles

The principle of virtual work is suitable for solving a wide range of problems. There are tasks however where different but related formulations might be more useful. Thus, two prominent variational principles will be extended here to take into account materials with electromechanical couplings. This novel approach to Dirichlet s principle of minimum potential energy will be employed later in Section 6.3.2. In comparison to the principle of virtual work, the extended general Hamilton s principle is considered to be equivalent and even more versatile, but only its derivation will be demonstrated here. 

1 Extended Dirichlet s Principle of Minimnm Potential Energy 

Another important principle of mechanics, see Dym and Shames [70] or Sokol-nikoff [166] for details, will be extended here to electromechanically coupled problems. Let there be a function Uq to establish the following relation between the fields of mechanical stress cr and electric flux density D on the one hand, and the fields of mechanical strain e and, for reasons to be clarified in Section 4.4.4, negative electric field strength E on the other  

In the virtual work of internal contributions given by Eq. (3.63), the field quantities appearing in virtual and actual form are in no way connected. With the substitution of Eq. (3.64) into Eq. (3.63), this independence from 

In the static portions of the virtual work of external contributions, the forces and charges acting on the constant volume and surface of the structure are not altered by the arbitrary variations bu of displacements and of electric potential respectively. Thus, the left-hand sides of Eqs. (3.45) and (3.53) may be written in the following form  

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On a related point, there have been other variational principles enunciated as a basis for nonequilibrium thermodynamics. Hashitsume [47], Gyarmati [48, 49], and Bochkov and Kuzovlev [50] all assert that in the steady state the rate of first entropy production is an extremum, and all invoke a function identical to that underlying the Onsager-Machlup functional [32]. As mentioned earlier, Prigogine [11] (and workers in the broader sciences) [13-18] variously asserts that the rate of first entropy production is a maximum or a minimum and invokes the same two functions for the optimum rate of first entropy production that were used by Onsager and Machlup [32] (see Section HE). 

In the quantum mechanics of atoms and molecules, both perturbation theory and the variational principle are widely used. For some problems, one of the two classes of approach is clearly best suited to the task, and is thus an established choice. Flowever, in many others, the situation is less clear cut, and calculations can be done with either of the methods or a combination of both. 

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

Stated in still other words this means that for any trial density p(r) - which satisfies the necessary boundary conditions such as p( ) - 0, J p( ) dr = N, and which is associated with some external potential Vext - the energy obtained from the functional given in equation (4-6) represents an upper bound to the true ground state energy E0. E0 results if and only if the exact ground state density is inserted into equation (4-8). The proof of the inequality (4-11) is simple since it makes use of the variational principle established for wave functions as detailed in Chapter 1. We recall that any trial density p(r) defines its own Hamiltonian H and hence its own wave function. This wave function can now be taken as the trial wave function for the Hamiltonian generated from the true external potential Vext. Thus, we arrive at... 

The utility of the Fukui function for predicting chemical reactivity can also be described using the variational principle for the Fukui function [61,62], The Fukui function from the above discussion, /v (r), represents the best way to add an infinitesimal fraction of an electron to a system in the sense that the electron density pv/v(r) I has lower energy than any other N I -electron density... 

The problem of time evolution for a Hamiltonian bilinear in the generators (Levine, 1982) has been extensively discussed. The proposed solutions include the use of variational principles (Tishby and Levine, 1984), mean-field self-consistent methods (Meyer, Kucar, and Cederbaum, 1988), time-dependent constants of the motion (Levine, 1982), and numerous others, which we hope to discuss in detail in a sequel to this volume. 

The second law of thermodynamics states that an isolated system in equilibrium has maximum entropy. This is the basis for a variational principle often used in determining the equilibrium state of a system. When the system contains several elements which are allowed to exchange mass with each other, the variational principle yields the condition that all elements must have equal chemical potential once equilibrium is established. 

When more satisfactory forms of diffusion coefficient for the hydro-dynamic repulsion effect become available, these should be incorporated into the diffusion equation analysis. The effect of competitive reaction processes on the overall rate of reaction only becomes important when the concentration of both reactants is so large that it would require exceptional means to generate such concentrations of reactants and a solvent of extremely low diffusion coefficient to observe such effects. This effect has been the subject of much rather repetitive effort recently (see Chap. 9, Sect. 5.5). By contrast, the recent numerical studies of reactions between uncharged species is a most welcome study of the effect of this competition in various small clusters of reactants (see Chap. 7, Sect. 4.4). It is to be hoped that this work can be extended to reactions between ions in order to model spur decay processes in solvents less polar than water. One other area where research on the diffusion equation analysis of reaction rates would be very welcome is in the application of the variational principle (see Chap. 10). 

Strictly, when the reaction of interest involves 3 or more species, the motion of all species cannot be accurately predicted, this if often unimportant, as in the case of reaction of one (the minority) species, A, with another (the majority) species, B. under such conditions, the B species can be regarded as a uniform distribution and if there are two more species, these are statistically independent. These points have already been discussed in Chap. 8, Sect. 2.3. When the densities of A and B species become similar, these approximations are no longer acceptable. Much of this chapter (Sects. 4—6) is devoted to a discussion of these effects, which may be important in the radiation chemical spur and other situations where the concentration of reactants is large. Further comments are also made in Chap. 10, Secs. 3—5 where the variational principle is used. 

Eucken discovered that the molecular heat of hydrogen falls at low temperatures from 5 to 3. This and other variations in specific heats with temperature can only be interpreted in terms of quantum dynamics, and the subjection of mechanical processes taking place among gas molecules to quantum principles must be taken into consideration in theories of chemical reaction mechanisms. 

Numerical models of conserved order-parameter evolution and of nonconserved order-parameter evolution produce simulations that capture many aspects of observed microstructural evolution. These equations, as derived from variational principles, constitute the phase-field method [9]. The phase-field method depends on models for the homogeneous free-energy density for one or more order parameters, kinetic assumptions for each order-parameter field (i.e., conserved order parameters leading to a Cahn-Hilliard kinetic equation), model parameters for the gradient-energy coefficients, subsidiary equations for any other fields such as heat flow, and trustworthy numerical implementation. 

Of all local motions, v(r), of an interface that pass the same amount of volume from one side to the other, the motion that is normal to the interface with magnitude proportional to the weighted mean curvature, v f) oc /c7n, increases the interfacial energy the fastest. However, fastest depends on how distance is measured. How this distance metric alters the variational principles that generate the kinetic equations is discussed elsewhere [14]. 

On the other hand, employing the Kohn variational principle [67] two lower bound estimates may be obtained [59, 60]... 

Because the variation principle is involved, certain matrix elements disappear as in ordinary SCF theory, and such relations have recently been referred to as generalized Brillouin theorems.38 A major review of the MCSCF method has been given by Wahl 39 the practical limit on the number of configurations seems to be around 50 at present, but energy results compare extremely favourably with those of traditional Cl calculations invoking many more configurations, and other calculated properties are encouraging. 

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