Lagrange multipliers enforcing

Unlike the case of an incompressible fluid where the Lagrange multiplier used to enforce incompressibility weakly can be associated with the pressure, there is no such correlation for the Lagrange multiplier enforcing charge neutrality. The resulting Lagrange multiplier at times appears to be associated with the electric potential part of the chemoelectric potential, and at other times has no clear correlation at all. 

Here f (r) is the Lagrange multiplier enforcing incompressibility a( ) + b( ) = 1- Equations [93]-[95] have been successfully couertions, which stem from the fart that the acmal value of solved by Matsen and co-workers using the spectral method. 

In eqn (5.17), PA(rA) and Oa are Lagrange multipliers enforcing constraints (5.12) and (5.15). All of this charge partitioning functional s parameters were theoretically derived, so its results can be regarded as non-empirical. 

K i) and i a( a)>0 and are Lagrange multipliers enforcing constraints (5.21) and (5.22) co is the unit sphere surface comprised of all possible endpoints for h, and integration over m means integration over all possible choices for h This spin partitioning functional has a unique minimum that is found by an iterative solution algorithm. ... 

Steps 1-3 are repeated until the maximum absolute error in the constraints falls below a target threshhold. Before the first iteration the Lagrange multipliers may be initialized to zero and the penalty parameter set to 0.1. The constraints are not fully enforced until convergence, and the energy in the primal program approaches the optimal value from below. 

Minimization of the functional (41) has to be performed under the orthonormality requirement in Eq. (4) for the NSOs, whereas the ONs conform to the N-representability conditions for D. Bounds on the ONs are enforced by setting rii = cos y, and varying y,- without constraints. The other two conditions may easily be taken into account by the method of Lagrange multipliers. 

These constraints can be enforced within the variational optimization of the energy function mentioned above by introducing a set of Lagrange multipliers jj, one for each constraint condition, and subsequently differentiating... 

Abstract In a thermodynamic framework which exploits the entropy inequality to obtain constitutive equations, it is common practice to assume charge neutrality and enforce this restriction using Lagrange multipliers. In this paper we show that the Lagrange multiplier used to enforce charge neutrality does not correspond to any known physical parameter, raising the question of whether charge neutrality can really be enforced. 

As a reference to something more familiar, consider the case of a fluid where incompressibility is enforced via a Lagrange multiplier. For a Stokesian fluid, it is assumed that the constitutive variables (stress, energy, heat flux) are a function of density, p, temperature, T, rate of deformation tensor, d, and possibly other variables (such as the gradients of density and temperature). Exploiting the entropy inequality in this framework produces the following constitutive restriction for the Cauchy stress tensor [10]... 

Next we present the results of exploiting the entropy inequality which involve the Lagrange multiplier A, and/or the charge of a constituent, zaA In each case, the first form is the result from enforcing charge neutrality ... 

An alternative interpretation of this equation is that y(t) is a time-dependent Lagrange multiplier, introduced to enforce constant normalization By the... 

This is transfer covariant if all quadratically integrable functions are represented in the same orbital basis. Requiring fps to be orthogonal to all radial factor riPa(r)) enforces a unique representation, but introduces Lagrange multipliers in the close-coupling equations. An alternative is to require... 

The effective hard core potential u (z) is a Lagrange multiplier that enforces the incompressibility condition... 

To maximize the current and thus the round-trip rate, this integral must be minimized. However, there is a constraint H E) is a probability distribution and must remain normalized which can be enforced with a Lagrange multiplier ... 

We have assumed here that the variations are performed within the domain of normalized densities. Alternatively, the minimization can be performed using the Euler-Lagrange procedure. Then the densities are allowed to vary also outside the normalization domain. This we shall do by relaxing the normalization constraint of the wavefunctions and by using the definition (3) of the density also in the extended domain. The normalization constraint is enforced by means of a Lagrange multiplier (pi),... 

Above we have assumed that the minimization is carried out within the domain of normalized of densities. Alternatively, we can perform the minimization, using the Euler-Lagrange procedure. Then we use the extension of the functionals valid also outside the normalization domain and enforce the normalization constraint by a Lagrange multiplier.5 For the Levy-Lieb energy functional (70) this leads to... 

The truncation of the series expansion (4) at second order allows the solution to be found by solving a linear system of equations. The only complication is the need to enforce the constraint (2), which can be taken care of with the method of Lagrange multipliers. In this context, the Lagrange multiplier can be interpreted as the chemical potential, and the solution to the constrained problem is the charge distribution and chemical potential which minimizes the free energy... 

We cannot solve (10.3.31) merely by forming the total differential wrt the mole numbers and setting that differential to zero, because the dN,- are not independent instead, they are related through (10.3.32). In the stoichiometric development in 10.3.2, the constraint (10.3.32) was included in the problem through stoichiometric coefficients and an extent of reaction t,. Here we impose the constraint in a different way namely, we allow the N, in the equilibrium condition (10.3.31) to vary independently and enforce the constraints (10.3.32) via Lagrange multipliers (see Appendix I). 

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