Lagrange expansion

The Lagrange expansion technique can also be applied to the calculation of the particle-scattering factors Px (q) of branched or linear polymers of DP = x from the path-weight generating function of the polydisperse system. In Chap. C.I1I we have shown the equivalence... 

If each element of a row is zero, D is zero (obvious from Lagrange expansion). 

Next, to determine the parameter f in Eq. (1.134), one appeals to the physical situation which is characteristic for constancy of the total energy which modulates in the thermodynamic function of the macrostate (formulated as a Lagrange expansion) in this respect we consider that the system exchanges the thermal energy with the environment after which is again isolated, so that the total number of particles of the system do not ever change. Thus we have... 

When the MFA is used in absence of the external field (J,- = 0) the Lagrange multipliers //, are assumed to give the actual density, p, known by construction. In presence of the field the MFA gives a correction Spi to the density p,. By using the linear response theory we can establish a hnear functional relation between J, and 8pi. The fields Pi r) can be expressed in term of a new field 8pi r) defined according to Pi r) = pi + 8pi + 8pi r). Now, we may perform a functional expansion of in terms of 8pi f). If this expansion is limited to a quadratic form in 8pj r) we get the following result [32]... 

The diagonal elements in the sum involving the Hamilton operator are energies of the corresponding deteiminants. The overlap elements between different determinants are zero as they are built from orthogonal MOs (eq. (3.20)). The variational procedure corresponds to setting alt the derivatives of the Lagrange function (4.3) with respect to the at expansion coefficients equal to zero. 

If there is only one determinant in the expansion (czo = 1), the last equation shows that the Lagrange multiplier is the (Cl) energy, A = . 

This expression is used as a trial-function expansion for T in much the same way as the Lagrange interpolation polynomial is in the polynomial collocation method of Villadsen and Stewart (10,1 1 There are four unknown constants associated with each node, giving a total of 4(n+l)(nH-1) unknowns in the expansion. 

Acrivos and Amundson (SO, 51) applied matrix algebra to the unsteady-state behavior of stagewise operations in chemical processes. Instead of using a characteristic vector expansion, they emphasize the use of the Sylvester-Lagrange-Buchheim formula (52). Even though this formula is equivalent to the characteristic vector expansion, it is more difficult to manipulate and is not easily related to physical concepts such as straight line reaction paths. 

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