The Legendre Transform

In practical applications we rarely have occasion to deal with isolated systems, i.e., those having constant U and V, although in many discussions of entropy, the system plus its surroundings are tacitly assumed to be equivalent to isolated systems. We will develop this thought further in a later section. 

It turns out, though, to be quite simple to develop additional thermodynamic potentials that give directionality information for systems having other types of constraints, using the Legendre Transform introduced in Chapter 2. This section follows 

To start with, we note that in theory we can extract U from the function 

Incidentally, although the exact nature of the functional relationship between U, S, and V is not needed here, it has been discussed in 5.3 as the fundamental equation, equations (5.10) and (5.11). 

We rely on the discussion and diagrams of the preceding section to convince the reader that U is a thermodynamic potential exhibiting a minimum for systems of given S 

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The enthalpy is useful in considering isentropic and isobaric processes, but often it becomes necessary to rather deal with isothermal and isobaric processes. In such case one needs a thermodynamic function of T and P alone, defining the Gibbs potential G = U(T, P, Nj) as the Legendre transform of U that replaces entropy by temperature and volume by pressure. This transform is equivalent to a partial Legendre transform of the enthalpy,... 

The statistical partition functions are seen to be related by Laplace transformation in the same way that thermodynamic potentials are related by Legendre transformation. It is conjectured that the Laplace transformation of the statistical partition functions reduces asymptotically to the Legendre transformation of MP in the limit of infinitely large systems. 

In 1982, Nalewajski and Parr took the thermodynamic analogy to its logical conclusion by extending the Legendre-transform structure of classical thermodynamics to DFT [8]. One of their results was the Maxwell relation for Equation 18.6,... 

The use of the electron density and the number of electrons as a set of independent variables, in contrast to the canonical set, namely, the external potential and the number of electrons, is based on a series of papers by Nalewajski [21,22]. A.C. realized that this choice is problematic because one cannot change the number of electrons while the electron density remains constant. After several attempts, he found that the energy per particle possesses the convexity properties that are required by the Legendre transformations. When the Legendre transform was performed on the energy per particle, the shape function immediately appeared as the conjugate variable to the external potential, so that the electron density was split into two pieces that can be varied independent the number of electrons and their distribution in space. 

Equation 19.17 is not the original way the N, p(r) ambiguity was resolved [20]. As mentioned previously, the original paper on the shape function in the isomorphic representation performed the Legendre transform on the energy per particle. This gives an intensive, per electron, state function [20] ... 

Here (N, F) (see Equation 30.17) stands for the Legendre transform of the BO potential energy surface W(N, Q), in which the nuclear-position coordinates Q are replaced by the corresponding forces F in the list of the parameters of state. Indeed, for the fixed number of electrons N,... 

Ozawa, Y. (1970) Application of the Legendre transformation to one-dimensional packed bed model. Chem. Engng. Sci. 25, 529-533. 

Unfortunately, this is mainly formal because neither the integral defining ip nor the Legendre transform are likely to be tractable. However, we show in Section II.C that Eq. (26) is equivalent to the more conventional form of the entropy of mixing as given by the second term in Eq. (5). From a conceptual point of view, it should be noted that the conventional form is normally derived by binning the distribution of particle sizes a and taking the number... 

The term (h — Xm), being multiplied by —N T, can now be interpreted as the entropy of mixing per particle in the small phase. It arises from the deviation of the (generalized) mean size m = m in the small phase from the mean size m of the parent, and it is given by the Legendre transform of the generalized cumulant generating function ... 

Although Eq. (33) looks somewhat different from the corresponding result (9) for/pr obtained by the projection method, the two methods are, mathematically, almost equivalent, as we now show. (For brevity we return to the case of a single moment m and continue to refer to the polydisperse feature o as size. ) First, consider the exact expression for the entropy of mixing (26) derived within the combinatorial method. This should be equivalent to the conventional result used as the starting point (5) in the projection method. To see this, write the Legendre transform conditions (24) out explicitly ... 

In this appendix, we discuss some interesting properties of the Legendre transform result (29) for the moment entropy of mixing, in particular its relation to large deviation theory (LDT). 

The temperature can be introduced as an independent variable by defining the Gibbs energy G with the Legendre transform... 

In this section we will consider the Legendre transforms that define the enthalpy H, Helmholtz energy A, and Gibbs energy G. 

Substituting the integrated fundamental equation for U (equation 2.2-14) in the Legendre transforms defining H, A, and G shows that... 

In this equation NH(j) is the number of hydrogen atoms in species j, and Ns is the number of different species in the system. The index number for species is represented by j so that the index number introduced later for reactants (sums of species) can be i. Substituting equation 4.1-2 and G = n-/r (equation 2.5-12) into the Legendre transform (equation 4.1-1) yields... 

Note that the Legendre transform has interchanged the roles of the conjugate intensive /r(H + ) and extensive nc(H) variables in the last term of equation 4.1-9. The number D of natural variables of G is Ns + 2, just as it was for G, but the chemical potential of the hydrogen ion is now a natural variable instead of the amount of the hydrogen component (equation 4.1-7). 

The treatment (Alberty, 1998a) of the binding of Mg2 +, or other metal ion that is bound reversibly by a reactant, follows the same pattern as the treatment of H +. A term c(Mg)/u(Mg) can be included in the Legendre transform with the term for hydrogen as follows ... 

When the chemical potentials of several species are held constant, it may be useful to write the Legendre transform in terms of matrices. For example, when the chemical potentials of several species are specified, the Legendre transform can be written as... 

The Legendre transform that defines the further transformed Gibbs energy G", which provides the criterion for spontaneous change and equilibrium in dilute... 

The equilibrium relations of the preceding section were derived on the assumption that the charge transferred Q can be held constant, but that is not really practical from an experimental point of view. It is better to consider the potential difference between the phases to be a natural variable. That is accomplished by use of the Legendre transform (Alberty, 1995c Alberty, Barthel, Cohen, Ewing, Goldberg, and Wilhelm, 2001)... 

Equation 8.5-3 indicates that the number of natural variables for the system is 6, D = 6. Thus the number D of natural variables is the same for G and G, as expected, since the Legendre transform interchanges conjugate variables. The criterion for equilibrium is dG 0 at constant T,P,ncAoi, ncA(3, /icC, and The Gibbs-Duhem equations are the same as equations 8.4-8 and 8.4-9, and so the number of independent intensive variables is not changed. Equation 8.5-3 yields the same membrane equations (8.4-13 and 8.4-14) derived in the preceding section. 

Thus, we can switch to n(r) as independent function, and introduce the Legendre transformation... 

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