Pfaffian form

Normally the apparatus of equilibrium thermodynamics can be used for the remoteness in the second and third sense and a corresponding choice of space of variables, though in each specific case this calls for additional check. Because for the spaces that do not contain the functions of state (in the descriptions of nonequilibrium systems these are the spaces of work-time or heat-time) the notion of differential loses its sense, and transition to the spaces with differentiable variables requires that the holonomy of the corresponding Pfaffian forms be proved. The principal difficulties in application of the equilibrium models arise in the case of remoteness from equilibrium in the first sense when the need appears to introduce additional variables and increase dimensionality of the problem solved. 

Equation (1.107) is more useful if it is integrated with the Pfaffian form however, this is not a straightforward step, since intensive properties are functions of all the independent variables of the system. The Euler relation for... 

Equation (1.4.24) is representative of the so-called Pfaffian forms to be introduced later. Note that Eqs. 

However, in general, W is not a function of state thus, one cannot ordinarily obtain a unique differential dW of W. In recognition of this fact we shall henceforth write 3W as the element of work. This symbol represents a shorthand notation of the quantity - (fxdx + fydy + fzdz) appearing on the right-hand side of (1.6.2) it represents another example of the so-called Pfaffian form introduced in Section 1.4(h). 

If every neighborhood of an arbitrary point xQ in a hyperspace contains points x not accessible from x0 via solution curves of the equation X(i) i xi then the Pfaffian form 3L — J jX Xi is holonomic. 

Before providing a proof of Carathdodory s Theorem we digress to specify necessary and sufficient conditions for establishing whether the Pfaffian form cIL — X(i) idxi is holonomic or not. 

We conclude that the Pfaffian form (1.11.1) is always integrable since (1.11.5) specifies q as well as R. However, if one were to examine a sum of three terms as XjdXi + X2dx2 + X3dx3 - cIL one would have to adjoin to (1.11.5) the additional relation X3 - q(3R/3x3). It is then no longer clear whether a function q(x1,x2,x3) and a second function R(x1,x2,x3) can be found that satisfy all three X - q(3R/3xi) equations. In fact, we shall now address ourselves precisely to this question. 

We see, then, that if an integrating factor 1/q exists that converts the Pfaffian form HL, into an exact differential, Eq. (1.11.7), then the coefficients X in the relation 3L -X]dx2 + X2dx2 + X3dx3 must obey (1.11.12). The latter relation may readily be generalized to the more general case n > 3, by replacing the subscripts 1, 2, 3 in Eq. (1.11.12) with i, j, k, respectively. 

In the Pfaffian form (1.11.13) let us first hold Xi.-.-.x, constant Eq. (1.11.13) then reduces to the form Xu.jdx,- + XjjdXjt this is certainly integrable as was demonstrated in part (a). Thus, one must be able to find a function H with a total differential of the form (1.11.7), as well as an integrating denominator , such that... 

The above immediately leads to the application of Caratheodory s theorem to the equation HQ - 0, which holds for adiabatic systems. Since the heat flow is related to changes in the thermodynamic coordinates of the system through the Pfaffian form HQ - this means there are states that... 

It is of interest to note that the First Law may be set up on a more mathematical basis by adapting the so-called Caratheodory theorem to the differential Pfaffian form for work dW = Yidyi, Eq. (1.5.11a). As explicitly developed in Chapter 9, when adiabatic conditions are imposed the theorem, under the constraint dW = 0, necessitates the existence of an associated function of state that is constant under these conditions. 

If a function L does not admit of an exact differential of the form (9.1.2) it may nevertheless be possible, under conditions established below, to set up functions q x, ..., Xi,..., Xn) such that the ratio dL/q = dR does constitute an exact differential. Pfaffian forms of this genre are of special interest they are said to be holonomic or integrable. For obvious reasons q is said to be an integrating denominator and 1/, an integrating factor. 

We digress here to specify necessary and sufficient conditions to establish whether or not the Pfaffian form dL = Yli Xi dxj is holonomic. 

We conclude that the Pfaffian form (9.2.1) is always integrable since Eq. (9.2.5) represents a set of simultaneous relations that may be solved for q and R. However, this scheme fails if a function of three or more variables, such as dL = X dx + X2dx2 + X idx i were to be examined. One would then have to adjoin to (9.2.5) the additional relation X3 = q(dR/dx2)dx. It is no longer clear whether this is possible in fact, we now address precisely this question. 

Next, we seek the inverse given a Pfaffian form... 

According to (9.3.17) we then require either that all f be zero or that all Fijn vanish. The first alternative cannot be correct since all the functions except fk may be chosen arbitrarily and fk is absent from the summation over i. This leaves only the alternative that all Fijk = 0. From the earlier discussion involving Eq. (9.2.20) it follows that the Pfaffian dL = dxi is integrable. We have thereby established the necessary condition for the Caratheodory theorem of Section 9.2 to hold. Given the fact that in the neighborhood of a point in phase space other points are inaccessible via solution curves of the form X, dxi — 0, the Pfaffian form is integrable. 

If this is the differential of a function, then M and N must be the appropriate derivatives of that function. Pfaffian forms exist in which M and N are not the appropriate partial derivatives of the same function. In this case du is called an inexact differential. It is an infinitesimal quantity that can be calculated from specified values of dx and dy, but it is not equal to the change in any function of x and y resulting from these changes. 

A Pfaffian form is also known as a total differential equation [10, pp. 326-330], This type of differential equation plays an important role in thermodynamics. Consider the vector function f(x) of the vector argument x. The scalar product f(x) dx is... 

For n = 3, the necessary and sufficient condition that the Pfaffian form Eq. (1.29) is integrable is... 

Spatial geminals for the triplet pairs, can also be expressed as linear combinations of basis functions, but their coefficient matrices must be antisymmetric. Clearly, the Pfaffian form reduces to the AGP wave function when = 0. 

The expression (3.3) is known as the Gibbs Jundamental form of the system (it represents a special Pfaffian form, as it is called in differential calculus). Every process that can be performed by the system must satisfy this differential form. On the left-hand side is the total differential of energy (because energy is a state function ) the energy forms located on the right-hand side of the equation generally do not constitute total differentials, although they can often be summed up into total differentials (see later). 

The above, relatively simplistic, formulation may be supplemented by a more formal and sophisticated approach that comes under the heading of Carathe odory s theorem. This provides a solid mathematical grounding for providing conditions under which the Pfaffian form drQ = may be converted... 

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