Pfaffian form

In mathematics, Pfaffian form is an expression which takes the form: [1]

 \sum_{k=1}^n X_i(x_1, \dots x_n) dx_i \,

(add)

History

The so-called Pfaffian form, supposedly, was derived in circa 1805 by German mathematician Johann Pfaff.

Truncated Pfaffian
The expression:

 dW = YdZ \,

where W is an amount of work done by the system, Y is an intensive property of the system, and Z is an extensive property of the system, is called a "truncated Pfaffian". [3]

Note
The Pfaffian expression is somehow related to the so-called Maxwell relations as found in the thermodynamic work of James Maxwell. [3]

Etymology
The names Pfaffian form, Pfaffian function, Pfaffian expression, truncated Pfaffian, etc., may have been introduced in Greek mathematician Constantin Caratheodory's 1908 "Studies in the Foundation of Thermodynamics", and what has since been called Caratheodory's theorem; although this needs to be fact-checked.

Another reference seems to state that the names Pfaffian function and Pfaffian form for these types of expressions were introduced in the 1970s by Russian mathematician Askold Khovanskii and named in honor of German mathematician Johann Pfaff.

Conjugate variables
See main: Conjugate variables
Pfaffian forms is a common formulation in thermodynamics, where the summation pairs, each taking the form of intensive (Xi) and extensive (xi) conjugate variable pair, act as quantifiable energy representations of transformations of the internal energy of the system. The right side of the Gibbs fundamental equation, for example, is a Pfaffian form. In short, the general use of the conjugate pairs perspective is that one can quantify the internal energy of a system as the sum of the conjugate variables. In short, with any extensity xi (extensive variable) it is always possible to associate a tension variable Xi (intensive variable):

X_i = \frac{\partial U}{\partial x_i}

which is called the "conjugate", whereby, according to the first law, the change in internal energy dU of a system is given by the summation of the product of the conjugate pairs:

 dU = \sum_{i=1}^k X_i dx_i

The combined law of thermodynamics:

 dU = TdS - PdV \,

being the simplest example, temperature T and pressure P being the intensive variables (dependent on position in the body) and S and V being the extensive variables (independent on position in the body).

References
1. Sychev, Viacheslav V. (1991). The Differential Equations of Thermodynamics (2.2: Pfaffian forms and Total Differentials, pgs. 11-). Taylor & Francis.
2. Caratheodory, Constantin. (1908). Studies in the Foundation of Thermodynamics (Untersuchungen uber die Grundlagen der Thermodynamik). Bonn; published in: Math. Ann., 67: 355-386, 1909.
3. Kestin, Joseph. (1966). A Course in Thermodynamics (truncated Pfaffian, pg. 531). London: Blaisdell Publishing Co.

Further reading
● Sychev, Viacheslav. (1981). The Differential Equations of Thermodynamics (Pfaffian, 5+ pgs). Taylor & Francis.
● Emanuel, George. (1987). Advanced Classical Thermodynamics (3.3: Legendre Transformation, pgs. 25-28). AIAA.
● Ott, Bevan J. and Boerio-Goates Juliana. (2000). Chemical Thermodynamics – Principles and Applications (2.2d: Caratheodory and Pfaffian Differentials, pgs. 63-). Academic Press.

External links
Pfaffian function – Wikipedia.
Pfaffian form – Wolfram MathWorld.

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