
Irish physicist Robin Giles' 1964 Mathematical Foundations of Thermodynamics, giving a depiction of the basics of mathematical thermodynamics. [10]

Irish physicist Robin Giles' 1964 Mathematical Foundations of Thermodynamics, giving a depiction of the basics of mathematical thermodynamics. [10]

In thermodynamics, mathematical thermodynamics is the study of the underlying mathematical structure of thermodynamics, in origin and application, or rather the mathematics of thermodynamics, in a general sense.

Overview
The history of the subject of the teaching and study of the mathematical underpinnings of chemistry, physics, chemical physics, physical chemistry, and in particular chemical thermodynamics seems to have emerged in the mid to late 19th century, when the subject intricacy began to compound.

The first to publish in this subject was German physicist Rudolf Clausius and his 1858 article “On the Treatment of Differential Equations which are Not Directly Integrable”, published in Dingler’s Polytechnics Journal, an article that was later expanded into the formation of chapter one “Mathematical Introduction” to both the first and second edition of his thermodynamics textbook The Mechanical Theory of Heat. The essence of the need for this introduction was to show the nuts and bolts of the proof, called the “condition for an exact differential” (note: the mathematician behind this proof still needs to be tracked down), of what constitutes a state function, with specific focus on making a state function for an element of heat, the formulation of which became “entropy”, an exact differential formulation of a quantity of heat. [1]

The first book devoted to the subject of the “higher mathematics” for students of chemistry and physics was the 1902 book Higher Mathematics for Students of Chemistry and Physics by English inorganic chemist Joseph Mellor who saw the need for the book in the eagerness of students to readily pursue research and work in the newly forming field of physical chemistry. [6] Mellor's book, however, according to the 1954 preface of the Dover edition by Donald Miller, is out-of-date to effect that:

“The discussions of heat in the sections on thermodynamics do not emphasize the fact that δq is really an exact differential.”

The next dominant higher mathematics publication was English physical chemist, chemical thermodynamicist, and chemistry historian James Partington’s 1911 Higher Mathematics for Chemical Students. [7]

The two-volume 1943 The Mathematics of Physics and Chemistry by Henry Margenau and George Murphy, according to Donald Miller (1954), is said to be “excellent” and “considerably more advanced” than Mellor’s Higher Mathematics.

In 1961, American chemists John Kirkwood and Irwin Oppenheim gave a decent six-page appendix on introductory mathematics in their chemical thermodynamics textbook. [4]

American physical chemist Howard Reiss, in his 1965 Methods of Thermodynamics, devotes chapter to what he calls “Mathematical Apparatus”, in which he gives an introduction to topics such as the total differential, exact differential (or complete differential), real functions, function of state (or state function), Pfaff differential expression (Pfaffian form), integrating denominator, Caratheodory’s theorem, transformation of variables, decomposition of a partial derivative, Euler’s theorem on homogenous functions, and the Lagrange method of undetermined multipliers. [3]

In 1973, physicist-engineer Robert Hermann is said to have given a noted overview of the mathematics of thermodynamics, in what seems to be directed towards surface science applications. [5]

The 2001-2003 work of American physical chemist Robert Alberty on the use of the Legendre transforms for use in chemical thermodynamics is more recent summary of one aspect of mathematical thermodynamics. [9]


The mathematics genealogy tree, from Harry Coonce's Mathematics Genealogy Project, according to which the Bernoulli family, in particular "Bernoulli brothers" (promoters of Gottfried Leibniz's version of differential calculus; as opposed to Isaac Newton's), Jacob Bernoulli and Johann Bernoulli (father to Daniel Bernoulli), the latter mentor to Leonhard Euler, who in turn mentored Joseph Lagrange, who in turn sprouted Joseph Fourier and Simeon Poisson, a group which gave birth to French heat theory school (Ecole polytechnique), which in turn gave birth to the science of thermodynamics, in particular the use of Euler's reciprocity relation to formulate "entropy" (S), the new state function (dQ/T) of a quantity of heat, as done by Rudolf Clausius (1850-1865), is the mathematical heritage of thermodynamics.

The mathematics genealogy tree, from Harry Coonce's Mathematics Genealogy Project, according to which the Bernoulli family, in particular "Bernoulli brothers" (promoters of Gottfried Leibniz's version of differential calculus; as opposed to Isaac Newton's), Jacob Bernoulli and Johann Bernoulli (father to Daniel Bernoulli), the latter mentor to Leonhard Euler, who in turn mentored Joseph Lagrange, who in turn sprouted Joseph Fourier and Simeon Poisson, a group which gave birth to French heat theory school (Ecole polytechnique), which in turn gave birth to the science of thermodynamics, in particular the use of Euler's reciprocity relation to formulate "entropy" (S), the new state function (dQ/T) of a quantity of heat, as done by Rudolf Clausius (1850-1865), is the mathematical heritage of thermodynamics.

Euler reciprocity relation

See main: Euler reciprocity relation

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Homogeneous functions
In circa 1750, Swiss mathematician Leonhard Euler introduced the concept of the homogeneous function, in his "theorem of homogeneous functions", which is said to be "of great use in thermodynamics". [4] Euler's theorem, supposedly, is what justifies or gives the mathematical form of the relation between extensive properties of a system and its internal variables.

Method of undetermined multipliers
In circa 1780s, Italian mathematician Joseph Lagrange introduced his so-called "method of undetermined multipliers", which seems to have something to do with thermodynamics.

Legendre transform
In circa 1790s, French mathematician Adrien-Marie Legendre introduced the so-called "Legendre transform", as it has come to be known.

Pfaff differential expression
In circa 1805, what as come to be called the "Pfaffian form" was introduced by German mathematician Johann Pfaff.

Inexact differential equations
In 1854, German physicist Rudolf Clausius introduced his theorem of the equivalence of transformations, in which he attempted to reformulate the relation between transformations of heat into work and work into heat in universal processes of volume expansion and contraction into what seems to have been a homogeneous function, of the Euler variety; although this is not specifically stated. This subject occupies a large part of mathematical thermodynamics, and seems to be the core of thermodynamics.

Clausius the followed this up with his 1858 article “On the Treatment of Differential Equations which are not Directly Integrable”, which became the opening thirteen page chapter to his 1865 The Mechanical Theory of Heat, which was expanded and reformatted into its final presentation in his second 1875 edition textbook, wherein the chapter became “Mathematical Introduction: on Mechanical Work, on Energy, and on the Treatment of Non-Integrable Differential Equations”, all centered around the justification as to why the integrating denominator of absolute temperature T makes or transforms the so-called inexact differential quantities of heat dQ or δQ into the so-called exact differential state function called entropy. [1]

Maxwell's relations
In circa 1871, Scottish mathematical physicist James Maxwell introduced his so-called reciprocity relations or Maxwell's relations, as they have come to be known, which are frequently discussed in thermodynamics.

Caratheodory's theorem
In 1908, Greek mathematician Constantin Caratheodory published his famous “Studies in the Foundation of Thermodynamics”, a treatise on a proof that an integrating denominator must exist for the state function of heat, or of entropy, a publication which seems to be the first attempt to take a look at the underlying mathematics of thermodynamics. [2]

References
1. (a) Clausius, Rudolf. (1858). “On the Treatment of Differential Equations which are not Directly Integrable.” Dingler’s Polytechnisches Journal, vol. cl. (pg. 29).
(b) Clausius, Rudolf. (1865). The Mechanical Theory of Heat (section: On the Treatment of Differential Equations which are not Directly Integrable, pgs. 1-13). London: Macmillan & Co.
(c) Clausius, Rudolf. (1875). The Mechanical Theory of Heat (section: Mathematical Introduction: on Mechanical Work, on Energy, and on the Treatment of Non-Integrable Differential Equations, pgs. 1-20). London: Macmillan & Co.
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. Reiss, Howard. (1965). Methods of Thermodynamics (ch. 2: Mathematical Apparatus, pgs. 20-31). Dover.
4. Kirkwood, John G. and Oppennheim, Irwin. (1961). Chemical Thermodynamics (Appendix: Introductory Mathematics, pgs. 249-54; 8). McGraw-Hill.
5. (a) Hermann, Robert. (1973). Geometry, Physics, and Systems (thermodynamics, 15+ pgs). M. Dekker.
(b) Bottomley, D.J., Makkonen, Lasse, and Kolari, Kari. (2008). “Incompatibility of the Shuttleworth Equation with Hermann’s Mathematical Structurre of Thermodynamics” (abs), Surface Science, 603(1).
6. Mellor, Joseph W. (1902). Higher Mathematics for Students of Chemistry and Physics: with special reference to the work of J.W. Mellor (exact differential, pg. 57-62). Longmans, Green, and Co.
7. Partington, James R. (1911). Higher Mathematics for Chemical Students. Methuen & Co.
9. (a) Alberty, Robert A. (2001). “Use of Legendre Transforms in Chemical Thermodynamics” (abs), IUPAC Technical Report, Pure Appl. Chem. Vol. 73, No. 8, pp. 1349-1380.
(b) Alberty, Robert A. (2003). Thermodynamics of Biochemical Reactions (2.5: Legendre Transforms for the Definition of Additional Thermodynamic Potentials, pgs. 26-30). Wiley.
10. Giles, Robin. (1964). Mathematical Foundations of Thermodynamics (Amz). Pergamon Press.

Further reading
● Owen, David R. (1984). A First Course in the Mathematical Foundations of Thermodynamics. Springer.
● Meixner, J. (1970). On the Foundation of Thermodynamics of Processes. Mono Book Corp.
● Coleman, B.D. (1974). “A Mathematical Foundation for Thermodynamics” (abs), Archive for Rational Mechanics and Analysis, 54(1): 1-104.
● Ruelle, David. (1978). Thermodynamic Formalism: the Mathematical Structures of Classical Equilibrium Statistical Mechanics (Encyclopedia of Mathematics and its Applications, Volume 5). Addison-Wesley; Thermodynamic Formalism (2nd ed). Cambridge University Press, 2004.
● Ott, J. Bevan, and Boerio-Goates, Juliana. (2000). Chemical Thermodynamics: Principles and Application (1.4: The Mathematics of Thermodynamics, pgs. 22-28; §A1: Mathematics for Thermodynamics, pgs. 593-616). Elsevier.
● Devoe, Howard. (2001). Thermodynamics and Chemistry (§: Mathematical Properties of State Functions, pg. 412-13). . Prentice Hall.
● Salamon, Peter, Andresen, Bjarne, Nulton, James, and Konopka, Andrzej J. (2006). “The Mathematical Structure of Thermodynamics”, Sci.Sdsu.edu.
● Olander, Donald R. (2007). General Thermodynamics (ch. 6: The Mathematics of Thermodynamics, pgs. 165-). CRC Press.
● Fink, Johannes K. (2009). Physical Chemistry in Depth (ch. 1: Mathematics of Thermodynamics (abs), pgs. 1-53). Springer.
● Fronsdal, Christian and Pathak, Abhishek. (2011). “On Entropy in Eulerian Thermodynamics” (abs), in: Second Law of Thermodynamics: Status and Challenges, San Diego (editor: Daniel P. Sheehan) (contents), Ca, 14-15 Jun. (pgs. 277-91). Section V. Classical and Kinetic Perspectives. AIP.
● Cooper, J.B. and Russell, T. (2011). “On the Mathematics of Thermodynamics” (abs), arXiv.org, Feb 08.

External links
● Mathematical and conceptual prerequisites of thermodynamics (2008) – PhysicsForums.com.
● Mathematical thermodynamics (2002) – Pruffle.MIT.edu.