Posts Tagged ‘Carathéodory’s theorem’


Before we begin

Here’s some news. My January 2014 blogpost “Carathéodory: the forgotten pioneer” has been translated into Greek by Giorgos Vachtanidis, and can be seen here.

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Two years on …

Despite the somewhat esoteric nature of Carathéodory’s axiomatic approach to thermodynamics via the geometric behavior of Pfaffians – or perhaps even because of it – my blogpost “Carathéodory: the forgotten pioneer” has received a surprisingly large number of hits, with plenty of brave individuals willing to click on the link to the English version of Carathéodory’s original paper published in 1909 in Mathematische Annalen under the title “Untersuchungen über die Grundlagen der Thermodynamik” [Examination of the foundations of thermodynamics].

Carathéodory’s second axiom “In the neighborhood of any equilibrium state of a system (of any number of thermodynamic coordinates), there exist states that are inaccessible by reversible adiabatic processes”, and the associated theorem giving the condition for dQ to be an integrable differential, constitute the real novelty of his approach.

My original post described Carathéodory’s theorem without going into the proof, since it is rather abstruse and would have appealed only to more avid students of his work. Two years on however, the statistics for this blogpost reveal that there are plenty of avid students wanting to know more. So as a supplementary post, here is a proof of Carathéodory’s theorem, due to Pierre Perrot. Enjoy.

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Carathéodory’s theorem

“If a differential dQ = ΣXidxi, possesses the property that in an arbitrarily close neighborhood of a point P defined by its coordinates (x1, x2,…, xn) there are points which cannot be connected to P along curves satisfying the equation dQ = 0, then dQ is integrable.”

In the following, use is made of a classical result known as the Clausius inequality



Cases n = 1 and n = 2 are trivial because a differential function of only one variable is necessarily total whereas a differential function of two variables is necessarily integrable. All points accessible to a given point P form a continuous domain around P. In an n-dimensional space (n≥3) around P, this domain fills a volume [n dimensions], or a surface [(n-l) dimensions], or a curve [≤ (n-2) dimensions].

The first possibility is excluded because it contradicts the hypothesis that around P there are points which are inaccessible. The third possibility is also excluded because the expression dQ = 0 already defines a surface element containing only points accessible to P. Therefore, points close to P and accessible to P define only a surface. If we now consider a point P’ on that surface, it is impossible to go from P to P’ by a curve satisfying the condition ∫dQ = 0 and not situated on this surface, otherwise every point situated within the immediate proximity of P would be accessible, which contradicts the hypothesis.

From a point P1 it is possible to define a surface S1, upon which all points are accessible to P1. Also, from a point P2 not situated on S1, it is possible to define a surface S2. Surfaces S1 and S2 have no common point between them, otherwise it would be possible to go from P1 to P2 by a path such that ∫dQ = 0. Therefore there is a family of surfaces where σ(x1, x2,…, xn) is constant, filling the space and having no common point among them. For this one-parameter family, dσ = 0 implies dQ = 0, from which, between dQ and dσ, there exists a relation of the type:


where, because dQ = ΣXidxi


Naturally, the family of surfaces for which σ is constant may also be expressed by S(σ) = constant, where S(σ) is an arbitrary function of σ:




(l/T) is the integrating factor. If a differential dQ has one integrating factor, it has an infinity, S being an arbitrary function of σ.

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Carathéodory’s theorem shows that if a differential dQ is integrable, the equation dQ = 0 characterizes in a space a family of surfaces sharing no common point. For any point P on one of these surfaces, it is always possible to find, immediately near that point, points which do not belong to the surface and which are therefore inaccessible by a curve solution of the equation dQ = 0. On the other hand, if dQ is not integrable, the equation dQ = 0 does not define any surface in the space and it will always be possible to link any two points with a curve solution of the equation dQ = 0.

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Constantin Carathéodory (1873-1950)

Constantin Carathéodory (1873-1950)


This post has been translated into Greek by Giorgos Vachtanidis, and can be seen here.

Readers may also be interested to know that my supplementary blogpost “Carathéodory revisited” contains a proof of Carathéodory’s theorem: “If a differential dQ = ΣXidxi, possesses the property that in an arbitrarily close neighborhood of a point P defined by its coordinates (x1, x2,…, xn) there are points which cannot be connected to P along curves satisfying the equation dQ = 0, then dQ is integrable.” The supplementary post can be seen here

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Back in the days when I was a college student – the era when we wore our hair long, when elbow patches were commonplace and Woodstock was still fresh in our minds, the teaching of thermodynamics took place along two main routes.

The first was the classical route focused on heat and its convertibility into work, led philosophically by Carnot, Mayer and Joule, and developed mathematically by Clausius, Thomson (later Lord Kelvin), Helmholtz and Rankine. The second was the statistical route founded on a molecular model, and associated especially with the names of Boltzmann and Maxwell.

Nobody mentioned the third route. None of us were taught anything about the axiomatic approach to thermodynamics, published in 1909 in Mathematische Annalen under the title “Untersuchungen über die Grundlagen der Thermodynamik” [Examination of the foundations of thermodynamics] by a 36-year-old Greek mathematician called Constantin Carathéodory, who at the time was living in Hannover, Germany.


The title page of Carathéodory’s 1909 paper in Mathematische Annalen

It is clear from the outset of his paper that Carathéodory had studied Gibbs’ magnum opus “On the Equilibrium of Heterogeneous Substances (1875-1878)”. And just like Gibbs, Carathéodory uses the internal energy U and the entropy S (introduced together with the absolute temperature T) as the fundamental building blocks upon which he constructs his version of thermodynamics.

But whereas Gibbs introduces entropy via the classical route taken by Clausius, Carathéodory finds it through a mathematical approach based on the geometric behavior of a certain class of partial differential equations called Pfaffians, named for the German mathematician Johann Friedrich Pfaff (1765-1825) who first studied their properties.

Carathéodory’s investigations start by revisiting the first law and reformulating the second law of thermodynamics in the form of two axioms. The first axiom applies to a multiphase system change under adiabatic conditions:

Ufinal – Uinitial + W = 0

Nothing original here, since this is an axiom of classical thermodynamics due to Clausius (1850). It asserts the existence of a form of energy known as internal energy U – an intrinsic property of a system whose changes under adiabatic conditions are equal and opposite to the external work W performed (for a closed system not in motion).

In Carathéodory’s approach however, heat is regarded as a derived rather than a fundamental quantity that appears when the adiabatic restriction is removed, i.e. ΔU+W ≠ 0.

The second axiom is a different matter altogether, and constitutes the real novelty of Carathéodory’s approach:


This can be rendered in English as “In the neighborhood of any equilibrium state of a system (of any number of thermodynamic coordinates), there exist states that are inaccessible by reversible adiabatic processes.”

For a single substance, this postulate is obvious enough since reversible adiabatic processes are isenthalpic – a known result of classical thermodynamics. For such processes, all attainable states are represented by points on a curve for which entropy S = constant. There are other points which do not lie on this curve, and which represent states which cannot be reached by adiabatic transition.

But Carathéodory’s arguments go further, making this axiom applicable to a system of multiple bodies and multiple independent variables.

He shows that if in the neighborhood of any given point, corresponding to coordinates x1, x2,…, there are points not expressible by solutions of the Pfaffian equation X1dx1 + X2dx2 +… = 0, then for the expression X1dx1 + X2dx2 +… itself there exists an integrating factor.

The significance of this discovery is that via Carathéodory’s first axiom, the equation of adiabatic condition dQ = 0 admits an integrating factor, which when multiplying dQ renders the product an exact differential of a function whose value is therefore independent of the path between sets of coordinates.

The integrating factor (denominator) in this case is the absolute temperature T, and the path-independent integral ∫dQrev/T is the entropy change ΔS. This conjugate force-displacement pair, whose product is heat, arises directly from the geometric behavior and solutions of Pfaffians.

Using these partial differential expressions, Carathéodory obtains a formal thermodynamics without recourse to peculiar notions such as the flow of heat, or cumbrous conceptions such as imaginary heat engines and cycles of operation. In short, Carathéodory reduces the argument to a clean-cut consideration of lines and surfaces, together with a pair of axioms regarding the possibility of reaching certain states by adiabatic means.

It sounds very neat and tidy, as well as highly original, so how come it didn’t figure on our college curriculum? The answer to that question can be found in the reception afforded to Carathéodory’s masterwork by the scientific establishment of the day.

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Carathéodory’s thermodynamic theory got off to a rather inauspicious start in that it was ignored for the first 12 years of its existence. World War I came and went. Then in 1921, the German mathematician and physicist Max Born took note of Carathéodory’s work and published a set of articles on it entitled Kritische Betrachtungen zur traditionellen Darstellung der Thermodynamik [Critical considerations on the traditional representation of thermodynamics] in Physikalische Zeitschrift.

That got the ball rolling, but only as far as Max Planck, who besides being the towering authority in thermodynamics at the time, also turned out to be a severe critic of the new axiomatic method:

“nobody has up to now ever tried to reach, through adiabatic steps only, every neighborhood of any equilibrium state and to check if they are inaccessible,” Planck wrote, adding “this axiom gives us no hint which would allow us to differentiate the inaccessible from the accessible states.”

Others, impressed by the elegance of Carathéodory’s method, tried to render its formal austerity palatable to a wider audience. But these efforts met with no success, and when Lewis and Randall’s hugely influential and curriculum-setting textbook Thermodynamics and the Free Energy of Chemical Substances appeared in 1923, there was not a mention of Carathéodory or his theory.

Although there have been some notable attempts down the decades to champion Carathéodory’s cause, the axiomatic theory of 1909 has failed to achieve inclusion in mainstream academic teaching, and has been consigned to the catalogue of interesting curiosities. Planck’s enduring criticism of the theory’s failure to provide a compelling physical concept of entropy, together with the equally enduring difficulty of the math, seem to have played the deciding role. cc01a

Constantin Carathéodory (left) looking dapper in the company of his father, brother-in-law and sister. Carlsbad, Czechoslovakia 1898. Photo credit Wikimedia Commons

Suggested further reading:

1. The Structure of Physical Chemistry, C.N. Hinshelwood, Oxford University Press (1951)
Chapter III, Thermodynamic Principles, contains a concise introduction to Carathéodory’s theory, together with a discussion comparing its strengths and weaknesses with the classical approach. This book has been reissued as part of the Oxford Classic Texts series.

2. Constantin Carathéodory and the axiomatic thermodynamics, L. Pogliani and M. Berberan-Santos, Journal of Mathematical Chemistry Vol. 28, Nos. 1–3, 2000
This paper reviews the development of Carathéodory’s theory and explores some aspects of Pfaffians, the mathematical tools of axiomatic thermodynamics. A brief biography is also included.
PDF downloadable from

and if you’re feeling brave…

3. Examination of the foundations of thermodynamics – English translation of Carathéodory’s 1909 paper
PDF downloadable from