Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:

In every neighborhood of any arbitrary initial state $P_{0}$ of a physical system, there exists neighboring states that are not accessible from $P_{0}$ along quasi-static adiabatic paths.

The above statement is taken from 'Heat and Thermodynamics' 8th Ed by Zemansky and Dittman, and it provides a very concise discussion on the topic which I did not find very illuminating. Moreover, Wikipedia states it slightly differently as:

In every neighborhood of any state $S$ of an adiabatically enclosed system there are states inaccessible from $S$.

With this formulation, he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics.

My questions are:

  • What is the exactly meant by Adiabatic Accessibility and how is this relevant to the formalism of The Second Law?
  • How is this formalism equivalent to the Kelvin Planck and Clausius statements of the Second Law?
  • Except for the satisfaction of having an axiomatic approach, does this provide any advantage over the Kelvin-Planck formalism using heat engines?

P.S. The text in italics has been copied from Wikipedia.

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    $\begingroup$ Try reading T. Frankel's 'The Geometry of Physics'. In chapter 6 he discusses Caratheodory's statement and relates it to Kelvin's, as well as places it in the broader context of holonomic and anholonomic constraints. $\endgroup$ – G. Paily May 22 '14 at 14:15

(1) Adiabatic accessibility means that by some purely mechanical, electrical, magnetic, etc. (but not thermal) method, an equilibrium state can be reached from another one. At the heart of Caratheodory's idea is the observation that given an equilibrium state A, all other states fall into 3 categories: (a) states that are mutually accessible, (b) states that are accessible but from which state A is not accessible, (c) states that are not accessible but from which state A is accessible. Caratheodory's idea is a broad generalization to Joule's paddle wheel experiment. A very good description of this is in Adkins: Equilibrium Thermodynamics.

(2) In all proofs, it is assumed that the infinitesimal work is representable as a 1st order differential form of the state parameters: $\delta W = y_1dX_1 + y_2dX_2 +...$, hence the apparent distinction between irreversible or reversible adiabatic process melts away into a reversible process.

(3) The classification of states into these categories, plus that the work is a 1st order differential form combined with a purely mathematical theorem by Caratheodory, which gives a result that for nonadiabatic processes, (i.e., one for which $dU-\delta W \ne 0$ there is a function $T$ for which $\frac{1}{T}(dU-\delta W)$ is a total differential) - hence the existence of entropy.

(4) Whether this approach is equivalent to the more classical approaches of Kelvin, Clausius, Planck, etc. is/was a source much debate, derision, praise, whatever. Some physicists love it, others despise it. A good review of the debate is in Truesdell: Rational Thermodynamics, 2nd edition; he does not like it...

(5) Caratheodory's is not the only way to axiomatic thermodynamics; it is possible to axiomatize on the basis of thermal engines, or Carnot cycles, as well - see again the Truesdell book.

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  • $\begingroup$ Can you elaborate on point 2? How does this distinction exactly melt away? $\endgroup$ – noir1993 May 22 '14 at 17:06
  • $\begingroup$ Just recall that energy conservation is $dU=\delta Q + \delta W$, so if you assume that $\delta W = y_1dX_1 + y_2dX_2$ then $\delta Q = dU - \delta W = dU- (y_1dX_1+y_2dX_2)$ so $\delta Q$ is also a 1st order differential form of the system parameters, say $U, X_1, X_2$ and the process by definition is reversible. Now you still need something else to show that for reversible process $\delta Q = TdS$ (with Caratheodory that would be adiabatic inaccessibility) but if it is irreversible then $\delta Q < TdS$ and neither $\delta Q$ nor $\delta W$ is a differential form of the system parameters. $\endgroup$ – hyportnex May 22 '14 at 17:24
  • $\begingroup$ I still can't see how the concept of increase of entropy comes out of this for an irreversible process. ( I'm getting the feeling that this works only for reversible processes ). $\endgroup$ – noir1993 May 22 '14 at 17:46
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    $\begingroup$ The increase of entropy in irreversible adiabatic process is a separate consideration and it follows from the asymmetry of accessible vs. inaccessible states and from the assumption that temperature (the integrating factor) is positive. I suggest that if you are really interested in these details then you read chapter 6. of Adkins's book. $\endgroup$ – hyportnex May 22 '14 at 17:59
  • $\begingroup$ I would definitely look it up. Thank you. This discussion has been very helpful. $\endgroup$ – noir1993 May 22 '14 at 18:09

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