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I heard that Schrödinger pointed out that (classical/statistical) thermodynamics is impaired by logical inconsistencies and conceptual ambiguities.

I am not sure why he said this and what he is talking about. Can anyone point some direction to study what he said?  

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Please read this post about how to write good question titles. – DanielSank Jan 5 '15 at 9:45

You asked, about logical inconsistencies and logical ambiguities in classical thermodynamics and classical statistical mechanics. As far as classical thermodynamics is concerned, there are no inconsistencies or ambiguities. Classical thermodynamics is based on several axioms (known as laws), and a bunch of definitions. All the rest is deduction. If this makes it seem like Euclidian plane geometry, it is.

Classical statistical mechanics (SM), on the other hand, has big problems. SM attempts to compute thermodynamics properties, i.e. macroscopic variables, from microscopic properties. Classical SM assumes that atoms and molecules obey classical mechanics. They don't. The result is that one cannot reproduce macroscopic reality that way without a number of additional assumptions. The problem of course is that on a microscopic level matter inherently obeys quantum mechanics.

For example classical mechanics assumes that identical particles are, in principle, distinguishable because you can mark them with a sufficiently small pen. In quantum mechanics, identical properties are really identical. You can't mark them, and if you interchange the positions and velocities of two of them, NOTHING changes -- so you can't even tell that the interchange occurred.

  ----- Paul J. Gans
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The following source discusses some inconsistency of quantum statistical mechanics indicated by Schrödinger (see the reference there):

The explanation is long and cannot be outlined here. I cannot be sure that Schrödinger did not discuss other issues with thermodynamics or statistical mechanics, so I cannot be sure this answer is useful for you.

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I think you need to be a little more specific as to the question. The inconsistency of classical mechanics with atomic physics is found by the attempted classical analysis of electron orbital behavior. If the electron orbit is modeled classically, then the electron should give off radiation in accordance to the Larmor formula.

It is this inconsistency that led Bohr to build his postulates of quantum mechanics in his model of the hydrogen atom. Schrodinger published his improvement in 1926 in four parts. Quantisation as a Problem of Proper Values, Part I, Part II, Part III, Part IV. You can also find his discussion in Physical Review in 1926.

An additional good paper is the translation of the "Schrodinger's Cat Paradox".

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