Gibbs' paradox
The Gibbs paradox is a term used in statistical mechanics , which refers to the mixing entropy relates. This is the increase in entropy that results from the mixing of two homogeneous, single-phase substances. Classical physics always predicts an increase in entropy here, while the experiments confirm this entropy of mixing only in the event that the two substances are different. Mixing two identical substances (e.g. chemically pure oxygen from two gas lines), however, leaves the entropy unchanged. The paradox was named after its discoverer Josiah Willard Gibbs, who at the end of the 19th century calculated with classical statistical physics by how much the achievable phase space volume increased by mixing , from which he derived the supposedly general formula for the entropy of mixing.
According to classical physics, the derivation by Gibbs is absolutely correct: Each atom (or molecule) could be marked by a number, e.g. B. before mixing, odd numbers for the particles of one amount of substance and even for those of the other. After mixing, particles with even and odd identification numbers would be able to distribute themselves arbitrarily. If two of them swap their places, a new (micro) state of the mixture arises with externally identical properties (macrostate). This results in a considerable increase in the possible (micro) states for each macrostate. In the context of classical physics, this necessarily leads to an increase in entropy.
Since one does not observe this increase in the case of mixing of the same particles, the argumentation has to be changed to the effect that when counting the possible states, the interchanging of two identical particles is inadmissible. This rule finds its deeper justification in quantum mechanics: All elementary particles of the same type and the atoms or molecules built up from them (provided they are in the same quantum state) are completely identical and therefore indistinguishable . Even the imaginary attachment of a number is a contradiction in terms. Accordingly, modern formulations of many-particle physics manage completely without the numbering of the particles (or their coordinates). Because of this, the paradox does not appear in modern physics.
Thought experiment
One looks at a structure that consists of two vessels that are only separated by a partition that can be opened and closed. Furthermore, let the same substance in both vessels be at the same pressure and temperature . After opening the partition, there is a mixing. If you now close the partition again, the initial state is restored: the same substance is again in both vessels at the same pressure and temperature.
According to classical physics, however, one is faced with the problem that the entropy must have increased due to the mixing. Either one assumes that the entropy has been reduced again by closing the partition, which would violate the second law of thermodynamics and, moreover, would only have to be assumed for the same substances, but not for different substances in the initial state. This idea for a solution is absolutely arbitrary and cannot be justified sensibly. However, if one assumes that the entropy has actually increased in this cycle, this reversible process could increase the entropy, which would make the term entropy nonsense.
To resolve the paradox one has to insert a correction term that compensates for the overcounting of the phase space volume by interchanging identical particles . From a quantum mechanical point of view, such a correction term results naturally, so that the paradox is solved by quantum mechanics. As a result, when two volumes of the same substance are mixed, the phase space volume does not increase and the entropy therefore also remains unchanged. In the case of different substances, however, the particles of one substance can be distinguished from those of the other substance, which means that the phase space volume continues to increase and with it the entropy. This also agrees with the experience that mixing different substances is an irreversible process.
In quantum mechanics , the (possible) degeneracy of many-particle states through permutation of the particles is known as exchange degeneracy . The observation shows that there is such an exchange degeneracy in nature is not there; that is the content of the exchange postulate . With his considerations on entropy of mixing, Gibbs had come across a very deep-seated principle which is one of the most important in modern physics.
The Gibbs Paradox - ET Jaynes (1996)
J. Willard Gibbs' Statistical Mechanics appeared in 1902. The American physicist ET Jaynes, in an analysis of an earlier text by Gibbs ( Heterogenous Equilibrium , 1875-78), concludes that Gibbs himself had already found satisfactory answers and the paradox there therefore actually none is. In particular, it indicates the application and scope of the concept of entropy:
"Nevertheless, we still see attempts to" explain irreversibility "by searching for some entropy function that is supposed to be a property of microstate, making the second law a theorem of dynamics, a consequence of the equations of motion. Such attempts, dating back to Boltzmann's paper of 1866, have never succeeded and never ceased. But they are quite unnecessary; for the second law that Clausius gave us was not a statement about any property of microstates. The difference in dS on mixing of like and unlike gases can seem paradoxical only to one, who supposes erroneously, that entropy is a property of the microstate. "
See also
literature
- Klaus Stierstadt, Günther Fischer: Thermodynamics: From Microphysics to Macrophysics (Section 5.5) . Springer, Berlin, New York 2010, ISBN 978-3-642-05097-8 ( limited preview in Google book search).