Gibbs paradox

The Gibbs paradox is a term used in statistical mechanics, which refers to the entropy of mixing. That is, the increase in entropy caused by the mixing of two homogenous, single phase materials. Classical physics says here always an increase in entropy ahead, while the entropy of mixing these experiments just in case confirm that the two substances are different. The mixing of two identical materials ( eg, chemically pure oxygen from two gas lines ), however, leaves the entropy unchanged. It was named after its discoverer, the paradox Josiah Willard Gibbs, the calculated end of the 19th century with the classical statistical physics to how much increased by the mixture, the achievable phase space volume, from which he derived the supposedly universal 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, eg before mixing odd numbers for the particles of a substance, quantity and especially for those of others. After mixing particles would be able to distribute any with even and odd identification number. Swapping two of their courses, to be a new (micro) state of the mixture with externally same properties (macro mode). Thus, a considerable increase of the potential (micro) states gives to each macrostate. This results in the framework of classical physics imperative to increase in entropy.

Since we do not observe this increase in the case of mixing the same particles, the argument needs to be amended so that, when counting the number of possible states that interchanging two identical particles is not allowed. This rule will apply in quantum mechanics their deeper foundation: All of the same elementary particles and the resulting constructed atoms or molecules (if they are in the same quantum state), are completely identical and thus indistinguishable. Even the thought of attaching a numbering is a contradiction in terms. Modern formulations of the many-body physics are therefore completely without the numbering of the particles ( or its coordinates). For this reason, the paradox in modern physics does not occur.

Thought experiment

Now consider a configuration which consists of two tubes which are separated by only a partition wall which can be opened and closed. It should also be in two vessels, the same material with the same pressure and temperature. After opening of the partition wall there will be a mixing. If we exclude now the partition again, the initial state is restored: in both vessels back is the same substance with the same pressure and temperature.

According to classical physics, but is faced with the problem that the entropy must be increased by the mixing. Either one assumes that one has reduced by closing the partition entropy again, which would violate the second law of thermodynamics and would have to be also accepted only for the same substances, but not for different substances in the initial state. This solution idea is completely arbitrary and not justified sense. But if we assume that in this cycle the entropy has actually increased, could be with this reversible process, the entropy increase, which would make the concept of entropy nonsensical.

To resolve the paradox, one must add a correction term, which compensates for the over counting of phase space volume by permutation of identical particles. Quantum mechanically, such a correction term results in a natural way, so that the paradox is thus resolved by quantum mechanics. Characterized in the mixture of two volumes of the same material is not the phase space volume increases, and thus the entropy remains unchanged. Wherein the particles of a different material fabric, however, are distinct from those of the other material, whereby this further increases the phase volume and with it the entropy. This is also consistent with the experience that the mixture of different substances is an irreversible process.

In quantum mechanics, the (possible ) degeneracy of Vielteilchenzuständen is known by permutation of the particles as exchange degeneracy. The observation shows that there is no such exchange degeneracy in nature; this is the content of the exchange postulate. Gibbs was thus met with his reflections on the entropy of mixing on a very low-lying principle, which is one of the most important of modern physics.

The Gibbs paradox - E. T. Jaynes (1996 )

J. Willard Gibbs' Statistical Mechanics appeared in 1902., The American physicist ET Jaynes comes in an analysis of an older text of Gibbs ( Heterogeneous Equilibrium, 1875-78 ) to the conclusion that Gibbs have already found satisfactory answers even there, and the paradox therefore clear that none actually. He draws particular attention to the application and scope of the Entropiebegriffes:

"Nevertheless, we tranquil lake Attempts to" explain irreversibility "by searching for some entropy function did is Supposed to be a property of microstate, making the second law a theorem of dynamics, a Consequence of the equations of motion. Search Attempts, dating back to Boltzmann 's paper of 1866, have never succeeded and never ceased. But They are quite unnecessary; for the second law did Clausius gave us something 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, did entropy is a property of the microstate. "

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