In a famous debate, Loschmidt criticized Boltzmann\'s new theory of statistical

Question

In a famous debate, Loschmidt criticized Boltzmann's new theory of statistical mechanics by asking what would happen if the velocities of all the atoms were reversed. Typical objections are that such a reversal would not be possible without a fine detailed control over all the atoms of the system. However, spin echo experiments are a clear counterexample, albeit one restricted to very special cases. Is there any theoretical proof that without a fine detailed control over most of the atoms of a totally isolated system, a Loschmidt reversal would be extremely improbable? Remember such a proof would have to cover all sorts of elaborate complicated contraptions. Are reversals impractical for theoretical reasons, or because of a lack of ingenuity?

Explanation / Answer

Lorschmidt reversal was an attack on naive Boltzmann ideas, based on a truncation of the full evolution to the Boltzmann equation. The Boltzmann equation is irreversible, it doesn't work backwards, and Lorschmidt is simply pointing out that it can't be right, because if you do a reversal, the Boltzmann equation must reverse in time, and it doesn't work backwards.

The reversal itself is not a problem for modern ideas--- you could probably actually do it. In modern ideas, the Boltzmann truncation is unnecessary, the entropy is directly defined on the unknown position of the state in phase space. So the entropy gain is only a measure of the loss of information. If you have a machine which can reverse the atoms' motion to approximately restore the original state, this means, by definition, that the information in the original state has not been lost yet, so the entropy has not gone up until the reversal no longer works.

There is no reason you can't build an (approximate) reverser, but it won't work perfectly. To make a perfect reverser would require infinite precision classically, it would require perfect knowledge of the particles (hence the reason you have a small entropy--- you know where everything is). Quantum mechanically, you would need to reverse outgoing photons, and do hopeless things.

The proof that you can't do Loschmidt reversal for a generic system is simply that it would reduces the entropy. Approximate reversals are fine, and the spin echo is an example.

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