# Theoretical proof forbidding Loschmidt reversal?

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?

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Collisions are very often irreversible due to soft radiation. It is impossible to reverse the soft radiation in a general case. It is an inelastic process like breaking a glass into large and small pieces. – Vladimir Kalitvianski Dec 12 '11 at 18:45
– Qmechanic Dec 12 '11 at 18:57
Quantum mechanics really forbids the kind of detailed microscopic control that would be needed for the Loschmidt reversal. While in classical physics one could say that for most initial states, the entropy will increase, while for a carefully selected special initial state, it may decrease, quantum mechanics says something else. It always predicts the probabilities only and the probabilities of a big decrease in entropy is small for every initial state, much like if one averages over initial states in classical physics. – Luboš Motl Dec 12 '11 at 19:00
@LubošMotl: Although QM deals with amplitudes of probabilities, it is not the reason for irreversibility. Irreversibility is first of all in creating and absorbing too many soft quanta, which are inelastic processes difficult to launch backwards. If one manages to create a gap in the energy spectrum, soft modes disappear and reversibility becomes manifest experimentally. – Vladimir Kalitvianski Dec 13 '11 at 10:01
Note that what @LubošMotl says is only true if you measure the system before you try to reverse it. For an unmeasured, isolated system it's always in-principle possible to do the reversal. Also note that by "state" he means "density matrix" and not "wavefunction." – Nathaniel May 16 '12 at 13:44

It's impossible to forbid Loschmidt reversal by any theorem. Poincare recurrence is the dreaded counterexample. Finite closed systems will return arbitrarily close to their original state. The only thing Loschmidt failed to anticipate is it would take far longer than expected: an exponential time.

Using the second law to "forbid" reversals is circular because this very same result is used to "prove" the second law in the first place.

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 Poincare recurrence needs strong assumptions that are violated for the universe as a whole. Thus there is no counterexample. – Arnold Neumaier May 16 '12 at 14:46

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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 Indeed, and there are some cases in which an approximate reversal is quite feasible - the most obvious being reversing light emanating from a point source through the use of a curved mirror. – Nathaniel May 16 '12 at 13:46

Rather more hand waving than a "proof", but...

Note that in quantum mechanics the meaning of time reversal symmetry is subtly different then it is classically{*} and they system can not be expected to retrace it's history in reverse.

Add to this any degree of sensitive dependence on initial conditions (which is certainly possible in classical systems), and you can magnify quantum effects to macroscopic differences even in systems that appear to be fully amenable to classical description.

{*} The cross-section for the time-reversed reaction is the same as for the forward reaction in all cases{+}, but you can't pass "backward" through the "collapse" of the wave function (however you understand the selection of eigenstates to proceed).

{+} Well, for the CPT-reversed reaction, anyway.

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I liked the following article on this issue: http://arxiv.org/abs/0802.0438 (published in Phys. Rev. Lett.). There is a partial correction at http://arxiv.org/abs/0912.5394 (I am not sure if it is reflected in the current version of the main article.)

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The Loschmidt paradox is valid only in bounded systems with finitely many degrees of freedom, as the proof of Poincare recurrence fails in an unbounded domain, or if the number of degrees of freedom is infinite.

As the true degrees of freedom are those of fields (though actually quantum fields), hence infinitely many, there is no Loschmidt paradox.

Actual reversal can be done experimentally ony for systems with a tiny number of degrees of freedom, typically just one.

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 While you're right about the "unreality" of the paradox for everyday large enough systems (and only for those, entropy is macroscopic), you're not right about the recurrences in principle (the first paragraph). First, it is not known whether the Universe is spatially closed and even if it is not, the Poincare recurrences do occur because the entropy of our Universe is at most $10^{120}$, as determined by the de Sitter cosmic horizon, and by the black hole complementarity applied to the cosmic horizons, it applies everywhere. After $\exp(10^{120})$ units of time, there have to be recurrences. – Luboš Motl May 21 '12 at 18:32 @LubošMotl: Why does a bound on the total entropy imply Poincare recurrences? – Arnold Neumaier May 21 '12 at 19:05