The question of the arrow of time


A recent paper by Lorenzo Maccone on Physical Review Letters (see here) has produced some fuss around. He tries to solve the question of the arrow of time from a quantum standpoint. Lorenzo is currently a visiting researcher at MIT and, together with Vittorio Giovannetti and Seth Lloyd, he produced several important works in the area of quantum mechanics and its foundations. I have had the luck to meet him in a conference at Gargnano on the Garda lake together with Vittorio. So, it is not a surprise to see this paper of him in an attempt to solve one of the outstanding problems of physics.

The question of the arrow of time is open yet. Indeed, one can think that Boltzmann’s H-theorem closed this question definitely but this is false. This theorem has been the starting point for a question yet to be settled. Indeed, Boltzmann presented a first version of his theorem that showed one of the most beautiful laws in physics: the relation between entropy and probability. This proof was criticized by Loschmidt (see here) and this criticism was sound. Indeed, Boltzmann had to modifiy his proof by introducing the so called Stosszahlansatz or molecular chaos hypothesis introducing in this way time asymmetry by hand.  Of course, we know for certain that this theorem is true and so, also the hypothesis of molecular chaos must be true. So, the question of the arrow of time will be solved only when we will know where molecular chaos comes from. This means that we need a mechanism, a quantum one, to explain Boltzmann’s hypothesis. It is important to emphasize that, till today, a proof does not exist of the H-theorem that removes such an assumption.

Quantum mechanics is the answer to this situation and this can be so if we knew how reality forms. An important role in this direction could be given by environmental decoherence and how it relates to the question of the collapse. A collapse grants immediately asymmetry in time and here one has to cope with many-body physics with a very large number of components. In this respect there exists a beautiful theorem by Elliot Lieb and Barry Simon, two of the most prominent living mathematical-physicists, that says:

Thomas-Fermi model is the limit of quantum theory when the number of particles goes to infinity.

For a more precise statement you can look at Review of Modern Physics page 620ff. Thomas-Fermi model is just a semi-classical model and this just means that this fundamental theorem can be simply restated as saying that the limit of a very large number of particles in quantum mechanics is the classical world. In some way, there exists a large number of Hamiltonians in quantum mechanics that are not stable with  respect to such a particle limit losing quantum coherence. For certain we know that there exist other situations where quantum coherence is kept at a large extent in many-body systems. This would mean that exist situations where quantum fluctuations are not damped out with increasing number of particles.  But the very existence of this effect implied in the Lieb and Simon theorem means that quantum mechanics has an internal mechanism producing time-asymmetry. This, together with environmental decoherence (e.g. the box containing a gas is classical and so on), should grant a fully understanding of the situation at hand.

Finally, we can say that Maccone’s attempt, being on this line of thought, is a genuine way to understand from quantum mechanics the origin of time-asymmetry. I hope his ideas will meet with luck.

Update: In Cosmic Variance you will find an interesting post and worthwhile to read discussion involving Sean Carroll, Lorenzo Maccone and others on the questions opened with Lorenzo’s paper.

KAM theorem and ergodicity


Integrable Hamiltonian systems have the property that, when they are slightly perturbed, their behavior gets only slightly modified. One can state this result in a more technical way through a beautiful theorem due to Kolmogorov, Arnold (see here) and Moser (KAM). In phase space, these systems move on tori and the effect of a small perturbation is to produce a small deformation of these tori. A condition on resonances must hold for KAM theorem to apply. Indeed, for a small set of initial conditions the motion is no more bounded. So, when the perturbation increases, invariant tori get progressively destroyed and chaos sets in. The system goes to occupy a large part of phase space and we are in a condition for ergodicity to be true.

One may ask what can happen when the perturbation becomes increasingly large. A first idea is that ergodicity is maintained and we keep on being in a situation of fully developed chaos. Indeed, this idea is plainly wrong. For an infinitely large perturbation, a dual KAM theorem holds and again we get invariant tori and bounded motion. I proved this in my recent paper (see here). Increasing the perturbation makes tori reform and we lose ergodicity again. Indeed, ergodicity appears to be there only for a limited range of parameters of the Hamiltonian system. This can make us think that this property, that appears to be essential to our understanding of thermodynamics and, more generally, of statistical mechanics, is not ubiquitous.

So, one may ask why all systems appear to behave as if ergodicity holds. The answer to this question is quite straightforward. What makes Hamiltonian systems behave ergodically is the fact that they are composed by a very large number of particles. It is this that provokes the correct working of our statistical approach and produces everyday reality we observe. This conclusion is quite important as makes clear that we do not need ergodicity at a very fundamental level but just at a macroscopic one. This intuition was already present in Boltzmann‘s Stosszahlansatz hypothesis. The existence of a dual KAM theorem makes all this very clear. Indeed, thermodynamic limit can make quantum system quite unstable with respect to coherent evolution producing a classical ergodic system.

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