The only assumption, while indeed crucial, is that the bath is regarded to be in the equilibrium state all the time. Of course, due to the finiteness of the contact between the system and the bath there are "boundary effects". For one, dissipation of energy stored in the system implies an energy flux into the bath, which almost by definition moves the bath out of equilibrium. Thus the validity of the assumption about equilibrium is determined by importance of such boundary effects. Usually the energy flux is comparably weak and the dynamics of the system is much slower comparing to typical bath's time scales. Hence the state of the bath can be approximated by equilibrium.

Claude,

The discrepancy here is the different governing time spans, short or long. Furthermore, decoherence might occur preferentially ('skewedly') towards some of the environment's variables.

Decoherence happens when some system's wave function starts to relate to outside variables not previously within their Schrödinger - the system's wave function begins to absorb and encompass new variables, and also might shed some prior variables: That's decoherence in a nutshell. These new variables of course are sourced from the system's environment. Now you can't really meaningfully relate average thermodynamical properties of a whole environment with the instant process of decoherence of a given system interacting with that environment, partly because the decoherence process is selective (it may relate to selected variables only within the environment, and ignore some other variables) and rapid (so that average thermodynamical values do not straightforwardly relate to the process of decoherence.) Remember also that all states of equilibrium only paper over vast differences within themselves, and intense statistical churning of values: a gas at, say, average room tempertaure inevitably contains clusters of particles at very cold temperatures and others at very high temperatures, all of which continuously shift and evolve dynamically.

How to prevent decoherence, and how and why certain contextual variables are taken on board by a given Schrödinger (when decoherence is permitted), and others not, is a very interesting question and theme of investigation in all such cases.

Actually nothing.

In this context, it is also important to mention that decoherence is reasonable and meaningful in everyday physics where we have well-defined distinctions between different devices. But it is completely worthless in the fundamental applications which is tried in the many worlds interpretation. Because in this fundamental case you have nothing to start with - you would need a fundamental decomposition of the universe into systems to start with and to define everything else, in particular the many worlds themself.

I am not convinced that the the validity of summing over the bath degrees of freedom depends in any way on the nature of the bath. If you really cannot control the degrees of freedom of the bath---in other words, if the only observables you can actually measure are of the type A tensor identity, where A is an arbitrary observable of the central system, then the pure state of the combined system (central system + bath) is indistinguishable from the partial trace over the bath.

This requires no assumptions on the bath whatever: it is just elementary QM formalism. On the other hand, whether we can, or not, derive a *closed* equation of motion for the state of the central system, is a non-trivial question, which does depend on issues of relative time scales. But I do not think the relevance of partial traces to describe imperfectly known systems depends on such issues.

Of course to sum over unobservable degrees of freedom is fine.  But this was known by the founding fathers too.  Schrödinger's cat and the measurement problem is about something completely different.  It is a problem solved, in particular, by de Broglie-Bohm theory.   

To some extent, I agree, one may solve the measurement problem by de Broglie-Bohm. But I believe decoherence brings clarity to the traditional picture: before, it was simply said that a measurement collapsed the wave function and that was it. Why it should do so, when does a measurement start, what type of system has the ability to measure, are all issues that are untouched by this ``postulate''. Decoherence, as far as I can tell, can tell us in quantitative terms when a system which is not fully controlled (and no system is, when it is exposed to an environment) loses phase coherence, so that measuring it involves no ``wave collapse''.

Whether all this was known, or  not, to the founding fathers, is a historical issue, which I would prefer no to get into. But we often see criticisms of textbook measurement theory precisely along the lines that ``measurement is ill-defined'' and so on. These criticisms, I believe, lose some of their force when the decoherence process which precedes measurement is taken into account.

F.Leyvraz, the collapse hasn't disappeared, it is hidden in the formalism of decoherence, and there is still no criterion allowing to know when it happens and when it doesn't. In fact, according to the "Heisenberg cut," the collapse can happen at any time, that doesn't change the final result, thus including during the proper measurement. Decoherence is then but a mathematical cloaking operation. It only describes loss of information by dissipation into random environment, which is trivial. Heat can destroy information both in classical and quantal case.

``there is still no criterion allowing to know when it happens and when it doesn't. ''

Indeed. Rather, decoherence shows how coherence is, in principle, gradually destroyed as time goes on, in the framework of ordinary unitary time evolution. It replaces the sudden collapse invoked by fiat with a continuous process happening within th eunitary evolution.

``the collapse hasn't disappeared, it is hidden in the formalism of decoherence, and there is still no criterion allowing to know when it happens and when it doesn't.''

What do you call collapse? If you observe a density matrix which corresponds to being in state psi_i with probability p_i, that is, a fully incoherent superposition, and you ``observe'' which of these states the system is actually in, does that constitute ``collapse''?

I would say not, or at least, I would say that this is not any more mysterious than finding out, by observation, which side a coin lands on.

Now the way I understand decoherence is that, at some stage, a microscopic system gets entangled with an uncontrollable macroscopic environment, and this leads to a state being, for all practicable measurements, equivalent to a mixture.

The advantage of this approach is that it is objective: it has  no reference to conscious beings. There is a certain amount of arbitrariness in saying when we are unable to observe the environment, but that is similar to what happens in statistical mechanics: we arbitrarily claim that, after a certain level of complexity, we cannot invert all velocities and undo irreversible processes.

So decoherence suggests that, before the measurement is actually performed, say by reading the instrument, the state has already been reduced, by physics alone, not by postulate, to a purely probabilistic mixture. Measurement then consists in observing a state in which coherence propoerties have already disappeared, so that there is no weird collapse involved. 

``[Decoherence] only describes loss of information by dissipation into random environment, which is trivial. Heat can destroy information both in classical and quantal case.''

That is slightly misleading. When ``heat'' or disordered energy, actually destroys classical information, a significant amount of energy is needed, whereas destruction of coherence requires very little coupling and happens, particularly in the case of coherent superpositions of macroscopic different states (cat states) on extremely fast time scales. Quantum coherence is thus infinitely more fragile than classical information. 

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