Bride of the Return of the Boltzmann Brain

Lubos Motl has returned to the subject of Boltzmann Brains, so how can I resist? Lubos makes a couple of arguments, which don't really seem compatible to me, but let me concentrate on the more important one. He wants to dispose of Poincare recurrences - the fact that a system eventually (after a veerrry long time) returns arbitrarily close to it's initial state.

We must be very careful to distinguish different types of description of physics. Below we will explain that Poincaré recurrences are only relevant for the exact, microscopic description of a physical system. When we describe a physical system microscopically, we really need to know the initial state completely accurately. When we know it accurately, we can say that it will return to the same point after the recurrence time.

What he is getting to here is the notion that you don't usually know the exact microscopic state, only the ensemble that it belongs to. He eventually gets to the point:

...it is simply not true that the microstates in the collection will return to the same initial state after a universal time. Individual states get restored but the precise time is different for each of them.

True enough, but so what? We didn't know the exact physical state of the initial system, but presumably it had one, and that state will sometime reach its Poincare recurrence - at that point it won't matter what happened to the other members of the ensemble, since they weren't real anyway.

Lubos makes this error because he made an earlier one:

But if we describe the initial state accurately, we can't talk about its entropy and we can't describe it by macroscopic words such as an "egg". Microstates can't be subjects of macroscopic assertions. Microstates have no emotions, if you wish.

If we want to say that the initial state has a certain nonzero entropy or that it is an "egg", it inevitably means that we only describe the state approximately. We only have incomplete information about the initial state. Most typically, we only specify certain macroscopic degrees of freedom - and even these degrees of freedom are specified with a nonzero error margin - and we ignore most microscopic details of the system i.e. allow them to have an arbitrary form.

When we talk about entropy, my statement is a tautology because the entropy is defined with respect to particular ensembles of microstates. If you say that a system has a certain entropy, it means that you only talk about the ensembles and in the same sentence, you simply cannot distinguish the individual microstates in the ensemble from each other.

Lumo is saying that entropy is only defined for ensembles, not for individual microstates. He is confusing the way we calculate the entropy with the entropy itself. A given physical system consists of a single microstate, and its entropy is a physical property of that microstate. When we calculate the entropy, however, we calculate the number of microstates compatible with the macroscopic configuration corresponding to the microstate - in effect, the entropy depends on the neighborhood, not the "emotions" of the microstate.

He gives us an example which will serve to refute his main claim:

OK, if you return to the same point of the phase space - for example a point where all oxygen atoms are located in the left half of the bedroom - is it correct to say that the entropy will have to drop sometime in the future, in order to return to the initial state that we started with and that could have a low entropy?

The answer is, of course, No. It is No even in ordinary physics of gas in a box.

Actually, the answer is the opposite. Consider the gas in the box. In order for the question to be meaningful, we need to have some way to measure where the atoms are in the box. We can do that with a sliding partition with which we regularly divide the box in half and count the atoms. If the number of atoms is small, the Poincare recurrence time won't even be terribly long, so when all the atoms wind up on one side is the entropy reduced? Yes it is. It's an elementary stat mech problem to show that it has decreased by a factor of N*log2, where N is the number of atoms.

Ironically, Lubos is a victim of the exact error he accuses others of:

...the inability of the authors to distinguish a microscopic description of a physical system from the macroscopic or approximate one i.e. their inability to distinguish an ensemble of microstates from its individual elements.

In statistical mechanics, everything is calculated from ensembles, a very useful technique. It's important to remember though, that those that things we calculate: temperature, pressure, entropy, are not properties of the ensemble - they are properties of the actual physical system, which in turn is just one point in that ensemble's phase space.

And that's this week's memo/lesson for the prof.