Friday, August 28, 2009

Time's Arrow

Time flies like an arrow,
but fruit flies like a banana


The arrow of time has been understood for 100 years says Lumo, except, apparently, in the Caltech Cosmology department. Forgetting that for a moment, is it understood by Motl?

I find him a bit hard to understand when he's in his rage against the heathens polemical mode, but he and I have been in this rodeo before, so I'm just going to say that he seems to utterly miss the point in his critique of Sean Carroll and colleagues. The problem Sean addresses, and the Lubos tries to sweep under the rug, is that of how a direction of time emerges from time symmetric laws of microphysics.

Boltzmann was perhaps the first to struggle with this, and his answer was the so-called H theorem. Loschmidt pointed out that his proof in fact assumed something equivalent to what he was trying to prove. Lubos discusses a more recent attempt by Weinberg to prove a quantum field theoretic version of the H theorem. I couldn't really follow his discussion but there was at least one really interesting point - Weinberg apparently relates it to decoherence.

I couldn't entirely follow his next point either, but he helpfully summed it up:

Whenever you calculate the probabilities of evolution from one "macroscopic state" (being identified with an ensemble of similar microscopic states) to another, it's important that you sum over the microstates in the future, but you take the average over the microstates in the past.

So what does this mean? I'm not sure, but the obvious interpretation leads to some strange statistics. In the classical consideration of the interaction of a system with a heat reservoir one derives the density of states leading to the famous Boltzman factor. It is intrinsic to this derivation that microstates be counted the same way in both the initial state (or past state) and the final (future) state. The Lumo prescription, averaging over the initial states, appears to count only one state for each macrostate (or energy value). This guarantees that entropy increases in the future but produces a very non-Boltzmann distribution function.

So perhaps this isn't what he intends. What is clear is that his method of counting states one way in the past and another in the future amounts to a very violent imposition of time assymmetry - imposing the the very condition that he purports to demonstrate as a precondition. He admits essentially as much in the comments. So what's his point? That he wants to dis Sean C?

That's about all I get out of it. One more turn chasing his tail around the circle.