January 10, 2009

But It's Only a Little Demon

Attention conservation notice: ~500 words of excessively cute foundations-of-statistical-mechanics geekery. Inspired by this post at The Statistical Mechanic.

I have here on the table before me my favorite classical-mechanical assemblage of interacting particles, with 2n degrees of freedom, n being a macroscopically large number. (The factor of 2 is both because there are always position and velocity degrees of freedom, and to avoid some factors of 1/2 later.) It is in turn part of a larger assemblage with many more degrees of freedom, say 2N. Both the smaller and larger assemblages are highly unstable dynamically, so I can expect statistical mechanics to work quite well. (Really, I can.) On the other hand, I presume that they are very thoroughly isolated from the rest of the universe, so I can ignore interactions with the outside. (Don't ask me how I know what's going on in there in that case, though.)

I have also an Aberdeen Mfg. Mk. II "Neat-fingered" Maxwellian demon, which is capable of instantaneously reversing all the velocities of the particles in the small assemblage (i.e., it can flip the sign of n velocity degrees of freedom). If I had a bigger research budget, I could have bought a Mk. V "Vast and Considerable" demon, which could reverse the whole assemblage's N velocity degrees of freedom, but I don't have to tell you about grants these days.

Now, with the Mk. V, I'd know what to expect: it's the old familiar myth about time's arrow running backwards: sugar spontaneously crystallizing out of sweetened coffee, forming granules and leaping out of the cup into the tea-spoon, etc. But the Mk. II isn't capable of reversing the arrow of time for the whole assemblage, just for part of it. And so there are N-n degrees of freedom in the larger assemblage whose arrow of time points the same way as before. So what happens?

My intuition is that at first the arrow of time is reserved in the small assemblage, leading to the local equivalent of coffee unsweetening. ("At first" according to who? Don't go there.) Eventually, however, interactions with the N-n unreversed degrees of freedom should bring the n degrees of freedom back into line. If interactions are spatially local, then I imagine the time-reversed region gradually shrinking. Mythologically: The sugar crystallizes and forms granules, say, and even starts to leap out of the cup, but neither the air molecules nor the spoon are in the right place at the right time to exactly take them back to sugar-jar, so they spill and make a mess, etc. More generally, an observer within the larger assemblage will first see a small region where, bizarrely, things happen in reverse, then a kind of hard-to-describe crawling molecular chaos, and then a restoration of the ordinary macroscopic natural order, albeit from a weird starting point. But this guess may be excessively shaped by the fluctuation-dissipation theorem. Does a single arrow of time have to get established at all? If so, how long does it typically take? (Intuition again, this time from large deviations: exponential in 2n-N.) Can the n reversed degrees of freedom ever impose their direction on the whole assemblage?

Somebody must have already looked into all this. Where?

Update, later that afternoon: I was probably unconsciously remembering this post by Sean Carroll. (Sean was very polite in pointing this out.) Also, John Burke answers my final "where" question "Budapest", which sounds about right.


Posted at January 10, 2009 13:45 | permanent link

Three-Toed Sloth