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 2*n* 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 2*N*. 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 2*n*-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 by crshalizi at January 10, 2009 13:45 | permanent link