Chalnoth said:
how did location A know to be at the same temperature as location B if there has never been any contact between the two locations?
I was actually thinking of a more specific argument, because it looks to me like the author of the paper has an answer to this one: that location A being at the same temperature as location B is the "natural" state anyway, so it doesn't need to be explained; what would need to be explained would be if they were
not at the same temperature.
The more specific argument I was thinking of goes like this: suppose I have a system S that is split into two subsystems, A and B, each of which contains an equal portion of S (equal volume, equal numbers of particles, etc.). (We make no assumptions at this point about the relationship between the two subsystems.) The total energy of S is fixed at E. The phase space of S includes a huge region, much huger than any other region, corresponding to thermal equilibrium.
Now suppose we pick, at random, a particular microstate in the phase space of S. The author's argument is that we are overwhelmingly likely to pick a microstate that is in the thermal equilibrium region of the phase space; and on that basis, he claims that it is overwhelmingly likely that the temperatures of the subsystems, A and B, are the same. However, that argument leaves out a crucial step. To see what it is, ask the question: in the particular microstate we picked, how is the total energy E split up between the energies of the two subsystems, which we can call EA and EB? The answer is that it is overwhelmingly likely that EA and EB are
not equal.
Now,
if the subsystems A and B are causally connected, the fact that EA and EB are not equal in a particular randomly chosen microstate doesn't affect the author's claim about the temperature, because that microstate will evolve to another microstate in which EA and EB are
different than the first one--and then to another where they are different again, etc., etc. And
on average, we will find that all of those differences cancel out, so that on average, the energy E is partitioned equally between EA and EB. In other words, the causal connection allows energy to be exchanged between the subsystems A and B, and this energy exchange is what makes their temperatures equal, on average.
But if the subsystems A and B are
not causally connected, they cannot exchange energy, and so if we start in a microstate where EA and EB are different (which is overwhelmingly likely), they will
stay different (because they will each be constant, since the subsystems are isolated). And because EA and EB are different, the temperatures TA and TB will be different as well (because everything else about the two subsystems is the same).
Another way of stating this argument is that the author has assumed a particular Hamiltonian for the total system S, one that induces an evolution on phase space that exchanges energy between the subsystems A and B. But there is another possible Hamiltonian which does
not exchange energy between A and B. (In fact there will be many possible Hamiltonians of each type.) So the real question is,
which Hamiltonian is the right one? And that depends on the causal connection, or lack thereof, between the two subsystems.
