Originally posted by HRG_new
1) The function f(t) = 1/(1+exp(-t)) has been strictly increasing for an infinite amount of time and hasn't reached its maximum yet. Thus the argument "if it had been going on for an infinite amount of time, it would be wound down by now" is faulty.
2) Even in our 4-dimensional world, the 2LoT has been formulated for systems within a region, thus with a boundary. It is highly dubious whether it is true in any infinite universe; there is no reason to believe that the boundary terms would in general go to zero as we inflate the region to cover the whole universe, and try to argue about an "entropy of everything".
In the higher-dimensional world of branes, we obviously can say even less about the "entropy of everything" (or whether the arrow of time has any meaning there).
I agree about your 2d point. Remember that all I was doing was "suspecting", not asserting. Tolman's result is for a closed (positive curvature, finite) space because within standard general relativity that's the only kind (at least among homogenous & isotropic spaces) for which speaking about a cyclic universe makes sense.
Your 1st statement is true but doesn't affect the argument. The semi-Newtonian way of stating Tolman's result (& this is what I worked out in the paper that I mentioned) is that - assuming you can get a closed universe to bounce at some minimum scale factor - entropy increases from one cycle to another. That means that there's an increase in gravitational mass which drives the oscillations to higher & higher amplitudes. (The universe doesn't "run down" but actually gets pumped up by dissipation - a counterintuitive result which is what got me interested in this in the first place.)
There is also a 1976 paper by G. Neugebauer & E. Meier (Annalen der Physik 7,33, 161) in which they develop an exact general relativistic model with bulk viscosity & show that if the parameters of the model obey a certain condition there will be a stage at which the universe will make a final bounce and take off, never to recontract.
Shalom,
George