John,
I think that I should not add to the confusion of the example problem, but as an aside, I would give a different explanation from the representation of entropy in the following:
"There are a lot of questionable assumptions in that model, such that any system, even one in which velocities are reversed, is presumably closed and thus subject to overall entropy, so the particles are not going to follow their original trajectories back to their original starting points, ..."
I will assume that your statement has to do with disorder entropy and the inclusion of a greater than sign in an equation that mimics the form of thermodynamic entropy. That 'disorder' equation is an assumed equation and not a derived equation. In other words, there is no derived thermodynamic foundational basis for it.
I have defined thermodynamic entropy. I have shown that Boltzmann's introduction of statistical entropy is not a continuation of nor a clarification of thermodynamic entropy. Boltzmann's entropy is the point where it is assumed, without justification, that the 'S' of thermodynamic entropy is a measure of disorder. But even leaving my work aside it is the case that:
For thermodynamic entropy, disorder has not been shown to increase. Its derivation by Clausius assumed ideal conditions that specified absorption of energy at a constant temperature. That assumption is tantamount to saying that disorder was not increased. While I would not argue that disorder does not increase, it certainly does, I would argue that the equal to and greater than signs for disorder entropy are arbitrarily borrowed and not derived.
The property Clausius derived as thermodynamic entropy is not a measure of disorder. It is something that physicists still do not know. I don't say this because of my work, I say it because no physicist can presently explain the property that was derived as Clausius' thermodynamic entropy.
Returning to the quote from the link, that work is in error for reasons that exist in the math. It no doubt is necessary to explain that the error that is in the math is an error of choice of model. Otherwise I might have to explain why the negative sign in the denominator of velocity can't be moved in front of the expression for velocity. In math it is a legal operation. In the model presented, it is not a correct move to make time negative and claim, as is represented by moving the negative sign, that that makes the velocity negative.
James Putnam
