I wouldn't have though it had to be. If the probability is 0, it means the particle or system isn't going to be in that state. It is outside the phase space, even if outside refers to a bubble. Meanwhile on your way to p=0 the integral of p Log(p) has long gone to 0 too. It is like trying to calculate the protein content in your block of swiss cheese, the holes may make you calculation more complicated, but you don't actually have to count the protein content of the holes.
You're right that as p tends to zero, p Log p does actually tend to zero instead of the ineffable negative infinity. I don't know why I never noticed that earlier. (But my lecturer said so, and he's really smart! Will an argument from authority suffice for now?
)
On the question of open and closed system, I wonder if a parable could help.
Special Cyclists claim bicycle tyres must have been specially inflated by God. The proof is that bicycle tyres contradict Boyle's Law. Boyle's law states that for a fixed mass of gas at constant temperature, the pressure and volume are inversely proportional. You increase the pressure and the volume decreases. But with bicycle tyres and balloons when you increase the pressure the volume increases. It is God that giveth the increase 1Cor 3:7. Theistic Inflaters (and atheistic cyclists) often reply that this is rubbish, Boyle's Law is only applies to ideal gases. Having to pump up your bicycle tyre is hardly ideal. Yet for the Special Cyclist, when God created the heavens and the earth and everything in them, he said that it was very good, so the gas must be ideal. Or Theistic Inflaters can argue that it applies at constant temperature, whereas you can feel the bicycle pump getting hot (fires of hell the Special Cyclists reply). The argument gets a bit closer to the problem when T.I.s say that pressure and volume are only inversely proportional for a fixed mass of gas, bicycle tyres are an open system because the pump is adding more gas. No reply the special cyclists, if it was an open system the tyre would leak. Or more aptly, they reply that according to Johnny 'Bicycle' Safari, from Answers in Cyclists, who should know because he has a Ph.D. in Physical Chemistry, Boyle's law applies to open systems too. So pressure and volume must be inversely proportional. Of course it does, the problem is taking a description of Boyle's law in a closed system and trying to apply it to an open system.
I am not sure if this is a separate problem or another aspect of the problem in the parable. The SLoT is very complex physics and most people who have come across it have been given a very simplified from, or an illustration of the SLoT in terms of a common effect, that disorder tends to increase (or in the parable the volume of gas in a piston decreases with pressure). And they mistake the simplified description, in some circumstances, for a universal law that applies everywhere in every circumstance.
Hehe, it's a cute little parable.
The first problem (with the creationist argument - not your parable) is that they aren't being clear just what they're talking about. If they said that
abiogenesis violated the SLoT, they might just have a shot at things. I haven't actually thought a lot about that situation yet, except to say that non-equilibrium chemistry is actually quite firmly known, if somewhat arcane - the Belousov-Zhabotinsky reaction is a fine example of a "cyclic" chemical reaction; the concentrations of the various reactants oscillate with a well-defined period (almost like the position of a pendulum), so that if one defines the up-swing as an increase in entropy, the concomitant down-swing must be a decrease in entropy!
Abiogenesis is then really a problem of unusually arcane non-equilibrium chemistry, a topic on which neither I nor Safarti has any experience (even though we both do / have done quite a bit of work in the general area of physical chemistry).
As an interlude, the SLoT really
is about disorder in a general sense. In its most utterly fundamental statement, the SLoT is a simple consequence of the fact that
causality respects probability. Interacting systems are driven into their most probable state; it is simply unfortunate that our cognitive circuits have evolved over the millenia to associate such most-probable-states with disorder. The genius of the SLoT is in recognizing that the state in which temperatures are equal is, in fact, the most probable state.
Thus, it really may not be too bad an analogy to say that the clutter in a room demonstrates the Second Law of Thermodynamics. In a sense, there are more immediate, proximate mechanisms that cause the disorder to build up; blaming it on the SLoT is a little like blaming it on quantum mechanics, like using a sledgehammer to pull out a tooth. But it is a good analogy in the sense that, say, the components of the immune system can be compared to the parts of an army: good homologous arguments.
The error is more subtle (and so is my thinking since two or three years ago when I first contributed to this thread). We'll compare two thought experiments. The first is a control, or placebo: it demonstrates normal thinking. Imagine a kitchen table and an egg. Because of gravity, the configurations in which the egg is on the table are far less likely than the configurations in which the egg is on the floor. Furthermore, again because of gravity (and the legendary frailty of egg shells) the configurations in which the egg is whole are again far less likely than the configurations in which the egg is cracked and messed up. As such, our intuition tells us that it's very likely for an egg to roll off the table, fall to the floor and smash; it is less likely but still thinkable for an egg to roll off the table, fall to the floor but not break; but it is quite ridiculous for the egg to be smashed on the floor, reconstitute itself into a whole egg, and then jump onto the table. Furthermore, the SLoT can (with a sufficient amount of tedium / funding / PhD slaves) be applied to the situation rigorously.
Now imagine loving parents having triplets. Unfortunately, though loving, they are not very responsible, and they end up being bitten by a radioactive spider while performing an experiment with gamma rays on a spaceship just before it gets hit by a solar flare. The ensuing genetic mutations affect all of their children. One child grows up with all sorts of enhancements like web glands and wall-crawling bristles; one child grows up with normal anatomy (though he occasionally turns green and strips naked when he's angry); and one child grows up with his internal organs deformed or absent altogether (although he gains the mystical ability to stretch himself and inexplicably stay alive).
Now our Spidey-boy's mutations are an example of what the creationists consider an increase in order, the addition of function via genetic change. Our poor old juvenile Reed Richard's mutations on the other hand are an example of what the creationists consider a decrease in order, the reduction of function via genetic change. Now
since the SLoT always causes order to decrease,
we should expect to see more Reed Richards than Peter Parkers,
and as generation after generation loses function, creatures should devolve, but they haven't, so if evolution explains today's biodiversity, it must violate the SLoT!
There are two errors in the creationist's argument, one of which is unique to the invocation of the SLoT, and one which is generic in all their arguments. You'll notice that their argument is predicated on Reeds being more common than Peters. The problem is in their invoking the SLoT to justify that. In our egg example, there is an energetic difference between an egg on table and an egg on floor. Boltzmann factors (and the assumption of thermal equilibrium) allow us to convert those energetic differences into probabilities, and the SLoT then unassumingly predicts that the more likely state is, well, more likely.
But what, precisely, are the energetics of giving birth to a Reed instead of a Peter? Does it actually take more energy to give birth to a slightly more "ordered" (whatever that might mean) offspring? What are the thermodynamics of an Australopithecus having young with a slightly larger cranium than a slightly smaller cranium? Can I make a perpetual motion machine out of a reptile whose young have feathers instead of scales? Clearly not, but I could certainly make a perpetual motion machine out of eggs that spontaneously jump onto tables without external energy input.
So the first error is using the SLoT to assign those probability distributions. The reasoning creationist will, in response, drop the SLoT. "Okay, so apart from abiogenesis where chemicals really do need energy to react," (hence my earlier caveat), "mutations don't have thermodynamic probability distributions. But surely they
still do have
some probability distribution! And harmful mutations simply
are more likely than beneficial mutations, even if that's not justified by the SLoT - it's just common sense!"
You see, the argument has stripped down and revealed that it was simply the "beneficial mutations are less likely" argument hiding behind a chemist's lab coat. And now all the usual evolutionist replies apply - beneficial mutations only need to happen once, the
retention of beneficial mutations is far more likely than the retention of harmful mutations, neutral mutations can be exapted, as can spandrels and vestiges, etc. etc. ad infinitum ad nauseam.
