posted
I’m reading Reinventing the Sacred, by Stuart Kauffman, and have a question about one of his arguments against the reducibility of thermodynamics to mechanics.
He uses the example of a drop of ink in a dish of water to demonstrate the argument for time’s arrow, stating that the droplet would diffuse to a uniform distribution naturally through time. At no point will it un-diffuse back into the droplet. Entropy in this closed drop-water-dish system increases or stays the same, and can never decrease. This one way flow from droplet to diffuse system demonstrates time’s arrow.
(He details it further by dividing the dish into tiny volumes, and states that any possible distribution of ink molecules in the dish is one microstate. The full arrangement of microstates that make up the droplet in the center is one macrostate, and so is the full arrangement of microstates that make up the fully diffused ink. He says “the ink in the center of the petri dish [as a drop] has fewer microstates, hence lower entropy, and is less probable than the higher-entropy, diffuse distribution to which the ink drop evolves. In other words, by random collisions, the ink system flows from the less probable to more probable macrostate.” I’ve never heard of entropy described as probabilities of macrostates before, but it seems to make sense.)
And here’s the part that threw me for a loop:
quote:But the philosopher David Alpert has pointed out a problem with this wonderful story. Given the time reversibility of Newton’s laws, if the system started as the improbable, highly ordered ink drop and Newton’s equations were solved backward in time, the drop would again diffuse across the petri dish from less-probably macrostates to more-probable macrostates. Entropy would appear to increase with time running backwards! What happened to the forward arrow of time, as reduced from classical thermodynamics to statistical mechanics? (emphasis in the original)
I have trouble wrapping my head around this argument. It seems to me that this “running backwards in time” is a non-physical experiment. It can’t be done, and there was clearly something putting the ink in the drop in the center at the beginning of the forward-time experiment. The ink, at no point in the past, was diffused throughout the water in the dish. It was added at time zero, and then diffusion kicked in and time’s arrow took over. Am I missing something here? Can you think about “running backwards in time” without thinking about how the drop really got to be a drop in the first place?
I’ll probably have more questions like this. He seems to have interesting ideas, that I’m not sure I agree with, but they’re making me think. He seems to really be stuck on this idea that reductionism (i.e. the concept that all of life, the universe, and everything can be reduced to a basic set of physical laws) is out to destroy morals, beauty, and everything wonderful in the universe. I don’t know many people who claim to be full reductionists (Raymond Arnold, I seem to remember you expressing something like this in a recent post…something about free will…), and I don’t quite see why he thinks this absolute reductionism is so horrible. I really hope he gets to that reasoning later in the book. I’ll let you guys know what I think as I keep reading.
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posted
The laws of physics are not, in fact, symmetric under time reversal. Common mistake by people who only read Newton, who is not forgotten but who, after all, described only a high-level approximation to the true laws.
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quote:Originally posted by The White Whale: I have trouble wrapping my head around this argument. It seems to me that this “running backwards in time” is a non-physical experiment.
I'm obviously reading the quote without the complete context, but I read that as being precisely his point. The fact that the math would seem to work in both directions (which is physically impossible) is proof that thermodynamics cannot be reduced to mechanics.
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quote:Originally posted by The White Whale: I’m reading Reinventing the Sacred, by Stuart Kauffman, and have a question about one of his arguments against the reducibility of thermodynamics to mechanics.
He uses the example of a drop of ink in a dish of water to demonstrate the argument for time’s arrow, stating that the droplet would diffuse to a uniform distribution naturally through time. At no point will it un-diffuse back into the droplet. Entropy in this closed drop-water-dish system increases or stays the same, and can never decrease. This one way flow from droplet to diffuse system demonstrates time’s arrow.
What if the ink pools at the bottom? Wouldn't that be un-diffusing?
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quote:Originally posted by The White Whale: I have trouble wrapping my head around this argument. It seems to me that this “running backwards in time” is a non-physical experiment.
I'm obviously reading the quote without the complete context, but I read that as being precisely his point. The fact that the math would seem to work in both directions (which is physically impossible) is proof that thermodynamics cannot be reduced to mechanics.
Ah, that helps.
I was stuck on the idea that you can't run time backward. But, according to classical mechanics, time can be run backward. Is that right?
So, since in the real world time cannot be run backward, the real world (or at least in this case the classical thermodynamic world) is not reducible to classical mechanics.
I really need to go over my foundations of science. I think I didn't pay attention enough the first time through.
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Herblay, the ink would pool at the bottom if it were more dense that the water, but in this case (which I didn't explicitly state) the ink is the same density as the water, and so doesn't pool.
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quote:So, since in the real world time cannot be run backward, the real world (or at least in this case the classical thermodynamic world) is not reducible to classical mechanics.
Well duh. We've known that since 1930. I say again: Classical mechanics is simply wrong on this point; the laws of physics are not symmetric under time-reversal. But since we in any case know that classical mechanics are wrong on any number of other points, this ought neither to shock anyone nor to form the basis of anyone's philosophy.
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posted
Thinking about it, we can rescue the Newtonian reduction by thinking about numbers of microstates. Since Newton's mechanics are fully deterministic, each of the ink-drop microstates must evolve into exactly one fully-diffused microstate. However, we've shown that there are many more fully-diffused microstates than ink-drop ones. Consequently, if you start with a fully-diffused state and run time backwards, it is really quite unlikely that you will see the ink-drop state; the reason being that your initial fully-diffused state has to be one of the ones that an ink-drop state can evolve into, and those are only a small fraction of the full set of diffused states. In other words, when the thought experimenter blithely says "We start with the fully-diffused microstate" he is, all unawares, injecting a huge amount of negative entropy into the problem. (That is, an unlikely state has, by construction, got low entropy.) Well, naturally, when you put in negative entropy you get back low-entropic states! But it does not follow that just any old fully-diffused state will give you back the ink drop if you reverse time. He has to select the state very carefully, and this injects the huge dose of negentropy that he then uses as his conclusion! A classic case of not noticing the circularity of one's argument.
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quote:Originally posted by The White Whale: Herblay, the ink would pool at the bottom if it were more dense that the water, but in this case (which I didn't explicitly state) the ink is the same density as the water, and so doesn't pool.
Generally speaking, this is not true. To diffuse, the two liquids need not have the same densities; they simply need to be mutually soluble. Oil floats on water because it is less dense, yes, but that's merely why it floats (rather than hovering or sinking). The reason it doesn't diffuse is because oil cannot dissolve in water. Alcohols tend to be less dense than water, and they dissolve/diffuse perfectly well.
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posted
Wait, hang on. He's not starting with a fully-diffused state and having it evolve backwards to an ink drop; he's starting with the ink drop and evolving backwards, and nonetheless getting the diffused state. I was addressing the wrong argument by sheer habit. My bad. I'll have to think a bit.
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posted
Slightly off topic, but I have read some interesting articles on the chances of there being reverse causality and how that can be shown through measuring weak forces.
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quote:He details it further by dividing the dish into tiny volumes, and states that any possible distribution of ink molecules in the dish is one microstate. The full arrangement of microstates that make up the droplet in the center is one macrostate, and so is the full arrangement of microstates that make up the fully diffused ink. He says “the ink in the center of the petri dish [as a drop] has fewer microstates, hence lower entropy, and is less probable than the higher-entropy, diffuse distribution to which the ink drop evolves. In other words, by random collisions, the ink system flows from the less probable to more probable macrostate.
This is a pretty reasonable explanation of statistical mechanics which is a widely accepted way to explain the second law of thermo dynamics in a quantitative manner.
The problem with the authors assessment is that he is ignoring the statistical implications of running time backwards. He is correct that from a strictly mechanics perspective, it would be possible for all the ink molecules to diffuse back to the center. But the probability of that ever happening is (without doing the actual calculations) great than 1 in 10 to 50th power. There hasn't been enough time since the beginning of the Universe to expect there to be even 1 in a trillion probability that this has ever happened, hence its a highly reasonable approximation to say it doesn't ever happen.
[ July 30, 2010, 08:46 AM: Message edited by: The Rabbit ]
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quote:Originally posted by Black Fox: Slightly off topic, but I have read some interesting articles on the chances of there being reverse causality and how that can be shown through measuring weak forces.
I've got some data that I can only explain through reverse causality and its really bothering me.
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posted
Ok, I've thought further. It seems to me (taking Newton's mechanics as a basis and noting that it is not perfectly accurate) that if you start with the ink drop and run time backwards, you do not get a fully-diffused state; rather you see the ink go back into the dropper whence it came, which is even less statistically likely than a droplet in a bowl of water. If you run time backwards and see a droplet become diffused, then you could run time forwards again and see a diffused state become a droplet; this is not very likely. Again, all this comes back to the basic point that physics is not time-symmetric, but you cannot demonstrate this fact using Newton alone.
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