Freethought & Rationalism Archive

Afghan · 07-20-2002, 03:37 PM

I don't see that the rate of fluctuation is particularly relevant, at least not to the principle of the argument. All you need to do is go further back in time to T-50 say and it still seems more likely that the present state evolved from a more entropic one, albeit further in the past and more slowly, just as, in fact, the increase of entropy in the future will take longer. The problem still remains though, it is likely that entropy is greater in the past as well as in the future. It is closing on time symmetric.

I am not sure an appeal to outside intervention is particularly helpful at all. How does one judge the likelihood of that? You can't. It strikes me as a rather, how shall I say, strong anthropic argument.

tronvillain · 07-20-2002, 09:32 PM

The rate of fluctuation is completely relevant. If the probability of a black/white switch is one in a hundred, and you walk into a room with ninety-nine white beans and three black beans, the most likely explanation is that you started with all or mostly white beans.

The simple fact is that any significant reduction in entropy by chance is so ridiculously improbable that the hypothesis is not worth considering in comparison to the hypothesis that the reduction was accompanied by an increase in entropy elsewhere. That is, in fact, what we observe.

If you merely wish to observe that given sufficient time, a significant decrease in entropy could occur by chance, I have no argument. In the real world, an appeal to chance to explain low entropy simply will not do, unless one proposes a truly staggering timescale, one which dwarfs the time which would dwarf the age of the universe.

Afghan · 07-21-2002, 04:23 AM

If I might use an analogy. Were I to be sitting in my house one day when all of a sudden a jet engine fell through my roof to my general consternation, would it be incredibly unlikely that it fell off a plane. Certainly, one would hope that such catastrophic aircraft failures are highly infrequent. The probability of it occurring precisely over my house seem even more vanishingly small. If I were to use your line of reasoning, I would conclude that is was not at all likely that the engine fell from an aircraft flying over my house because this is an incredibly unlikely event.

However, it is a conditional probability problem: what are the chances that an engine would off a plane flying over my house? Very small. But what are the chances of the same happening given that an engine falls through my roof. Very much higher. The two events are not independent. The fact of one prejudices the probability of the other.

If we have 100 beans, there are 100 times as many configurations of one black bean than no black beans. There are about 50 times as many again of two black beans. Whilst it might be more likely that a low entropy configuration might result from a configuration of 100 whites than any particular configuration within a higher entropy state, there are simply a lot more of these configurations that it could result from.

Please don't make me do the maths rigorously. Please!

tronvillain · 07-21-2002, 06:30 PM

Permit me to quote myself:

Quote:

The simple fact is that any significant reduction in entropy by chance is so ridiculously improbable that the hypothesis is not worth considering in comparison to the hypothesis that the reduction was accompanied by an increase in entropy elsewhere. That is, in fact, what we observe.

In the case of your jet engine, it is far more likely that it fell from a plane passing overhead than it is that it spontaneously assembled in mid-air. This is the difference between someone setting up the hundred white beads for you to observe and a hundred beads just happening to be white when you observe them.

Now, your argument appears to be that while there is only one way of having one hundred white beads, there are something like 4895 ways of having ninety-eight white and two black beads, and that this difference in P(H) makes up for the difference in P(E/H). Is that right?

tronvillain · 07-22-2002, 11:17 AM

Anyway, I've been discussing this with a friend of mine, and this is what we came up with:

I've had some more thoughts about Afghan's beans.

Yes?

Considering the case of 99 white/1 black we have some information about the possible prior state of 98/2: One of those black beans is the one we're holding. So there are only 99 possible prior cases, not 4000+

Not necessarily.

Absolutely, if we're considering the model where we call a state change one bean changing color.

Ah, I was using actual probability again.

The state change model preserves the relevant features. Once we figure out that model, we can extend it to more complex models if necessary.

What did we calculate 100 choose 2 to be though?

4 thousand and something. 50 * 99 = 4950

Right.

But that's not accurate, because we know ahead of time one of the two beans that were black in the previous state. So there are only 99 prior states.

If we use the actual probabilities and multiply by the number of state for each (99 and 4950) we get 0.91 (0.18 + 0.73) which is still above 0.34

Let's use the simpler model. However we're still at 1.98:1 in favor of the prior state being 98/2

Pardon?

There's 1 prior state of 100/0, with probability 1 of changing to 99/1. There's 99 prior states of 98:2 with each having a probability of 1/99 of changing to 99:1. Hmm... so the odds are the same of having either state being prior. But that's starting to make sense because when you're numbering the beans, all states are equally probable.

Right...

Hmmmm... hang on.

Looking at the more complex model 99 is more likely to come from 98 than 100, but 98 is still more likely to go to 97 than 98, which is all entropy requires. And 100 can't go anywhere but down to 99, and once it's at 99 it's more likely to go to 98 than 100 again and so on.

Wait... The Afghan Bean problem just proves that when you evolve the system, you *always* go to higher entropy. That means, in effect, that the past lower entropy state is unrecoverable. Which proves the time-asymmetry of thermodynamics.

Interesting.

So the AB problem isn't even a paradox. It just proves you can't thermodynamically capture the past. Which you can't, not by examining the microworld anyway.

Trying to switch it around the way Afghan does to find the past doesn't work. If you do it his way and then play it forward you will have a succession of UNLIKELY events all occurring, like tails coming up every time.

Which is our actual observation.

I could have sworn we observe likely events occurring. After all, tails generally does not come up every time.

If we observe the past in time-reversal (via memory or via a macroscopic movie played backwards), we see unlikely events happening.

Ah, in the sense that the particular sequence of heads and tails is unlikely.

Given the present, the actual past is unlikely. The future is more likely, which is why we see an arrow of time.

Now that's an interesting thought. Moving randomly from the present will produce future, simply because reproducing the past is unlikely.

Right. And, unsurprisingly, when we label the beans to really examine the situation closely, we lose thermodynamics altogether and we get time symmetric behavior

Anyway, I'll see what I can do with all of this.

Ok. Well, perhaps we can understand thermodynamics a little better. I wish I could run it by my friend with the PhD. in physics though.

Well, if we're wrong, we're wrong.

[ July 22, 2002: Message edited by: tronvillain ]</p>

tronvillain · 07-22-2002, 11:25 AM

Afghan:

Quote:

Please don't make me do the maths rigorously. Please!

Do you think you could? I suppose you could use Bayes' Theorem but I think it would require something like 150 calculations to work out P(e/~h).

tronvillain · 07-22-2002, 11:31 AM

Here's another version:

Quote:

Consider a jar with 100 beans in it, some white, some black. Every once in a while a state transition, where one arbitrary bean changes color.

It's obvious that, according to thermodynamics, regardless of the initial state, you'll eventually see about 50 white beans and 50 black beans.

I pick up such a jar, examine the beans and I find 99 white beans and 1 black bean. Here's the apparent paradox: It is asserted that it's more likely that the state prior to this state was 98 white : 2 black than it was 100 white : 0 black. Being mathematically challenged, I haven't actually looked at the numbers.

I maintain, however, that this is not a "paradox"; it just shows that thermodynamcis is time-asymmetric, which matches our observations: The actual past is unlikely given the present. Furthermore, it shows that the specific past is not recoverable from the present merely by examining the laws of the microworld. These observations are in accord with our experience.

Indeed, if the beans are numbered, then it turns out that the prior state of 100w : 0b is equally probable to the prior state of 98w : 2b. Here's the proof:

At the previous state, since the beans are numbered, we know one of two situations obtained: Either 1) the black bean was white and it changed, or 2) one of the 99 white beans was black, and it changed. In situation 1, the probability that it would have changed to the current 99w : 1b state is unity. The probability that one of the 99 possible cases of situation 2 would have changed to 99w : 1 black is 1/99, since we know our own black bean didn't change. Therefore if we weigh the probabilities, we find that 1 * 1 = 1, and 99 * 1/99 = 1 and either situation is equally probable.

But this just means (again, as observed) that if we actually enumerate and separately identify the beans, all possible cases are equally probable, which is again in accord with our experience.

Afghan · 07-22-2002, 11:41 AM

I'm actually working on the proof right now. I'll put it up on my webpage when it is done and link to it. It'll probably take a couple of days to get there. My background is, I'm afraid, in mathematics rather than physics so my definition of entropy owes more to information theory than thermodynamics but that was already expected.

It has to be said though, the conversation with your friend that you posted captured the essence of what my proof shows - I was just having a really hard time elocuting my thoughts on the subject. It is also quite true that time symmetric entropy (if that is the phrase) is not really all that surprising nor contradictory. Low entropy states are unlikely to develop and usually we would expect to see high entropy states. Therefore when we do find low entropy states we would expect the entropy of the states close to them in time to have a higher entropy.

Interestingly enough, I've also got another type of magic bean called "Afghan's Double Magic Bean" that displays entropic properties that are recognisably closer to those of the real world than the simple Magic Bean. Storing the new variety is proving to be problematic, however, as they reproduce by binary fission every hour or so.

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07-20-2002, 03:37 PM	#21
Afghan Senior Member Join Date: Mar 2001 Location: Afghan is a non-local variable Posts: 761	I don't see that the rate of fluctuation is particularly relevant, at least not to the principle of the argument. All you need to do is go further back in time to T-50 say and it still seems more likely that the present state evolved from a more entropic one, albeit further in the past and more slowly, just as, in fact, the increase of entropy in the future will take longer. The problem still remains though, it is likely that entropy is greater in the past as well as in the future. It is closing on time symmetric. I am not sure an appeal to outside intervention is particularly helpful at all. How does one judge the likelihood of that? You can't. It strikes me as a rather, how shall I say, strong anthropic argument.

07-20-2002, 09:32 PM	#22
tronvillain Veteran Member Join Date: Oct 2000 Location: Alberta, Canada Posts: 5,658	The rate of fluctuation is completely relevant. If the probability of a black/white switch is one in a hundred, and you walk into a room with ninety-nine white beans and three black beans, the most likely explanation is that you started with all or mostly white beans. The simple fact is that any significant reduction in entropy by chance is so ridiculously improbable that the hypothesis is not worth considering in comparison to the hypothesis that the reduction was accompanied by an increase in entropy elsewhere. That is, in fact, what we observe. If you merely wish to observe that given sufficient time, a significant decrease in entropy could occur by chance, I have no argument. In the real world, an appeal to chance to explain low entropy simply will not do, unless one proposes a truly staggering timescale, one which dwarfs the time which would dwarf the age of the universe.

07-21-2002, 04:23 AM	#23
Afghan Senior Member Join Date: Mar 2001 Location: Afghan is a non-local variable Posts: 761	If I might use an analogy. Were I to be sitting in my house one day when all of a sudden a jet engine fell through my roof to my general consternation, would it be incredibly unlikely that it fell off a plane. Certainly, one would hope that such catastrophic aircraft failures are highly infrequent. The probability of it occurring precisely over my house seem even more vanishingly small. If I were to use your line of reasoning, I would conclude that is was not at all likely that the engine fell from an aircraft flying over my house because this is an incredibly unlikely event. However, it is a conditional probability problem: what are the chances that an engine would off a plane flying over my house? Very small. But what are the chances of the same happening given that an engine falls through my roof. Very much higher. The two events are not independent. The fact of one prejudices the probability of the other. If we have 100 beans, there are 100 times as many configurations of one black bean than no black beans. There are about 50 times as many again of two black beans. Whilst it might be more likely that a low entropy configuration might result from a configuration of 100 whites than any particular configuration within a higher entropy state, there are simply a lot more of these configurations that it could result from. Please don't make me do the maths rigorously. Please!

07-22-2002, 11:17 AM	#25
tronvillain Veteran Member Join Date: Oct 2000 Location: Alberta, Canada Posts: 5,658	Anyway, I've been discussing this with a friend of mine, and this is what we came up with: I've had some more thoughts about Afghan's beans. Yes? Considering the case of 99 white/1 black we have some information about the possible prior state of 98/2: One of those black beans is the one we're holding. So there are only 99 possible prior cases, not 4000+ Not necessarily. Absolutely, if we're considering the model where we call a state change one bean changing color. Ah, I was using actual probability again. The state change model preserves the relevant features. Once we figure out that model, we can extend it to more complex models if necessary. What did we calculate 100 choose 2 to be though? *4 thousand and something. 50 99 = 4950 Right. But that's not accurate, because we know ahead of time one of the two beans that were black in the previous state. So there are only 99 prior states. If we use the actual probabilities and multiply by the number of state for each (99 and 4950) we get 0.91 (0.18 + 0.73) which is still above 0.34 Let's use the simpler model. However we're still at 1.98:1 in favor of the prior state being 98/2 Pardon? There's 1 prior state of 100/0, with probability 1 of changing to 99/1. There's 99 prior states of 98:2 with each having a probability of 1/99 of changing to 99:1. Hmm... so the odds are the same of having either state being prior. But that's starting to make sense because when you're numbering the beans, all states are equally probable. Right... Hmmmm... hang on.** Looking at the more complex model 99 is more likely to come from 98 than 100, but 98 is still more likely to go to 97 than 98, which is all entropy requires. And 100 can't go anywhere but down to 99, and once it's at 99 it's more likely to go to 98 than 100 again and so on. Wait... The Afghan Bean problem just proves that when you evolve the system, you always go to higher entropy. That means, in effect, that the past lower entropy state is unrecoverable. Which proves the time-asymmetry of thermodynamics. Interesting. So the AB problem isn't even a paradox. It just proves you can't thermodynamically capture the past. Which you can't, not by examining the microworld anyway. Trying to switch it around the way Afghan does to find the past doesn't work. If you do it his way and then play it forward you will have a succession of UNLIKELY events all occurring, like tails coming up every time. Which is our actual observation. I could have sworn we observe likely events occurring. After all, tails generally does not come up every time. If we observe the past in time-reversal (via memory or via a macroscopic movie played backwards), we see unlikely events happening. Ah, in the sense that the particular sequence of heads and tails is unlikely. Given the present, the actual past is unlikely. The future is more likely, which is why we see an arrow of time. Now that's an interesting thought. Moving randomly from the present will produce future, simply because reproducing the past is unlikely. Right. And, unsurprisingly, when we label the beans to really examine the situation closely, we lose thermodynamics altogether and we get time symmetric behavior Anyway, I'll see what I can do with all of this. Ok. Well, perhaps we can understand thermodynamics a little better. I wish I could run it by my friend with the PhD. in physics though. Well, if we're wrong, we're wrong. [ July 22, 2002: Message edited by: tronvillain ]</p>

07-22-2002, 11:41 AM	#28
Afghan Senior Member Join Date: Mar 2001 Location: Afghan is a non-local variable Posts: 761	I'm actually working on the proof right now. I'll put it up on my webpage when it is done and link to it. It'll probably take a couple of days to get there. My background is, I'm afraid, in mathematics rather than physics so my definition of entropy owes more to information theory than thermodynamics but that was already expected. It has to be said though, the conversation with your friend that you posted captured the essence of what my proof shows - I was just having a really hard time elocuting my thoughts on the subject. It is also quite true that time symmetric entropy (if that is the phrase) is not really all that surprising nor contradictory. Low entropy states are unlikely to develop and usually we would expect to see high entropy states. Therefore when we do find low entropy states we would expect the entropy of the states close to them in time to have a higher entropy. Interestingly enough, I've also got another type of magic bean called "Afghan's Double Magic Bean" that displays entropic properties that are recognisably closer to those of the real world than the simple Magic Bean. Storing the new variety is proving to be problematic, however, as they reproduce by binary fission every hour or so.

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