I How strong is the justification for low probability extreme thermal fluctuations?

[syed]

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I asked a question related to a table levitating but I am going to try to be specific about my question after one of the forum mentors stated I should make my question more specific (although I'm still not sure why one couldn't have asked if a table levitating is possible according to physics).

Specifically, I am interested in knowing how much justification we have for an extreme low probability thermal fluctuation that results in a "miraculous" event compared to, say, a dice roll. Does a low-probability dice roll carry the same level of experimental evidence or justification as a low-probability thermal fluctuation (assuming the probabilities are the same)? For this question, let's assume that the event we are considering is a marble statue of a human spontaneously moving its hand, for a moment in time.

Presumably, the chance of a marble statue spontaneously moving its hand would involve a lot of atoms suddenly moving in a particular direction. The probability of this would be extremely low. Let's call it P. Now, imagine a fair dice rolling on 6 many, many times in a row, such that the probability of this is also the same value P.

Is the experimental evidence that we have for establishing these probabilities the same for both events? I ask because let's assume that for some as of yet unknown reason, the state of particles corresponding to the marble statue spontaneously moving (which would be an extreme thermal fluctuation) was exactly 0. If so, from what I can gather, all the experiments that have been done to confirm statistical mechanics would still be just as correct. In other words, how do we know that the probability of states like this occurring is extremely minimal rather than 0? Can there be an experiment that would distinguish between these two? And more importantly, is the justification of assigning a minimal non zero probability to this kind of event on par with the justification we have for assigning the same probability to a dice rolling many times such that its probability was P?

 

[PeterDonis]

I am interested in knowing how much justification we have for an extreme low probability thermal fluctuation that results in a "miraculous" event compared to, say, a dice roll.

Our justification is that our current known laws of physics say this is possible, just extremely, extremely, extremely unlikely.

Is the experimental evidence that we have for establishing these probabilities the same for both events?

I'm not sure what you mean. Nobody has ever done that many die rolls. We have no direct experimental evidence for either probability, and we're not making use of such evidence to compute either probability. We're making use of our current known laws of physics, as they apply to each type of scenario.

The experimental evidence for our current known laws of physics is extremely strong. That's why we listen to what those current known laws of physics tell us, even about questions like the one you're asking, where the answer they give seems outlandish. Our current known laws of physics tell us lots of things that seem outlandish if you don't understand how the laws work. Try explaining to someone in the 19th century that a device you can carry in your hand can let you speak to anyone on the planet anywhere by punching in some numbers, can let you look up all kinds of factual information in seconds, and can tell you where you are anywhere on the planet to an accuracy of a few meters. They'd think such claims outlandish. But those are all things your smartphone can do because the people who designed it understood how our current known laws of physics can make such outlandish things possible.

let's assume that for some as of yet unknown reason, the state of particles corresponding to the marble statue spontaneously moving (which would be an extreme thermal fluctuation) was exactly 0.

This assumption makes no sense unless you have an alternate theoretical model to our current laws of physics, that makes such a prediction. You don't.

how do we know that the probability of states like this occurring is extremely minimal rather than 0? Can there be an experiment that would distinguish between these two?

Of course. Just roll enough dice, or do the equivalent with the marble statue. Such an experiment might take an unworkably long amount of time, but it's possible in principle, according to our current known laws of physics.

is the justification of assigning a minimal non zero probability to this kind of event on par with the justification we have for assigning the same probability to a dice rolling many times such that its probability was P?

Both are based on our current known laws of physics, so yes.

 

[PeterDonis]

This assumption makes no sense unless you have an alternate theoretical model to our current laws of physics, that makes such a prediction. You don't.

@syed, I strongly recommend that you read and re-read this particular statement in my post until you thoroughly understand it. Claims like the one you made that I made this statement in response to are personal speculation and are off limits here. We can't discuss hypotheticals that violate the laws of physics, and that's what your hypothetical does, because it makes a claim that's different from what our current known laws of physics tell us. We can't discuss such hypotheticals here, and if you continue to try to pose them, the thread will be closed and you'll receive a warning. Please take heed.

 

[syed]

Our justification is that our current known laws of physics say this is possible, just extremely, extremely, extremely unlikely.


I'm not sure what you mean. Nobody has ever done that many die rolls. We have no direct experimental evidence for either probability, and we're not making use of such evidence to compute either probability. We're making use of our current known laws of physics, as they apply to each type of scenario.

The experimental evidence for our current known laws of physics is extremely strong. That's why we listen to what those current known laws of physics tell us, even about questions like the one you're asking, where the answer they give seems outlandish. Our current known laws of physics tell us lots of things that seem outlandish if you don't understand how the laws work. Try explaining to someone in the 19th century that a device you can carry in your hand can let you speak to anyone on the planet anywhere by punching in some numbers, can let you look up all kinds of factual information in seconds, and can tell you where you are anywhere on the planet to an accuracy of a few meters. They'd think such claims outlandish. But those are all things your smartphone can do because the people who designed it understood how our current known laws of physics can make such outlandish things possible.


This assumption makes no sense unless you have an alternate theoretical model to our current laws of physics, that makes such a prediction. You don't.


Of course. Just roll enough dice, or do the equivalent with the marble statue. Such an experiment might take an unworkably long amount of time, but it's possible in principle, according to our current known laws of physics.


Both are based on our current known laws of physics, so yes.

When you say that we can't actually do the experiment with respect to dice rolls, this wouldn't be correct, atleast in terms of running an experiment that would be an analogue to it. For example, we can run a random generator and have it output a sequence, such that the probability of that particular sequence is about the same as the probability of the particles in the marble statue moving at the same time. Let's suppose the computer program generates a random digit between 0 - 9 for each element in the sequence. We can then actually, physically, observe this output. The "kind" of output that we do observe (which will be random looking) is no different in kind from a sequence of all 6s for example.

On the other hand, we have never observed an extreme thermal fluctuation of the kind proposed in the marble statue event. So, isn't there a different level of justification for the two kinds of scenarios I mentioned?

And again, I am not saying that physics is incorrect. I am asking if there is a difference in the level of justification for both of these kinds of results. More generally, I am asking for the empirical justification for the kinds of microstates that I mentioned being physically possible. Please don't misinterpret what I'm writing.

 

[PeterDonis]

When you say that we can't actually do the experiment with respect to dice rolls, this wouldn't be correct

I said it would take an unworkably long amount of time. That's because of the number of rolls that would be required to test for a small enough probability P to be comparable to the probability of the table levitation scenario that was discussed in the previous thread.

For example, we can run a random generator

That would still take an unworkably long amount of time, since it's just the computer equivalent of rolling dice.

we have never observed an extreme thermal fluctuation of the kind proposed in the marble statue event

Nor have we observed a sequence of all 6's on dice that is long enough to have the same tiny probability. Do you have any idea how long such a sequence of all 6's would have to be? Have you tried to calculate it? Note that @Nugatory gave a back of the envelope estimate of a probability in the previous thread. Have you tried to calculate how many 6's in a row it would take to test that such a sequence occurs with that same probability?

I am not saying that physics is incorrect. I am asking if there is a difference in the level of justification for both of these kinds of results.

And I've given you the answer: no, there isn't. Both are justified by the laws of physics as applied to those particular scenarios.

More generally, I am asking for the empirical justification for the kinds of microstates that I mentioned being physically possible.

And I've given it to you: the laws of physics say so.

Please don't misinterpret what I'm writing.

I'm not. You're just refusing to accept the answers I've given you. There's no point in continuing discussion on that basis: I'd just keep repeating "that's what the laws of physics say" and you'd keep asking "what's the justification?", round and round in a circle. That would be a waste of my time and yours.

Thread closed.

 

[Nugatory]

For example, we can run a random generator and have it output a sequence, such that the probability of that particular sequence is about the same as the probability of the particles in the marble statue moving at the same time.

You have not thought through the implications of the question in post #16 of your other thread. Say that we set this random number generator to work. How will observing its output permit us to distinguish between “probability zero” and “probability very small but not zero”?

 

[Dale]

> the probability of this is also the same value P.
>
> Is the experimental evidence that we have for establishing these probabilities the same for both events?

They are the same by definition. Experimental evidence is not relevant.

> In other words, how do we know that the probability of states like this occurring is extremely minimal rather than 0? Can there be an experiment that would distinguish between these two?

There cannot be a repeatable experiment that distinguishes between the two probabilities that you described earlier because you constructed them to be equal.

However, there can be an experiment which distinguishes between any two different probabilities.

[Here][1] is an online statistical power calculator for this purpose. To detect the difference between P=0.2 and 0.1 requires a sample size of 398. To detect the difference between P=0.02 and P=0.01 requires a sample size of 4636. To detect the difference between P=0.001 and P=0.002 requires a sample size of 47020. Etc.

The online calculator may run into numerical issues with excessively small probabilities, but the principle remains: for any two distinct probabilities there exists a sample size such that an experiment with that sample size would repeatably detect the difference (with 95% confidence and 80% statistical power).

[1]: https://clincalc.com/stats/samplesize.aspx