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  • BRADY HARAN: You got another number for us?

  • TONY PADILLA: I have, yeah.

  • Yeah.

  • It's the--

  • well, I'll just write it down, shall I?

  • That's probably the best thing to do.

  • So it is 10 to the 10 to the 10 to the 10 to the 10--

  • and then this is the strange bit-- to the 1.1.

  • OK?

  • This has been claimed to be the largest finite time that

  • has ever been calculated by a physicist

  • in a published paper.

  • This is the paper.

  • It's a bit of a weird paper.

  • It's not had a huge impact or anything.

  • It's about black hole information loss

  • and conscious beings.

  • We won't go down that route.

  • He actually calculates something called the Poincare

  • recurrence time for a certain type of universe within a

  • certain cosmological model.

  • And this is the number that he gets out.

  • So this is the one on I'm on about.

  • Let me just check I got the number of 10's right.

  • Yeah.

  • I did.

  • Here it is, equation 16.

  • So he's put Planck times, millennia, or whatever,

  • because basically, it don't really make any difference

  • whether you use seconds, Planck times, millennia, years

  • when the number is this big.

  • But there's other interesting numbers in here,

  • which is this one here.

  • You got three 10's to the 2.08.

  • That's the Poincare recurrence time for our universe.

  • BRADY HARAN: What's all this about, dude?

  • TONY PADILLA: OK.

  • So Poincare recurrence time.

  • This is something that's arisen

  • from statistical mechanics.

  • Very simply, if we had a pack of cards, and we've only got a

  • finite number of cards in that pack.

  • And let's say we keep dealing each other

  • hands of five cards.

  • Eventually, if we do it for long enough, you're going to

  • get a royal flush, Brady.

  • That's guaranteed to happen.

  • And if you wait long enough again you'll get another one.

  • And so on, and so on, And that's true because there's a

  • finite number of cards.

  • Now what Poincare realized is that if you take a gas, the

  • particles, you can put the gas in a box.

  • Now you can put all the particles in one corner of the

  • box, and then they'll disperse and then they'll move around.

  • But Poincare recurrence tells us is that after a very, very,

  • very, very long time, those particles will eventually

  • return to the corner of the box.

  • You always get a repetition.

  • And it's basically because the thing that controls the

  • evolution of that system only has a finite region of what we

  • call [INAUDIBLE] space, of solution space, that's

  • accessible to it.

  • And so eventually, you always come back to it arbitrarily

  • close to where you started.

  • The time scales are truly enormous before you start

  • expecting it to happen.

  • So you can apply this quantum mechanically as well.

  • So what you're doing in quantum mechanics, what you're

  • really talking about is the evolution of microstates

  • within the quantums, so these sort of quantum building

  • blocks of your system.

  • And they will eventually return--

  • they will evolve, but eventually return to their

  • initial states.

  • And what Don Page in this paper has tried to do is he's

  • actually applied that to various types of universe,

  • models of various types of universe.

  • It's a bit of a cheat, because a universe is what we call a

  • macroscopic object.

  • It's a large object.

  • It's not really a quantum micro state, in some sense.

  • What it is, is it an ensemble, an average, of all the

  • microstates.

  • So what he's tried to do is he's tried to say, OK, I'm

  • going to treat the universe as this average

  • of all these states.

  • But then I'm going to count up all the possible averages and

  • treat them as a sort of microstate in itself.

  • So it's a bit of a cheat, but he gets an extra exponential

  • out from that.

  • BRADY HARAN: Are we talking about, Jupiter's over there,

  • the Andromeda Galaxy's over there, I'm in this room

  • filming you.

  • TONY PADILLA: Yeah.

  • Yeah.

  • All that.

  • OK?

  • Now there's a very large number of possibilities that

  • you can have, but it's finite.

  • And so as the system evolves, it's only got to--

  • and it can only evolve, the system can only evolve through

  • a finite number of possibilities.

  • And eventually, it evolves back to where it started.

  • So actually, it's--

  • I've often heard it said that the universe, for example,

  • will evolve, will expand.

  • Eventually, everything will be spread out very far because of

  • the expansion of the universe.

  • That all objects will have collapsed to form black holes.

  • That then those black holes will evaporate from Hawking

  • radiation, and all you will have is this very sort of

  • bleak landscape of just radiation that's come out

  • these evaporated black holes, that's uniformly distributed,

  • and it'll be very boring.

  • OK?

  • But that isn't the end state of the universe.

  • The end state of the universe is that after these truly epic

  • time scales, you will eventually have a Poincare

  • recurrence, and you'll wind up back where you started from.

  • And it's quite easy to see how it might happen.

  • Imagine you have this sort of bleak universe, right?

  • Just have a little fluctuation.

  • That little fluctuation sort of gathers together, builds up

  • other things.

  • Eventually, it sort of forms a sort of galaxy, even.

  • From that galaxy, you get planets, stars.

  • And you keep going, you keep going.

  • Eventually, you'll get back to the situation where it looks

  • like it is today.

  • Now, what I think is fair--

  • I think it's a fair point--

  • is that there's no way of ever being aware of these

  • repetitions over these large times.

  • And the reason is, you could never build a device.

  • You could never be an observer that could measure this.

  • And that's because over these huge time scales, such a

  • device or such an observer would definitely thermalize,

  • would definitely become part of this recurrence itself.

  • And so there's never anybody or anything that

  • could measure it.

  • It's this sort of idea, this sort of notion within physics,

  • is that if you can't measure it, it's

  • irrelevant in some sense.

  • So.

  • BRADY HARAN: But according to this, in this number of--

  • in this time, in this number of years, you and I are going

  • to make this video again?

  • You and I are going to make this video again?

  • TONY PADILLA: Oh, I know this is--

  • less than that.

  • This is for a special type of universe that's

  • particularly large.

  • The number for us is at least less than this other number

  • that he's written down here, which is 10 to the 10 to the

  • 10 to the 10 to the 2.08.

  • I'll just say years.

  • So this one is the one that applies to us in our physical

  • universe, what we call our causal patch.

  • This one applies to seeing what is the Poincare

  • recurrence time for a truly vast domain of universe that

  • you can get out of certain models of cosmology.

  • So they're all based, of course, on the same idea.

  • We can work through where these numbers come from.

  • So the Poincare recurrence time of any system is roughly

  • proportional to the number of states in that system.

  • Because we're applying this to the universe, this is really

  • the number of macrostates, the number of these averages of

  • microstates that you could talk about.

  • So let's call this Nmacro.

  • This, then, where we'd expect it to be about the exponential

  • of the number of microstates.

  • Now why is that?

  • Well, this doesn't really have to be an e.

  • It could be a 2 or whatever.

  • Basically, any microstate is either in or out of the

  • averaging with some weighting.

  • And so this is the number that you get out.

  • The number of microstates when you relate to the entropy is e

  • to the entropy.

  • Let's look at some volume of the universe,

  • of some radius r.

  • Then the entropy--

  • we've done this before.

  • This is the same arguments you have before--

  • the entropy that you could possibly have in this region

  • of space, basically it's proportional to--

  • I'll just be a bit sloppy with factors--

  • r squared over the Planck length squared.

  • So the next step is e to the e to the--

  • So now let's apply it.

  • So let's apply it to the really big

  • number that he does.

  • The question is, what's r?

  • Well, the radius of the universe is about 10 to the 26

  • meters, our visible universe, so far as we can see.

  • What we call our causal patch.

  • The Planck length is about 10 to the minus 34 meters.

  • So r over n l Planck squared is about 10 to the 120, which

  • is a number you often see in physics, actually.

  • This is sort of the number that's associated with

  • cosmological constant problems.

  • But anyway.

  • Well, 120 is about 10 to the 2.08.

  • That's where that is coming from.

  • I don't know why he's so precise about this 2.08,

  • because what he's going to do next is--

  • the number we have is e to the e to the 10 to

  • the 10 to the 2.08.

  • But what he does here is he just approximates

  • these e's as 10's.

  • Which is fine, really, in the broader scheme of things.

  • e, 10.

  • For the sake of cosmology, they're more or

  • less the same thing.

  • All the e's become 10's, and you're got 10 to the 10 to the

  • 10 to the 2.08.

  • Which hopefully is what he's got there, and it is.

  • So that's where that number comes from.

  • So this is basically the Poincare recurrence time for

  • our visible universe, for our patch.

  • BRADY HARAN: Why is the other number bigger?

  • TONY PADILLA: Well, because the other number--

  • there, he's looking at--

  • he's trying to get a big number is what

  • I guess he's doing.

  • He's trying to get a bigger number.

  • What he's looking out there is a model of inflation.

  • Now, inflation is a model of the very early universe, where

  • the universe grew really quickly out of a very small

  • patch of the universe.

  • The amount by which it blows up depends on various

  • parameters in the model.

  • But basically, the thing that you get is you get to the size

  • of the universe.

  • So r, for that case, is of order e to

  • the 4 pi over M squared.

  • So this is actually r over l Planck.

  • He's done everything in Planck units.

  • But M here is the mass of some of this influx, what we call

  • the inflaton field.

  • It's just some field in the model

  • that causes the expansion.

  • 1 over M squared is 10 to the 12.

  • I think this might be 13, actually.

  • 10 to the 12.

  • So this is about 10 to the 13 overall,

  • because 13 is about 10.

  • Bear with me.

  • So this is e to the 10 to the 13, roughly.

  • The 13, well, is about 10 to the 1.1.

  • That's where the 1.1 comes from.

  • He's very precise about this 1.1, and yet he's very sloppy

  • with some of the other factors.

  • You get e to the e to another e to the 10 to

  • the 10 to the 1.1.

  • And then we do the same thing.

  • We turn all the e's into 10's.

  • BRADY HARAN: Help me understand, because obviously,

  • you have bit numbers there.

  • How long a time is this?

  • TONY PADILLA: OK.

  • So this is truly vast.

  • Like I said, there's no device, there's no observer,

  • anything that could survive this kind of

  • length of time scale.

  • In fact, you would probably say that the universe is more

  • likely to tunnel out of the current state before this

  • could happen, in some sense.

  • This is such a long time scale that one might say that

  • actually the probability of tunneling to a new phase of

  • the universe, completely different, is actually going

  • to dominate over this.

  • That that would occur first.

  • So maybe this is kind of an irrelevant point.

  • Yeah.

  • It's truly vast.

  • It's bigger than a googol, clearly.

  • Way bigger.

  • Is it bigger than a googolplex?

  • I think so.

  • So you know, let's just check that.

  • Yeah, clearly it is.

  • It's enormous.

  • It won't be as big as Graham's number, but you know, Graham's

  • number's the daddy, right?

  • So.

  • Well, I didn't actually know about this paper.

  • But yeah, I was just--

  • I just thought, oh, big numbers.

  • Let's see what's interesting about big numbers.

  • And then I stumbled across this paper.

  • What he claims-- in fact, he says it--

  • he claims, "So far as I know, these are the longest finite

  • times that have been explicitly calculated by any

  • physicist." So whether somebody has calculated a

  • longer one--

  • and I'm sure some of the viewers will try to calculate

  • a longer one and then claim that they've--

  • but this is in a published paper.

  • So you've got to get the paper published.

  • But the challenge is on, I guess, to find a longer one.

  • There probably has been, too, but he was the one that

  • pointed it out.

BRADY HARAN: You got another number for us?

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最長的時間 - Numberphile (The LONGEST time - Numberphile)

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    簡宇謙 發佈於 2021 年 01 月 14 日
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