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.
