Maxwell’s demon | Thermodynamics | Physics | Khan Academy

December 4, 2019 0 By Kody Olson


The second law of thermodynamics
tells us that the entropy of the universe
is always increasing. So the change in entropy for
the universe, when it undergoes any process,
is always greater than or equal to 0. And we showed in the previous
video that it has a lot of implications. Regardless of how you define
your entropy, whether you define entropy as equal to
some constant times the natural log of the number of
states your system could take on, or whether you define change
in entropy to be equal to the heat added to the
system divided by the temperature at which it is
added, either of these descriptions, combined with
our second law of thermodynamics tell us things
like, when you have a hot body next to a cold body– so say
this is T1, and then I have T2 over here– that heat will
flow from the hot body to the cold body. And we showed that mathematically in the last video. That heat will flow
in this direction. Now, one of the commenters on
the last video said, hey, could you cover Maxwell’s
demon? And I will. Because it’s an interesting
thought experiment that seems to defy this principle. It seems to defy the second
law of thermodynamics. And it has a very tantalizing
name, Maxwell’s demon. Apparently, though, it was not
Maxwell who called it a demon. It was Kelvin. All these guys, you know, they
all meddle in everything. So Maxwell’s demon. And this is the same Maxwell
famous for Maxwell’s equation, so he obviously dealt with
a lot of things. He’s actually also the first
person to ever generate a color image. So this is in the mid-1800s. So all around a fairly
sharp individual. But what’s Maxwell’s demon? So when we say something has
a higher temperature than something else, what
are we saying? We’re saying that its average
kinetic energy of its molecules bumping around
here– that the average kinetic energy of the molecules
here– is higher than the average kinetic energy
of the molecules here. Now notice, I said its average
kinetic energy. And we’ve talked about
this multiple times. Temperature is a macro state. We know that at the micro level,
all of these molecules have different velocities. They’re bumping into each other,
transferring momentum to each other. You know, this guy might
be going super fast in that direction. This guy might actually
be going quite slow. This guy might be going
super fast like that. That guy might be going
quite slow. It’s just a hodgepodge
of things. You could actually draw
a distribution. If you knew the micro states
of everything, you could actually draw a little
histogram. We could say, for T1–
let’s say this is on the Kelvin scale. So you could say, look, my
average temperature is here, but I have a whole distribution
of particles. So let’s say this is number
of particles. And I won’t put a scale there. You’ll get the idea. So I have a bunch of particles
that are at T1, but I have some particles that could be
really close to absolute 0. I mean, it’d be very few, but. And then you have a bunch that
are maybe at T1, and then you have a bunch of particles that
could have actually kinetic energy higher than T1. Higher than the average
kinetic energy. Maybe that’s this one here. Maybe the guy down here is
this guy with barely any kinetic energy. It means there’s some guy
who’s almost completely stationary, who’s, you
know, sitting right around there someplace. So there’s a whole distribution
of particles. Likewise, this T2 system right
here, on average, these molecules have a lower
kinetic energy. But you know, there might be one
particle here that has a really high kinetic energy. But most of them on
average are lower. So if I were to draw the
distribution of T2, my average kinetic energy is lower, but
my distribution might look something like this. It can’t go backwards
like that. It might look something
like this. Oh, I don’t know, maybe it looks
something like that. Let me try it a little
different. I’ll make it go just as high. Maybe it looks something
like that. right? So notice, there are some
molecules in T1 that are below the average kinetic
energy of T2. Right? There are these molecules
here. These are these slow
guys right there. And notice, there are some guys
in T2 that have a higher kinetic energy than
the average in T1. So these are these
guys right here. And so the fast guys in T2–
so even though T2 is, quote unquote, colder, it has lower
average kinetic energy, there’s some molecules, if you
look at the micro state, that are actually moving around quite
rapidly, and there are some molecules here that are
moving around quite slowly. So what Maxwell said is, hey,
what if I had my– and he actually didn’t use the word
demon, but we’ll use the word demon, because it makes it
seem very interesting and metaphysical on some level, but
it really isn’t– what if I had some dude, let’s call him
the demon, with a little trapdoor here? Let me draw a little
bit neater. So between those two
systems, let’s say that they’re insulated. Let’s say that they’re separated
from each other. So this is T1 where I have a
bunch of particles, you know, with their different
kinetic energies. And then, here is T2. And I’m making them separated,
and maybe they’re connected only by this little connection
right here. T2. These guys have a slower
kinetic energy. And what Maxwell, his little
thought experiment was, hey, let me say that I have some
dude in charge of a door– maybe the door is right
here– and he has control over this door. And whenever a really fast
particle in T2, one of these particles over here, come near
the door– so let’s say this guy is flying– let’s say
that guy right there. He’s going super fast. He has
super high kinetic energy, and he’s just going perfectly
for the door. So the demon says, hey. I see that guy. He’s coming for the door. He’s going to lift his hatch,
and he’s going to allow this particle to get into T1. So after he lifts the hatch,
that particle will just keep going, and it’ll be in T1. And then he closes the hatch
again, because he just wants the fast particles to
go from T2 to T1. And then when he sees a little
slow, you know, pokey little particle coming here, one of
these guys down here, he opens the trapdoor again, and he
allows that one to go. So then that guy shows
up in here. So if he just kept doing that,
what’s it going to look like at the end? Well, at the end, you’re going
to segregate– and it could take a while. But you’re going to segregate
all the slow particles on– let me draw it. I’ll make the boundary in brown,
because now it’s not clear which one is– well. We’ll talk about it
a little bit. So that’s the boundary. That’s his door. What’s going to happen
at the end? All the fast particles– some
of them are going to be the original fast particles
that are in T1, right? There are some original fast
particles in T1 are going to be still on the side
of the barrier. Let me draw– make sure you
don’t get these two confused. This is a separate picture. Now all of the fast particles
from T2 are also going to be stuck there. Because eventually they’re all
going to get close to that door, if you wait long enough. So then this guy’s also going
to have a bunch of the, what would originally, in the T2 side
of the barrier, they’re also going to be there. So you’re going to have a
bunch of fast particles. Likewise, all the slow T2
particles are going to be remaining on the side
of the barrier. So these are the slow guys. And he would have let all the
slow T1– I shouldn’t even call them T1 anymore. I’ll call them side 1. Side 1 particles here. Slow side 1 particles. So what just happened here? This was the hot body, this
was the cold body. The second law of thermodynamics
would have told us that heat would have gone
from here to here. That their temperatures
should have equalized to a certain degree. So the hot bodies should get
colder, the cold bodies should get hotter. They should kind of average
out a little bit. But using this little demonic
figure, what did he do? He made the hot body
hotter, right? Now the average kinetic energy
here is even higher. He transferred all of these high
kinetic energy particles to that distribution, so now
that distribution is going to look– the way you could think
about it, if you transferred all of these guys to this guy
over here, the distribution will now look something like–
let me see if I can do it. It will look something
like that for T1, instead of the old one. And T2– and he took all the
hot ones away, all the cold ones away, from T1. So these guys are going
to disappear. They’re not going to
be there anymore. And he added them to T2. So the distribution of T2 is
going to look like that, and he erased, of course,
these from T2. He took all of these
guys out of T2. Let me erase this right here. That was the old distribution
of T1. So the T2 distribution now looks
something like this. So T2, the new average might
be something like here. This is my new T2. And my new T1 is going to move
to the right a little bit. The average is going
to be higher. So this demon seems to have
violated the second law of thermodynamics. Let me box off this
right here. My little diagrams
are overlapping. This example shows that the hot
got hotter and the cold got colder. So Maxwell’s thought experiment
says, hey, we violated the second law
of thermodynamics. And this actually
was a conundrum for many, many years. Even in this century, people
kind of, hey, you know, there’s something fudgy
about here, something not quite right. And the thing that’s not quite
right, and I’m not going to prove it to you mathematically,
is– and it’s kind of analogous to the
refrigerator example– is, to have something here– to have
some dude, perhaps he’s a demon, here, pulling this
little door when it’s convenient, when the fast
particles are going from this side or the slow particles are
going from that side– in order for him to do it
correctly, he’s going to have to keep track of where all
the particles are. He’ll need to keep track
of particles. I mean, these aren’t balls,
like, you know, macro balls. These are micro molecules or
atoms. He’s going to have to bounce light off of them, or
he’s going to have to bounce electrons, use an electron
microscope. He’s going to have to keep
track of these gazillion particles that are there. And I mean, think about it. He might have to have a super
duper– if it doesn’t occur in his head, he might have to have
some kind of hardcore computer microchip that’s
churning away. And this would be, actually,
for a computer to do this, this would be intensive
computation power. And let your computer run for
a little bit and feel the microchip– this is generating
a lot of heat. His bouncing off light, or
whatever he’s trying to bounce off of the different molecules
to be able to measure how fast they’re going, that’s also
going to generate heat. He’s going to have to
do work to do that. He’s going to have to
measure everything. There’s a lot of stuff that’s
going on that he’s going to have to do. So the current answer is– and
it’s not easy to prove mathematically– but the current
answer is, if you actually wanted to build a demon
like this– and probably in our world today, you’d use
some type of computer with some type of sensors– to
attempt to do this– and there are people who have attempted
to do this, on some level. This computer and this whole
system is going to generate more entropy– so the delta S
here is going to generate more entropy than the entropy that’s
lost by making the cold side colder and the
hot side hotter. So Maxwell’s demon,
and I didn’t do anything rigorous here. I didn’t prove it to you. But Maxwell’s demon, it’s
an interesting thought experiment, because it gives
you a little bit more intuition about the difference
between macro states and micro states. And what happens at the
molecular level in terms of temperature, and how you can
make a cold body colder and a hot body hotter. But answer is, it really
isn’t a paradox or anything like that. When you think about the entropy
of the entire system, you have to include
the demon himself. And if you include the demon
himself, he’s generating more entropy every time he opens that
door– and maybe there’s some energy required to
open the door itself. But he generates more entropy
when he does all of this than the entropy that might be lost,
when say, for example, one of these slowpoke particles
kind of just traverses onto that side
of the barrier. Anyway, I thought I would just
expose you to that, because it’s a really neat thought
experiment. So I’ll see in the next video.