There's another sense in which this slogan is true
which is that all physical systems contain information.
I gave specific examples of this, switch open or closed contains information,
the presence or the absence of a photon - a particle of light - contains information.
Photon polarized like this or photon polarized like that contains information.
But this discovery, that all physical systems contain information,
is actually a very old discovery.
If I have a physical system that has n bits, then that implies it has two to the n,
which I'll represent as being capital n,
because capital n is bigger than little n.
It has two to the N possible states.
Conversely, if I have a system that has N states, then it can be used to represent
log to the base two of N bits.
The logarithm is basically the total number of bits we need to represent this number.
So, for instance, if N is equal to eight, then log to the base two of eight is equal to three.
It's the number of bits I need to write out eight,
because eight is equal to 1 0 0 0. It's just the number of zeros in the bits.
So when we have a system that has not just two states or four states or eight states or
sixteen states or a thousand and twenty-four states
it still represents information.
If I have a system that has three states,
for example, yes, no and maybe, then it represents log to the base two of three,
which is equal to one plus change bits.
It's in between log to the base two of two, which is equal to one bit,
and log to the base to of four, which is equal to two bits.
So something that has three different states,
it's not binary, it's tertiary,
then it has a number of bits in between one and two,
and this observation that any physical system that has a different number of distinguishable
states can register a certain amount of information
was actually realized back in the 19th century.
Back in the 19th century, physicists such as Maxwell and Boltzmann, in Vienna,
Maxwell is in Cambridge, or Gibbs at Yale, they realized that there was this funny quantity called entropy.
Now what is entropy?
Entropy is something that messes up your ability to do work.
It's some measure of disorder.
Or of randomness.
The way that people came up with the concept of entropy in the middle of the 19th century
was that they were looking at how energy could be converted to other forms of energy.
These scientists and others were interested in how heat could be transformed into work.
There's a famous result called the Second Law of Thermodynamics.
The Second Law of Thermodynamics basically says that entropy increases, or entropy tends
to increase, let's say.
But what is entropy?
So back in the mid-19th century, these scientists realized that energy and heat was actually the
kinetic energy, the microscopic energy of moving molecules.
Molecules bouncing off of each other, so they had a certain kind of kinetic energy,
the energy of motion, and that we could identify the energy in heat with this
energy of motion with all the individual molecules in a gas, like the gas in this room.
Now, that's a certain amount of energy, they were able to measure this amount of energy,
and they also were able to know that you could turn this energy into work, using, for instance,
the steam engine.
You got a piston with a cylinder, here's all these steam molecules bouncing off back
and forth, you move out the piston, the molecules gradually slow down,
energy gets smaller, and energy goes into work in the steam engine.
But not all of this energy could be converted into work.
Why is that?
It's because these molecules bouncing off of each other,
they have some degree of disorder, and this degree of disorder could not be decreased.
So somehow, to have this degree of disorder, even when the piston goes out,
they still have to be moving around,
and so you can't extract all of the kinetic energy from these molecules.
Now because these guys, they were guys,
by the way, in the course of these little lectures, I'm going to use the word "guys"
to refer not only to people, men, but also to elementary particles and to women as well.
So these guys figured out that if they wanted to come up with a formula for entropy,
to describe this quantity that never decreased.
They said, well let's let N equal the number of possible states or configurations
or as Boltzmann put it, "complexions", of these molecules in the gas.
If they defined a quantity, S, which is equal to the logarithm of N, which you may note
is the amount of information that's contained in these molecules in the gas,
just be counting the number of possible states, configurations or complexions,
this quantity right here, the log of N, is the number of bits that's required to label
each of the possible different complexions, or states of the molecules in the gas,
and they said if we define S to be proportional to this, they defined a constant k, which is now
called Boltzmann's constant.
If they defined this S to be k log N, then this was actually this quantity entropy,
this thermodynamic quantity that gunged up the works of heat engines and prevented you
from getting all the energy of molecules.
So famous, in fact, that it actually appears on Boltzmann's grave.
He died in 1906 by his own hand, he was a depressive person, after a visit to the US,
I'm not sure if there's any correlation between this.
But back in those days if you had a famous formula, people popped it on your grave.
Actually there is an interesting story, which was that this constant k that's on Boltzmann's
grave was actually by the famous German physicist Max Planck,
and it was originally called Planck's constant.
But Planck also had another constant named after him, h,
which we'll see again, because it governs the quantum mechanical behavior of things.
So after a while people just said, hey let's just call it Boltzmann's constant.
What Boltzmann would feel if he saw some other man's constant on his grave, I don't know,
maybe he be turning over in his grave.