Before we begin, it is important to emphasize that Maxwell’s Demon is not evil.
It is not her fault that she was born looking this way.
Although James Maxwell is famous for his equations on Electromagnetism,
he is also infamous for his idea on how to violate the Second Law of Thermodynamics.
It was Lord Kelvin who first referred to this idea as “Maxwell’s Demon.”
It took over 100 years to figure out the real reason why Maxwell’s Demon
does not actually violate the Second Law of Thermodynamics,
and this reason involved a revolutionary discovery:
Information is entropy,
and regardless of what physical system is used to store information…
Whether it be the biological neurons of a brain…
The electronic logic gates of a computer…
Or the quantum states of subatomic particles…
It is an unavoidable consequence of the fundamental laws of physics that
erasing information increases the entropy of the surrounding environment.
It is not when we acquire information that we must pay a price,
but when we attempt to forget it.
To review the fundamentals,
we can’t reduce the entropy of a gas by simply compressing it into a smaller volume,
because this transfers energy to the gas particles, thereby increasing their entropy.
Maxwell’s Demon avoids this problem because she is always very careful
to never touch any of the particles, and hence no energy is transferred to them.
The demon can open and close a door,
thereby selecting when she wants a particle to pass through.
This door does not contain any friction or mass,
and hence no energy is transferred when the door is opened or closed.
If the demon is clever enough, she can operate the door in a way
such that more and more particles gather together on the same side.
This decreases the entropy of the particles,
in apparent violation of the Second Law of Thermodynamics,
which states that the entropy of the Universe can only increase.
What is shown here is actually a slight variation of Maxwell’s original idea,
which involved operating the door such that all the fast particles end up on one side,
and all the slow particles end up on the other side,
but the general concept is the same.
The reason Maxwell’s Demon does not actually violate the
Second Law of Thermodynamics is because in order for the
demon to accomplish her task, she can’t open and close the door at random.
The demon needs to acquire information
about the positions and velocities of the particles,
and the information in her memory is entropy.
To understand why information is entropy consider the following.
The ball can be located on any of these squares.
Each possible location for the ball is what we will call a “micro-state.”
To simplify the analysis, for the remainder of this video,
we will assume that every micro-state is equally likely to occur.
Suppose we don’t know the exact location of the ball,
but we know that it is located somewhere inside this rectangle.
This rectangle is an example of what we call a macro-state.
This other rectangle is another example of a macro-state.
Macro-state “B” has a larger number of possible micro-states
than does macro-state “A.”
We refer to this by saying that macro-state “B” has a higher entropy.
Another way of looking at this is in terms of “bits” of information,
where each “bit” can have a value of either “zero” or “one.”
If we know that we are in macro-state “A”,
then we only need one additional bit of information
to describe which micro-state we are in.
If we know that we are in macro-state “B”,
then we need two bits of information to describe which micro-state we are in.
Since macro-state “B” involves more bits of information,
macro-state “B” contains more entropy.
In this way, we can define entropy in terms information.
Suppose we start somewhere in macro-state B,
and the demon then always moves the ball to the red square.
Initially, the macro-state of the system is described by
this rectangle, which has 4 possible micro-states.
At the end, the system is described by this new rectangle,
which contains only 1 possible micro-state.
Does this mean that the entropy of the universe has decreased?
The answer is no, because we also have to consider
the entropy of the information in the demon’s memory.
Although we ourselves don’t know the original position of the ball,
we know that the demon had to observe this information to perform her task.
Therefore, at the end we are in a macro-state where the information
in the demon’s memory can be in one of four different possible configurations.
Hence, the entropy of the information in the demon’s memory has increased, and this
has offset the decrease in entropy of forcing the ball into one specific location.
But if information is a form of entropy, what if we try to decrease the entropy
of the universe by simply asking the demon to forget this information?
As it turns out, this will not decrease the entropy of the universe,
because if the demon tries to erase her memory, she will dissipate energy as heat,
thereby increasing the entropy of her surroundings.
Let’s now return to our original example.
When we begin, the demon’s memory can be a blank slate and contain no information.
But, as she observes the positions and velocities of the particles
to know when to open the door, her memory starts to fill up with information.
The more her memory fills up,
the more the entropy of the information in her brain increases,
thereby offsetting the decrease in entropy of the particles in the box.
If her memory fills up completely,
she will have to erase some information in her memory in order to continue.
But, as the demon erases information in her memory,
she has to dissipate heat, thereby increasing the entropy of the surroundings.
Therefore, the overall entropy of the universe never decreases,
and the second law of thermodynamics is never violated.