I enjoy the paintings of Édouard Cortès because while he's called a 'postimpressionist', he captures for me a certain essence of the impressionist ideal. His paintings, or at least the ones I like best, are rather limited in their range of subject matter. But he seems to have perfected a way of capturing the feel of wet or snowy Paris streets, which makes the warmth of indoor lights even more beckoning.
Above is one of his paintings of the Place Vendome. Below is his painting of Le Quai de la Tournelle and Notre Dame.
Go back to my September 26th entry to see where we left off.
We'll start by working out its partition function. And we'll only do a particle in a one-dimensional box. It's easy enough:
But the whole idea raises some questions....
Some people get freaked out by the concept of entropy for a single particle — I guess because it involves probability theory for a single particle, and they think probability only applies to large numbers of things.
I sometimes ask them "how large counts as "large"?"
In fact the foundations of probability theory are just as mysterious for large numbers of things as for just one thing. What do probabilities really mean? We could argue about this all day: Bayesian vs. frequentist interpretations of probability, etc. I won't.
Large numbers of things tend to make large deviations less likely. For example the chance of having all the gas atoms in a box all on the left side is less if you have 1000 atoms than if you have just 2. This makes us worry less about using averages and probability.
But the math of probability works the same for small numbers of particles.
Even better, knowing the entropy of one particle in a box will help us understand the entropy of a million particles in a box — at least if they don't interact, as we assume for an 'ideal gas'.
But why a one-dimensional box?
One particle in a 3-dimensional box is mathematically the same as 3
noninteracting distinguishable particles in a one-dimensional box!
The
So, we start with one particle in a 1-dimensional box. I showed
you how to compute its partition function and get
Yesterday we worked out the partition function of a particle in a
1-dimensional box. From this we can work out its expected energy.
Look how simple it is! It's just
Why so simple?
We can use the chain rule
More generally, the partition function always decreases with
increasing
But when is the partition function of a system proportional to
We've already seen an example with 2 degrees of freedom: the classical
harmonic oscillator. On September 25th we saw that in
this example
In fact, what we're seeing now is just another view of the
equipartition theorem! I proved it a different way on
August 18th. But the
partition function for the system below is proportional to
Here's another thing to consider:
While our particle in a 1d box has
So here's a puzzle for you. Say we have a harmonic oscillator with
spring constant
In other words: how can a particle care so much about the difference between an arbitrarily small positive spring constant and a spring constant that's exactly zero, making its expected energy twice as much in the first case?
I'll warn you: this puzzle is deliberately devilish. In a way it's a trick question!
Today we'll see how entropy works for a single classical particle in a
1-dimensional box of length
We've already worked out the expected energy
The answer looks a lot like the entropy of an ideal gas! That's no coincidence — we're almost there now.
One of the most amazing discoveries of 20th-century physics: particles are waves. The wavelength of a particle is Planck's constant divided by its momentum!
This was first realized by Louis de Broglie in his 1924 PhD thesis, so it's called the de Broglie wavelength.
Why am I telling you this? Because I want to explain and simplify the formula for the entropy of a particle in a box. Even though I derived it classically, it contains Planck's constant! So, it will become more intuitive if we think a bit about quantum mechanics.
This week we'll see an intuitive explanation for our formula of the
entropy of a particle in a 1d box. We'll use this intuition to
simplify our formula. That will make it easier to generalize to
It helps us visualize the world.
We get this approximate formula from a blend of ideas. Classical
mechanics says kinetic energy is
We derived
Worse,
So, we say the 'root mean square' of
Similarly, even if the root mean square of
So, we're really getting that some kind of 'harmonic root mean square'
of the de Broglie wavelength
So, we can think of
Particles are also waves! Even in classical mechanics you can understand the entropy of a particle in a 1d box as the log of the number of wavelengths that fit into that box...
...times Boltzmann's constant, plus a small correction.
We've already worked out the entropy of a classical free particle in a
1d box, so expressing it in terms of the so-called 'thermal
wavelength'
Still,
The partition function of a classical particle in a box is incredibly
simple and beautiful! For a 1-dimensional box, it's just the length
of the box divided by the thermal wavelength
The calculation works the same way in any dimension. Integrate over
position and you get the (hyper)volume of your box. Integrate over
momentum and you get
Once we know the partition function
It matters whether we can tell the difference between the particles or not. Only if they're indistinguishable will we get the experimentally observed answer.
Let's do it!
For distinguishable particles, the total entropy increases a lot when we open a tiny door connecting two boxes of gas - because it takes more information to say where each particle is. For indistinguishable particles, this doesn't happen!
Now let's see why.
The key to computing entropy is the partition function.
For
Why is this?
For
For indistinguishable particles, we integrate the same symmetrical function over a space that's
From the partition function we can compute the expected energy in the usual way. Here we get the same answer for distinguishable and indistinguishable particles!
Indeed, we can get this answer from the equipartition theorem. But I'll just remind you how it works:
From the partition function we can also compute the free energy. Here the two gases work differently!
Since the partition function for the indistinguishable particles is
As usual, from the expected energy and free energy we can compute the entropy.
As you'd expect, the gas of indistinguishable particles has less entropy:
Finally, we can use Stirling's formula to approximate
This gives a wonderful approximate formula for the entropy of an ideal gas of indistinguishable particles. In this approximation, doubling
And so this course is done. 'Twas fun!
Or almost done:
There's an open-book final for all of us! We can now compute the entropy of helium, and with a bit more work solve the mystery that got us started.
Another painting by Édouard Cortès. This is one of the Rue de Lyon. So luminous!
If you turn a cube inside out like this, you get a shape called a rhombic dodecahedron. It's great — there's a lot to say about it. But if you do the analogous thing in 4 dimensions you get something even better: a 4d Platonic solid called the 24-cell!
The 24-cell has 16 corners at those of the original hypercube:
We can fill 3d space with rhombic dodecahedra! Start with a cubic lattice with every other cube filled, and then 'puff out' the filled cubes until they become rhombic dodecahedra. They completely fill 3d space, forming the rhombic dodecahedral honeycomb:
Similarly, we can fill 4d space with 24-cells. We can do this the same way: take a hypercubic lattice with every other hypercube filled in, and then 'puff out' those filled-in hypercubes until they become 24-cells that fill all of 4d space. We get something called the 24-cell honeycomb.