Thanks to the precise mathematical analogy between classical mechanics and thermodynamics that we saw last time, we can define Poisson brackets in thermodynamics just as in classical mechanics. That is, we can define Poisson brackets of quantities like entropy, temperature, volume and pressure. But what does this mean?
Furthermore, we can get quantum mechanics from classical mechanics by replacing Poisson brackets by commutators. That’s a brief summary of a long story, but by now this story is pretty well understood. So, we can do the same maneuver in thermodynamics and get some new theory where thermodynamic quantities like entropy, temperature, volume and pressure no longer commute! But what does this mean physically?
The obvious guess is that we get some version of thermodynamics that takes quantum mechanics into account, but I believe this is wrong. I want to explain why, and what I think is really going on. Briefly, I think what we get is more like statistical mechanics—but with a somewhat new outlook on this subject.
In getting to this understanding, we’ll bump into a few thorny questions. They’re interesting in themselves, but I don’t want them to hold us back, because we don’t absolutely need to answer them here. So, I’ll leave them as ‘puzzles’. Maybe you can help me with them.
In the classical mechanics of a point particle on a line, I considered a 4-dimensional vector space where a point describes the position momentum time and energy of a particle. This is usually called the extended phase space, since it’s bigger than the usual phase space where a point only keeps track of position and momentum. We saw that the extended phase space naturally acquires a symplectic structure
We can define Poisson brackets of smooth functions on any symplectic manifold. Instead of reviewing the whole well-known procedure I’ll just turn the crank and tell you what it gives for the coordinate functions on the extended phase space. The Poisson brackets of position and momentum are nonzero, and similarly for time and energy. All the rest vanish. We have
so we call these pairs of functions conjugate variables.
Puzzle 1. The first equation above says that momentum generates translations of position. The second says that energy generates time translations, but in reverse, thanks to the minus sign. Why the difference? Is this a problem?
To go to quantum mechanics, we can build a noncommutative algebra generated by elements I’ll call and We require that these commute except for two, which have commutators that mimic the Poisson brackets above:
Here is Planck’s constant, or more precisely the reduced Planck constant. This has dimensions of action, making the above equations dimensionally consistent: momentum times position has dimensions of action, and so does energy multiplied by time.
Puzzle 2. Does this mean the well-known Poisson bracket equations is not dimensionally consistent? Does this cause problems in classical mechanics? If not, why not?
We can work with the algebra having these generators and relations; this is called a Weyl algebra. But physicists usually prefer to treat this algebra as consisting of operators on some Hilbert space. The Stone–von Neumann theorem says there’s a unique ‘really nice’ way to do this. I don’t want to explain what ‘really nice’ means—it’s too much of a digression here. In fact I wrote a whole book about it with Irving Segal and Zhengfang Zhou. But this article is better if you just want the basic idea:
• Wikipedia, Stone–von Neumann theorem.
The main thing I need to say is that in the ‘really nice’ situation, and are unbounded self-adjoint operators, and the spectrum of each one is the whole real line. That’s not all there is to it: more must be true. But a problem immediately shows up. This is fine for the usual position and momentum operators in quantum mechanics, but the spectrum of the Hamiltonian is usually not the whole real line! In fact we usually want its spectrum to be bounded below: that is, we don’t want to allow arbitrarily large negative energies.
As a result, in quantum mechanics we usually give up on trying to find a ‘time operator’ that obeys the relation
There is more to say about this:
• John Baez, The time-energy uncertainty relation.
But this is not what I want to talk about today; I mention it only because we’ll see a similar problem in thermodynamics, where volume is bounded below.
In any event, there’s a standard ‘really nice’ way to make and into operators in quantum mechanics: we take the Hilbert space of square-integrable functions on the line, and define
We may try to copy this in thermodynamics.
So, let’s see how it works! There are some twists.
In my explanations of classical mechanics versus thermodynamics so far, I’ve mainly been using the energy scheme, where we start with the internal energy as a function of entropy and volume and then define temperature and pressure as derivatives, so that
As we’ve seen, this makes it very interesting to consider a 4-dimensional space—I’ll again call it the extended phase space—where and are treated as independent coordinates. This vector space has a symplectic structure
and the physically allowed points in this vector space form a ‘Lagrangian submanifold’—that is, a 2-dimensional surface on which the symplectic structure vanishes:
The symplectic structure lets us define Poisson brackets of smooth functions on the extended phase space. Again I’ll just turn the crank and tell you the Poisson brackets of the coordinate functions. The math is just the same, only the names have been changed, so you should not be surprised that two of their Poisson brackets are nonzero:
and all the rest vanish.
Puzzle 3. What is the physical meaning of these Poisson brackets?
Again we may worry that these equations are dimensionally inconsistent—but let’s go straight to the ‘quantized’ version and do our worrying there!
So, we’ll build a noncommutative algebra generated by elements I’ll call and We’ll require that these commute except for two, namely the pair and the pair What relation should these obey?
A first try might be
However, these equations are not dimensionally consistent! In both cases the left side has units of energy. Entropy times temperature and volume times pressure both have units of energy, since the first is related to ‘heat’ and the second is related to ‘work’. But has units of action.
I declare this to be intolerable. We are groping around trying to understand the physical meaning of the mathematics here; dimensional analysis is a powerful guide, and if we don’t have that going for us we’ll be in serious trouble.
So, we could try
where is some constant with units of energy.
Unfortunately there is no fundamental constant with units of energy that plays an important role throughout thermodynamics. We could for example use the mass of the electron times the speed of light squared, but why in the world should thermodynamics place some special importance on this?
One important quantity with units of energy in thermodynamics is where is temperature and is Boltzmann’s constant. Boltzmann’s constant is fundamental in thermodynamics, or more precisely statistical mechanics: it has units of energy per temperature. So we might try
but unfortunately the temperature is not a constant: it’s one of the variables we’re trying to ‘quantize’! It would make more sense to try
but unfortunately these commutation relations are more complicated than the ones we had in quantum mechanics.
Luckily, the fundamental constant we want is sitting right in front of us: it’s Boltzmann’s constant. This has units of energy per temperature—or in other words, units of entropy. This suggests quitting the ‘energy scheme’ and working with the second most popular formulation of thermodynamics, the ‘entropy scheme’.
Here we start by writing the entropy of our system as a function of its internal energy and volume A simple calculation starting from then shows
It will help to make up short names for the quantities here, so I’ll define
and call the coldness, for the obvious reason. I’ll also define
and call the boldness, for no particularly good reason—I just need a name. In terms of these variables we get
This is the so-called entropy scheme. This equation implies that times has dimensions of entropy, and so does
If we go through the procedure we used last time, we get a 4-dimensional vector space with coordinates and symplectic structure
The physically allowed points in this vector space form a Lagrangian submanifold
which is really just the submanifold we had before, now described using new coordinates.
But let’s get to the point! We’ll build a noncommutative algebra generated by elements I’ll call and We’ll require that these commute except for two, namely the pair and the pair And what commutation relations should these obey?
We could try
This is now dimensionally consistent, since in each equation both sides have dimensions of entropy!
However, there’s another twist. Quantum mechanics is about amplitudes, while statistical mechanics is about probabilities, which are real. So I actually think I want
Let me give some evidence for this!
I will ignore the second equation for now and focus on the first. Suppose we’re doing statistical mechanics and we have a system that has probability of having internal energy We would like to define an internal energy operator and coldness operator Say we set
Then it’s easy to check that they obey the commutation relation
Moreover—and this is the exciting part—they make physical sense! If the system definitely has internal energy then vanishes except at It follows that is an eigvenvector of the internal energy operator with eigenvalue
On the other hand, suppose the system definitely has temperature and thus coldness Then statistical mechanics tells us that is given by the Boltzmann distribution
where is a normalizing constant called the partition function. It follows that is an eigenvector of the coldness operator with eigenvalue
This makes me very happy. We are seeing a nice analogy:
• Internal energy eigenstates in statistical mechanics are like position eigenstates in quantum mechanics.
• Coldness eigenstates in statistical mechanics are like momentum eigenstates in quantum mechanics—except for a missing factor of i in the exponential, which makes the coldness eigenstates decrease exponentially instead of oscillate.
We can also define a temperature operator
at least if we ignore the eigenvectors of the coldness operator with eigenvalue zero. But the coldness operator is more fundamental in this approach.
A lot of further questions and problems appear at this point, but I think I’ll stop here and tackle them later.
• Part 1: Hamilton’s equations versus the Maxwell relations.
• Part 2: the role of symplectic geometry.
• Part 3: a detailed analogy between classical mechanics and thermodynamics.
• Part 4: what is the analogue of quantization for thermodynamics?