Lecture 9: The Phase Space Fluid and the Gibbs-Liouville Theorem

Lenny loved watching the river, especially watching little bits of floating debris making their way downstream. He tried to guess how they would move between the rocks or get caught in eddys. But the river as a whole—the large-scale current, the volume of water, the shear, and the divergence and convergence of the flow were beyond him.

The Phase Space Fluid

Focusing on a particular initial condition and following it along its specific trajectory through phase space are very natural things to do in classical mechanics. But there is also a bigger picture that emphasizes the entire collection of trajectories. The bigger picture involves visualizing all possible starting points and all possible trajectories. Instead of putting your pencil down at a point in phase space and then following a single trajectory, try to do something more ambitious. Imagine you had an infinite number of pencils and used them to fill phase space uniformly with dots (by uniformly, I mean that the density of dots in the q, p space is everywhere the same). Think of the dots as particles that make up a fictitious phase-space-filling fluid.

Then let each dot move according to the Hamiltonian equations of motion, so that the fluid endlessly flows through the phase space.

(1)

The harmonic oscillator is a good example to start with. In Lecture 8 we saw that each dot moves in a circular orbit with uniform angular velocity. (Remember, we are talking about phase space, not coordinate space. In coordinate space, the oscillator moves back and forth in one dimension.) The whole fluid moves in a rigid motion, uniformly circulating around the origin of phase space.

Now let’s return to the general case. If the number of coordinates is N, then the phase space, and the fluid, are 2N-dimensional. The fluid flows, but in a very particular way. There are features of the flow that are quite special. One of these special features is that if a point starts with a given value of energy—a given value of H(q, p)—then it remains with that value of energy. The surfaces of fixed energy (for example, energy E) are defined by the equation

(2)

For each value of E we have a single equation for 2N phase-space variables, thus defining a surface of dimension 2N – 1. In other words, there is a surface for each value of E; as you scan over values of E, those surfaces fill up the phase space. You can think of the phase space, along with the surfaces defined in Eq. (2) as a contour map (see Figure 1), but, instead of representing altitude, the contours denote the value of the energy. If a point of the fluid is on a particular surface, it stays on that surface forever. That’s energy conservation.

Figure 1: Contour plot of energy surfaces of a harmonic oscillator in phase space.

For the harmonic oscillator, the phase space is two-dimensional and the energy surfaces are circles:

(3)

For a general mechanical system, the energy surfaces are far too complicated to visualize, but the principle is the same: The energy surfaces fill the phase space like layers and the flow moves so that the points stay on the surface that they begin on.

A Quick Reminder

We want to stop here and remind you of the very first lecture, where we discussed coins, dice, and the simplest idea of a law of motion. We described those laws by a set of arrows connecting dots that represented the states of the system. We also explained that there are allowable laws and unallowable laws, the allowable laws being reversible. What is it that characterizes an allowable law? The answer is that every point must have exactly one incoming arrow and one outgoing arrow. If at any point the number of incoming arrows exceeds the number of outgoing arrows (such a situation is called a convergence), then the law is irreversible. The same is true if the number of outgoing arrows exceeds the number of incoming arrows (such a situation is called a divergence). Either a convergence or divergence of the arrows violates reversibility and is forbidden. So far we have not returned to that line of reasoning. Now is the time.

Flow and Divergence

Let’s consider some simple examples of fluid flow in ordinary space. Forget about phase space for the moment, and just consider an ordinary fluid moving through regular three-dimensional space labeled by axes x, y, z. The flow can be described by a velocity field. The velocity field is defined by going to each point of space and specifying the velocity vector at that point (see Figure 2).

Figure 2: Velocity field.

Or we may describe the velocity field to be the components of the velocity: v_x(x, y, z), v_y(x, y, z), v_z(x, y, z). The velocity at a point might also depend on time, but let’s suppose that it doesn’t. In that case the flow is called stationary.

Now let’s suppose the fluid is incompressible. This means that a given amount of the fluid always occupies the same volume. It also means that the density of the fluid—the number of molecules per unit volume—is uniform and stays that way forever. By the way, the term incompressible also means indecompressible. In other words, the fluid cannot be stretched out, or decompressed. Consider a small cubic box defined by

Incompressibility implies that the number of fluid points in every such box is constant. It also means that the net flow of fluid into the box (per unit time) must be zero. (As many points flow in as flow out.) Consider the number of molecules per unit time coming into the box across the face x = x₀. It will be proportional to the flow velocity across that face, vx(x₀).

If v_x were the same at x₀ and at x₀ + dx, then the flow into the box at x = x₀ would be the same as the flow out of the box at x = x₀ + d x. However, if v_x varies across the box, then the two flows will not balance. Then the net flow into the box across the two faces will be proportional to

Exactly the same reasoning applies to the faces at y₀ and y₀ + d y, and also at z₀ and z₀ + d z. In fact, if you add it all up, the net flow of molecules into the box (inflow minus outflow) is given by

The combination of derivatives in the parentheses has a name: It is the divergence of the vector field and is denoted by

(4)

The divergence is aptly named; it represents a spreading out of the molecules, or an increase in the volume occupied by the molecules. If the fluid is incompressible, then the volume must not change, and this implies that the divergence must be zero.

One way to think about incompressibility is to imagine that each of the molecules, or points, of the fluid occupies a volume that cannot be compromised. They cannot be squeezed into a smaller volume, nor can they disappear or appear from nowhere. With a little bit of thought, you can see how similar incompressibility is to reversibility. In the examples that we examined in Lecture 1, the arrows also defined a kind of flow. And in a sense the flow was incompressible, at least if it was reversible. The obvious question that this raises is whether the flow through phase space is incompressible. The answer is yes, if the system satisfies Hamilton’s equations. And the theorem that expresses the incompressibility is called Liouville’s theorem.

Liouville’s Theorem

Let’s go back to the fluid flow in phase space and consider the components of the velocity of the fluid at every point of the phase space. Needless to say, the phase-space fluid is not three-dimensional with coordinates x, y, z. Instead it is a 2N-dimensional fluid with coordinates p_i, q_i. Therefore, there are 2N components of the velocity field, one for each q and one for each p. Let’s call them v_qi and v_pi.

The concept of a divergence in Eq. (4) is easily generalized to any number of dimensions. In three dimensions it is the sum of the derivatives of the velocity components in the respective directions. It’s exactly the same in any number of dimensions. In the case of phase space, the divergence of a flow is the sum of 2N terms:

(5)

If the fluid is incompressible, then the expression in Eq. (5) must be zero. To find out, we need to know the components of the velocity field—that being nothing but the velocity of a particle of the phase space fluid.

The flow vector of a fluid at a given point is identified with the velocity of a fictitious particle at that point. In other words,

Moreover, and are exactly the quantities that Hamilton’s equations, Equations (1), give:

(6)

All we have to do is plug Equations (6) into Eq. (5) and see what we get:

(7)

Recalling that a second derivative like does not depend on the order of differentiation, we see that the terms in Eq. (7) exactly cancel in pairs:

Thus the phase space fluid is incompressible. In classical mechanics, the incompressibility of the phase space fluid is called Liouville’s theorem, even though it had very little to do with the French mathematician Joseph Liouville. The great American physicist Josiah Willard Gibbs first published the theorem in 1903, and it is also known as the Gibbs-Liouville theorem.

We defined the incompressibity of a fluid by requiring that the total amount of fluid that enters every small box be zero. There is another definition that is exactly equivalent. Imagine a volume of fluid at a given time. The volume of fluid may have any shape—a sphere, a cube, a blob, or whatever. Now follow all the points in that volume as they move. After a time the fluid blob will be at a different place with a different shape. But if the fluid is incompressible, the volume of the blob will remain what it was at the beginning. Thus we can rephrase Liouville’s theorem: The volume occupied by a blob of phase space fluid is conserved with time.

Let’s take the example of the harmonic oscillator in which the fluid moves around the origin in circles. It’s obvious that a blob maintains its volume since all it does is rigidly rotate. In fact, the shape of the blob stays the same. But this latter fact is special to the harmonic oscillator. Let’s take another example. Suppose the Hamiltonian is given by

You probably don’t recognize this Hamiltonian, but it is completely legitimate. Let’s work out its equations of motion:

What these equations say is that q increases exponentially with time, and p decreases exponentially at the same rate. In other words, the flow compresses the fluid along the p axis, while expanding it by the same amount along the q axis. Every blob gets stretched along q and squeezed along p. Obviously, the blob undergoes an extreme distortion of its shape—but its phase space volume does not change.

Liouville’s theorem is the closest analogy that we can imagine to the kind of irreversibility we discussed in Lecture 1. In quantum mechanics, Liouville’s theorem is replaced by a quantum version called unitarity. Unitarity is even more like the discussion in Lecture 1—but that’s for the next installment of The Theoretical Minimum.

Poisson Brackets

What were the nineteenth-century French mathematicians thinking when they invented these extremely beautiful—and extremely formal—mathematical ways of thinking about mechanics? (Hamilton himself was an exception—he was Irish.) How did they get the action principle, Lagrange’s equations, Hamiltonians, Liouville’s theorem? Were they solving physics problems? Were they just playing with the equations to see how pretty they could make them? Or were they devising principles by which to characterize new laws of physics? I think the answer is a bit of each, and they were incredibly successful in all these things. But the really astonishing degree of success did not become apparent until the twentieth century when quantum mechanics was discovered. It almost seems as if the earlier generation of mathematicians were clairvoyant in the way they invented exact parallels of the later quantum concepts.

And we are not finished. There is one more formulation of mechanics that seems to have been very prescient. We owe it to the French mathematician Poisson, whose name means “fish” in French. To motivate the concept of a Poisson bracket, let’s consider some function of q_i and p_i. Examples include the kinetic energy of a system that depends on the p’s, the potential energy that depends on the q’s, or the angular momentum that depends on products of p’s and q’s. There are, of course, all sorts of other quantities that we might be interested in. Without specifying the particular function, let’s just call it F (q, p).

We can think of F (q, p) in two ways. First of all, it is a function of position in the phase space. But if we follow any point as it moves through the phase space—that is, any actual trajectory of the system—there will be a value of F that varies along the trajectory. In other words, the motion of the system along a particular trajectory turns F into a function of time. Let’s compute how F varies for a given point as it moves, by computing the time derivative of F:

By now the routine should be obvious—we use Hamilton’s equations for the time derivatives of q and p:

(8)

I don’t know exactly what Poisson was doing when he invented his bracket, but I suspect he just got tired of writing the right hand side of Eq. (8) and decided to abbreviate it with a new symbol. Take any two functions of phase space, G (q, p) and F (q, p). Don’t worry about their physical meaning or whether one of them is the Hamiltonian. The Poisson bracket of F and G is defined as

(9)

Poisson could now save himself the trouble of writing Eq. (8). Instead, he could write

(10)

The amazing thing about Eq. (10) is that it summarizes so much. The time derivative of anything is given by the Poisson bracket of that thing with the Hamiltonian. It even contains Hamilton’s equations themselves. To see that, let F (q, p) just be one of the q’s:

Now, if you work out the Poisson bracket of q_i and H, you will discover that it has only one term—namely, the one where you differentiate q_k with respect to itself. Since , we find that the Poisson bracket {qk, H} is just equal to , and we recover the first of Hamilton’s equations. The second equation is easily seen to be equivalent to

Notice that in this formulation the two equations have the same sign, the sign difference being buried in the definition of the Poisson bracket.

The French obsession with elegance really paid off. The Poisson bracket turned into one of the most basic quantities of quantum mechanics: the commutator.