In 2009, Erik Verlinde argued that gravity is an entropic force. This created a big stir—and it helped him win about $6,500,000 in prize money and grants! But what the heck is an ‘entropic force’, anyway?

Entropic forces are nothing unusual: you’ve felt one if you’ve ever stretched a rubber band. Why does a rubber band pull back when you stretch it? You might think it’s because a stretched rubber band has more energy than an unstretched one. That would indeed be a fine explanation for a metal spring. But rubber doesn’t work that way. Instead, a stretched rubber band mainly has less entropy than an unstretched one—and this too can cause a force.

You see, molecules of rubber are like long chains. When unstretched, these chains can curl up in lots of random wiggly ways. ‘Lots of random ways’ means lots of entropy. But when you stretch one of these chains, the number of ways it can be shaped decreases, until it’s pulled taut and there’s just one way! Only past that point does stretching the molecule take a lot of energy; before that, you’re mainly decreasing its entropy.

So, the force of a stretched rubber band is an entropic force.

But how can changes in either energy or entropy give rise to forces? That’s what I want to explain. But instead of talking about force, I’ll start out talking about pressure. This too arises both from changes in energy and changes in entropy.

Entropic pressure — a sloppy derivation

If you’ve ever studied thermodynamics you’ve probably heard about an ideal gas. You can think of this as a gas consisting of point particles that almost never collide with each other—because they’re just points—and bounce elastically off the walls of the container they’re in. If you have a box of gas like this, it’ll push on the walls with some pressure. But the cause of this pressure is not that slowly making the box smaller increases the energy of the gas inside: in fact, it doesn’t! The cause is that making the box smaller decreases the entropy of the gas.

To understand how pressure has an ‘energetic’ part and an ‘entropic’ part, let’s start with the basic equation of thermodynamics:

What does this mean? It means the internal energy of a box of stuff changes when you heat or cool it, meaning that you change its entropy but also when you shrink or expand it, meaning that you change its volume Increasing its entropy raises its internal energy at a rate proportional to its temperature Increasing its volume lowers its internal energy at a rate proportional to its pressure

We can already see that both changes in energy, and entropy, can affect Pressure is like force—indeed it’s just force per area—so we should try to solve for

First let’s do it in a sloppy way. One reason people don’t like thermodynamics is that they don’t understand partial derivatives when there are lots different coordinate systems floating around—which is what thermodynamics is all about! So, they manipulate these partial derivatives sloppily, feeling a sense of guilt and unease, and sometimes it works, but other times it fails disastrously. The cure is not to learn more thermodynamics; the cure is to learn about differential forms. All the expressions in the basic equation are differential forms. If you learn what they are and how to work with them, you’ll never get in trouble with partial derivatives in thermodynamics—as long as you proceed slowly and carefully.

But let’s act like we don’t know this! Let’s start with the basic equation

and solve for First we get

This is fine. Then we divide by and get

This is not so fine: here the guilt starts to set in. After all, we’ve been told that we need to use ‘partial derivatives’ when we have functions of several variables—and the main fact about partial derivatives, the one that everybody remembers, is that these are written with with curly d’s, not ordinary letter d’s. So we must have done something wrong. So, we make the d’s curly:

But we still feel guilty. First of all, who gave us the right to make those d’s curly? Second of all, a partial derivative like makes no sense unless is one of a set of coordinate functions: only then we can talk about how much some function changes as we change while keeping the other coordinates fixed. The value of actually depends on what other coordinates we’re keeping fixed! So what coordinates are we using?

Well, it seems like one of them is and the other is… we don’t know! It could be or or or perhaps even This is where real unease sets in. If we’re taking a test, we might in desperation think something like this: “Since the easiest things to control about our box of stuff are its volume and its temperature, let’s take these as our coordinates!” And then we might write

And then we might do okay on this problem, because this formula is in fact correct! But I hope you agree that this is an unsatisfactory way to manipulate partial derivatives: we’re shooting in the dark and hoping for luck.

Entropic pressure and entropic force

So, I want to show you a better way to get this result. But first let’s take a break and think about what it means. It means there are two possible reasons a box of gas may push back with pressure as we try to squeeze it smaller while keeping its temperature constant. One is that the energy may go up:

will be positive if the internal energy goes up as we squeeze the box smaller. But the other reason is that entropy may go down:

will be positive if the entropy goes down as we squeeze the box smaller, assuming

Let’s turn this fact into a result about force. Remember that pressure is just force per area. Say we have some stuff in a cylinder with a piston on top. Say the the position of the piston is given by some coordinate and its area is Then the stuff will push on the piston with a force

and the change in the cylinder’s volume as the piston moves is

Then

gives us

So, the force consists of two parts: the energetic force

and the entropic force:

Energetic forces are familiar from classical statics: for example, a rock pushes down on the table because its energy would decrease if it could go down. Entropic forces enter the game when we generalize to thermal statics, as we’re doing now. But when we set these entropic forces go away and we’re back to classical statics!

Entropic pressure—a better derivation

Okay, enough philosophizing. To conclude, let’s derive

in a less sloppy way. We start with

which is true no matter what coordinates we use. We can choose 2 of the 5 variables here as local coordinates, generically at least, so let’s choose and Then

and similarly

Using these, our equation

becomes

If you know about differential forms, you know that the differentials of the coordinate functions, namely and form a basis of 1-forms. Thus we can equate the coefficients of in the equation above and get:

and thus:

which is what we wanted! There should be no bitter aftertaste of guilt this time.

The big picture

That’s almost all I want to say: a simple exposition of well-known stuff that’s not quite as well-known as it should be. If you know some thermodynamics and are feeling mildly ambitious, you can now work out the pressure of an ideal gas and show that it’s completely entropic in origin: only the first term in the right-hand side above is nonzero. If you’re feeling a lot more ambitious, you can try to read Verlinde’s papers and explain them to me. But my own goal was not to think about gravity. Instead, it was to ponder a question raised by Allen Knutson: how does the ‘entropic force’ idea fit into my ruminations on classical mechanics versus thermodynamics?

It seems to fit in this way: as we go from classical statics (governed by the principle of least energy) to thermal statics at fixed temperature (governed by the principle of least free energy), the definition of force familiar in classical statics must be adjusted. In classical statics we have

where

is the energy as a function of some coordinates on the configuration space of our system, some manifold But in thermal statics at temperature our system will try to minimize, not the energy but the Helmholtz free energy

where

is the entropy. So now we should define force by

and we see that force has an entropic part and an energetic part:

When the entropic part goes away and we’re back to classical statics!