The Universe is a giant energy minimization machine
Physics, at its worst, can be tremendously complicated. Sometimes learning physics (or any other branch of science) can seem having to memorize a giant catalogue of phenomena, each discovered by some guy who got an equation named after him. It can seem like if I want to solve a problem, then I need to begin by preparing a huge cheat sheet of possible relations and dependencies that have to be taken into account. For example, if I were solving a problem in atmospheric physics, I might feel compelled to make a list like this:
- Remember that charged particles drift in crossed electric and magnetic fields (…formula…)
- Remember the coriolis acceleration (…formula…)
- Remember relativistic effects (…formula…)
- Remember that ionization depends exponentially on temperature (…formula…)
Apparently, solving a problem in atmospheric physics requires me to remember and understand all these effects, plus dozens of others, and decide how each will affect the problem. What an ugly mess! I would never have chosen to become a scientist if I knew it would be like that. And sometimes, for sure, it is. I have definitely read some ugly scientific papers, in which a bunch of different effects (each with its own formula) get patched together to form some awful and unintelligible Frankenstein theory.
That’s not to say that all the different effects in physics aren’t interesting. They usually are, and discovering a new phenomenon is the rush that everyone studies physics for. But no one wants to approach every problem by making a long list of things they have to remember and then making sure that each one gets its say in the answer.
Luckily, physics doesn’t actually have to be like this. Strangely enough, I find that the more I study physics and expand my “catalogue of effects”, the more I end up approaching every problem the exact same way. That is, instead of adhering to a big messy list, I am guided by a single principle:
- Remember that the universe is a giant energy minimization machine
I’ve talked before about how energy is the most important concept in physics. But I’ve realized lately that the degree to which I rely on it for solving problems is kind of astonishing. In fact, these days I approach just about every problem using the same three steps:
- Draw a big picture and write out all the things that can possibly happen (usually by defining some mathematical variables).
- Decide how much energy the various possibilities require (usually by writing down some equations).
- Find out which possibility has the lowest energy (usually by minimizing those equations). This is what will actually happen.
Now, these steps can be deceptively hard, but the approach is remarkably simple. If you want to know what will happen to some object or set of objects in the future, you just have to remember this: of all the options available, the universe will choose the one with the lowest (free) energy. It may be hard to describe how that thing will happen, but you can rest assured that eventually it will.
In this post and the next one, I want to give an example or two of surprising realizations you can reach by adhering to the doctrine that the universe will somehow find a way to minimize its total energy.
Example #1: a mysterious force of suction
Imagine performing the following experiment: you take two flat plates of material and give them equal and opposite electric charges (if you want, you can do it the way they did it in the 1600s: by rubbing a piece of glass against a piece of resin). Once you’ve done that, take your plates to a big reservoir of water, and hold them parallel to each other so that their bottom edges just barely touch the surface. Like so:
The lines here indicate the presence of electric field: since there are positive charges next to negative charges, there is an electric field going from positive to negative. If the plates are fairly large relative to the area between them, then the electric field lines will be pretty much parallel and horizontal.
This is the end of the experimental setup. What will happen?
At first, it looks like nothing should happen at all. The plates attract each other, but you are holding them in place so that they can’t move. The only other possibility is that maybe the water could be pushed around by the electric field. But it doesn’t seem like this should happen either, since the electric field goes in straight lines and apparently doesn’t even touch the water.
But try approaching this from the perspective I advocated in the introduction. Don’t ask “what forces are pushing things around?”. Instead, ask “how could the universe lower its energy?”. The answer to that question is a little more interesting.
In this case, there is some energy to be saved by bringing water in between the charged plates. That’s because water has a high dielectric constant (about 80 times larger than air), so if there is water in between the plates then the strength of the electric field is diminished. And electric field is a type of energy (with energy density ), so if you reduce its magnitude then you lower the energy of the universe. You can also think of it this way: water molecules are polar objects, with a positive end and a negative end. Each of them would gain some energy by being in the presence of en electric field, where they can point their positive ends toward the negative charge and their negative ends toward the positive charge.
Apparently, then, there is some energy to be gained if the water spontaneously jumps up in between the two plates. And, in fact, it does, like this:
This is a conclusion that we might not have reached from the beginning. There don’t seem to be any strong forces pushing the water upward. But the water gets sucked up nonetheless, because the energy of the universe can be lowered when it does.
If you want to figure out how high the water level rises in between the plates, then you have to balance this electrostatic energy savings with the cost of picking up all that water against the earth’s gravity. You can do so by writing down the total energy as a function of the water height and then finding out which value of minimizes the total energy. For anyone following along at home, I’ll give the final answer [UPDATED]:
Here, is the charge on the plates, is their area, is the vacuum permitivity, is the acceleration due to gravity, and is the density of water.
You may find this answer pretty unsatisfying; certainly I did when I first encountered it. There seems to be a contradiction: at the beginning of the problem there is no force acting on the water molecules, but somehow they rise up to fill the space between the plates. How is that possible?
The answer is that there is, in fact, a force on the water molecules. My description above of the electric field — that it goes in straight lines and exists only between the two plates — isn’t really correct. The electric field bends a little bit at the edges and at the boundary of the water surface. This bending is enough to provide a force that pushes the water upward and holds it there once it has risen. This a subtle point. But, amazingly, we didn’t need to know it in order to figure out what would happen (or even to calculate the magnitude of the force). Just knowing the general behavior of the electric field, and how much energy it stored, was enough to figure everything out.
For those of my readers with a background/inclination toward economics, I pose the following question: do you think about economics problems in a similar way? Do you approach them with a similar guiding principle, like “a person or population will do whatever results in the maximum income?”, even when it’s not clear how or why they should do that thing?
When I taught introductory physics, I liked to state the parallels between energy and money explicitly: it isn’t created or destroyed, and everyone is trying to get as much (or for energy, as little) of it as possible.