## Wednesday, December 29, 2010

### mechanics, thermodynamics, and gravity

Apparently there is a new physical theory that gravity arises out of entropy (link from instapundit):
A few month's ago, Erik Verlinde at the the University of Amsterdam put forward one such idea which has taken the world of physics by storm. Verlinde suggested that gravity is merely a manifestation of entropy in the Universe. His idea is based on the second law of thermodynamics, that entropy always increases over time. It suggests that differences in entropy between parts of the Universe generates a force that redistributes matter in a way that maximises entropy. This is the force we call gravity.
This is interesting because I've always thought physics should concentrate more on thermodynamic-style theories rather than mechanical-style theories, and up to now mechanical-style theories have been far more dominant.

Consider a box divided into two chambers, A and B. Chamber A has high-pressure air and chamber B is a vacuum. You can extract useful work from this system by putting a fan between the two chambers and allowing air to run through the fan from chamber A to B. The moving air turns the fan and you can use a belt attached to the fan to turn something else. This system does work. I'm going to be a bit coy about how “work” is defined here, but think of it as doing something like running a car or generating electricity or cooling a house.

This two-chamber system will do work until the pressure in the two chambers is equalized. After that, there is no longer any organized motion of air from one chamber to another to turn the fan (there may be random microscopic motions of air, but these cannot be used to perform work). As the two chambers become more equal in pressure the system loses some of its potential to do work.

There are many other examples of two-chamber setups that can do work. For example, you can have a chamber with hot water in one side and cold water in the other. You can put a heat engine between the two sides and extract energy from it until the temperature of the two sides is the same. You can also get work from a system where there is fresh water on one side and salt water in the other. It will do work until the salinity equalizes. You can get work from a system having one side full of oxygen and the other full of nitrogen at the same temperature and pressure. You can extract work until the two sides have the same mixture.

Now the reason that you can extract work from these systems is that in each case, there is some sort of force or tendency that tries to change the state of the two chambers until the two chambers are the same, are uniform. We call this tendency potential energy. By controlling the tendency of the two chambers to become uniform, we release potential energy and get work.

Entropy is sort of the opposite of potential energy. Entropy is at a minimum at the beginning when the two chambers have the greatest difference. This is also when potential energy is at a maximum. Once the two chambers are uniform, the entropy is at a maximum and the potential energy is zero.

The systems I've talked about up to now are classical thermodynamic systems. Now lets think about a different kind of system, what would be considered a classical mechanical system. Consider a two-chamber system of astronomical proportions. There is an entire planet sitting in chamber A and a chunk of space rock sitting in chamber B. The rock will want to fall towards the planet and you can extract work from the falling rock.

Classical mechanics is about masses and forces and acceleration. Classical thermodynamics is about heat and energy. You can talk about masses and forces in thermodynamic system by bringing in a very complex theory called statistical mechanics. Or you can go the other way and use the notions of heat and energy to talk about mechanical systems. In this way of talking, we don't talk about the force of gravity, instead we talk about the potential energy that exists between the rock and the planet. This potential energy can be extracted as work, much like the potential energy of the other systems can be extracted as work. Once the rock is sitting on the planet, there is no potential energy left.

Throughout most of modern physics, there has been a strong preference for the mechanical type of theory over the thermodynamic type of theory. This is because there has been a strong tendency to view mechanical-type theories as being explanatory while thermodynamic-type theories were merely descriptive. For example, in the two-chamber experiment with high pressure air in one chamber and a vacuum in the other chamber, physicists have felt that what is really happening, the real explanation goes something like this: the chamber with high pressure has a lot of gas molecules bouncing around in it. Once you open a hole in the wall, the molecules that are headed in the right direction to hit that part of the wall will now pass through and hit the fan blades instead. Each molecule that hits the fan blade will bounce off of the blade, imparting a tiny bit of momentum. The sum of all of those molecule-sized momentum changes add up to enough to cause the fan to turn.

This is plausible as an explanation, but now consider applying the same idea to the planet/rock example. By similar reasoning we might say that the reason that the rock falls is because there is a gravitational force between the rock and the planet and that potential energy is just a mathematical fiction. Is that plausible? Why not say that the potential energy is real and the force is just a mathematical fiction? What makes one of these descriptions more real, more explanatory than the other? Frankly, I don't think that there is any rational basis to choose. Both force and potential energy are theoretical entities, as are molecules and momentum changes.

Now, as I said above, potential energy is sort of the opposite of entropy, so when you apply the thermodynamic theories to the mechanical system, it amounts to the statement that the planet/rock system has low entropy when the planet and rock are distant, and that once the rock is resting on the planet the system has maximum entropy. This notion seems to be incompatible with the statistical notion of entropy.

Recall from my original examples with thermodynamics that high entropy is always associated with uniformity –a state where the contents of both chambers have the same uniform contents. This has led physicists to associate entropy with just this state. They say that entropy is equivalent to disorganization and by their theories uniformity is equivalent to disorganization (I know that isn't entirely intuitive, but that's how it is defined).

Now consider the planet/rock system again. In this system the lowest entropy was when there was matter in both boxes. The highest entropy is when there is matter in only one box (which contains both the planet and the rock once all of the potential energy has been expended). This seems to contradict the theory that identifies entropy with disorganization unless you define disorganization as just that state of matter that has less potential energy. In other words, you could make a definition like this: A state S1 is more organized than a state S2 if and only if state S1 has higher potential energy than state S2. However, if you define it that way, then your definition is not an independent theory –it is just a set of code words to talk about potential energy. It hasn't added anything to our knowledge.

An independent theory of entropy (or potential energy) in terms of disorder would let you decide how ordered or disordered a system is before you know anything at all about how potential energy works in that kind of system. That would be exciting because it would show a genuine mathematical relationship between energy and organization. I don't know if such a thing exists or not since I never studied that area, but I find this new theory of gravitation interesting because it seems to imply that there is a genuinely independent definition of entropy that applies not only to traditional thermodynamic systems, but also to mechanical systems involving gravity.

UPDATE: after doing some more reading, it seems that I had the basic idea wrong. This new theory is about microscopic events, statistical mechanics, rather than macroscopic thermodynamics.

Anonymous said...

Hi Dave,
I do not know the new physical theory, but let me say that since Hawkin gravitation and entropy have an interesting connection by the surface of black holes.

And let me elaborate on your view of thermodynamics, statistical mechanics and classic mechanics. Thermodynamics connects even in its simplest application, the gas law, microscopic quantities with macroscopic once. Ok, here you do not have to know those connections. But the interesting part is to connect those two sets of quantities, it leads to an amazing predictive power: a few quantities can explain many phenomenons, quantitative. The connection is just much simpler if the 'particles' do not interact (your thermodynamics?), like the ideal gas, or interact in a simple way. Interacting 'particles' are leading to such interesting things like phase transitions e.g. water/ice (your statistical mechanics). And yes, since classical mechanical systems can describe situations better, there are used when possible. But it is not possible if you have more than lets say 10 'particles'. Then you use statistical methods, with success. It is even a big bend to use statistical methods on small systems. There is even a branch of physics which tries to deal with systems where is number of particles is too big for a classical mechanical analysis and too small for a statistical one.

So it is not about preferences, it is about what methods are fitting to which problem.

Anonymous said...

Hi Dave,

entropy as gravitation? Gravitation and energy are so connected (G prop T) that energy can only be defined as global quantity. The difference is between 'is the same' and 'is connected via a constant multiplier'. Btw., gravitation seem to decrease entropy (order), not increase. Anyway, no news to the great question, how to perform the quantization of gravitation and bind it together with quantum field theories. Certainly, there is nothing to sniff about any consistent extension or refinement to gravitation.