Over at the Virtuosi, there’s a nice discussion of the physics of letting air out of tires. Jesse opens the explanation with:

Have you ever noticed how when you let air out of a bike tire (or, I suppose, a car tire) it feels rather cold? Today we’re going to explore why that is, and just how cold it is. Many people consider the air escaping from a tire as a classic example of an adiabatic process. What is an adiabatic process? It is a process that happens so quickly there is no time for heat flow to occur. For our air in the bike tire this means we’re letting it out of the tire so quickly that no energy can move into it from the surrounding air.

The rest of the explanation lays out the math, including an expreimentally testable prediction. I’m not going to talk about that, though. What caught my eye was the word “adiabatic” in there.

The definition of “adiabatic” given above, namely a process that occurs very quickly, is amusing to me because it’s almost exactly the opposite of the definition I usually encounter. In my corner of cold-atom physics and quantum optics, when we talk about something being “adiabatic,” we almost always means that it’s a process that takes place slowly.

This comes up in a lot of contexts, but the most common has to do with changing the width of a trap and looking at the distribution of trapped states. In quantum mechancis, we almost always talk about particles in terms of their energy, so a particle that’s held in some sort of trap gets represented by a picture like the one on the left below:

i-01b71c36fc9ef434ca3711f2f6939b4e-wells.jpg

The blue line represents the potential energy of the particle. This increases as you move out from the center, and puts limits on the motion of a particle with finite total energy– if you want to get very far away from the center, you need to start with a ton of kinetic energy in the middle of the potential energy well so that you can convert it to the huge amount of potential energy the particle will have when it’s far away.

The red lines represent the wavefunctions for some of the different states a trapped particle can occupy given that potential energy, offset vertically by an amount that roughly corresponds to the total energy for that wavefunction (that is, the lowest red curve has the lowest total energy, the next one up has slightly more energy, and so on). If you squared those functions, you would find the probability of finding the particle at a given spot in the well, and that probability is high in the well, and drops off to zero on the outside.

So, let’s imagine that you have an atom in a well like this, prepared in the lowest total energy state available. Now, let’s imagine that you want to double the width of the well, turning it into the picture on the right. What happens when you do that?

Well, if you make the change suddenly, the wavefunction of the particle in the new well still looks like the lowest red curve on the left, which is re-drawn on the right-hand well. This doesn’t look like any one of the allowed states of the well on the right, though (green dotted lines)– it’s sort of similar to the lowest energy state, but more sharply peaked. It looks a little like the third state up, but it only has one lump instead of three. And so on.

To make something that looks like that red wavefunction, you need to add together a bunch of the green wavefunctions. Which means there’s a good chance of finding the particle in a state with a higher energy than the lowest energy available, which is a very different situation than the system you started with.

On the other hand, if you expand the well very slowly, the atom has time to make small changes to its wavefunction, and can move smoothly from the lowest-energy red state to the lowest-energy green state. If you do the expansion slowly enough (“slow” here being defined as “long enough that the atom bounces back and forth many times inside the well while the expansion is taking place”), you can make the probability of staying in the lowest energy state as low as you like.

This is what we call an “adiabatic” expansion in cold-atom physics– it’s an expansion of the trap that keeps the system always in the lowest energy state (technically, I suppose it keeps the relative populations of all the states the same, but what we usually care about is the ground state, or lowest energy state). You can also do adiabatic compression of states, which works the same way.

So why does the same term mean different things? The results are the same in the most important sense, namely that the system being considered does not gain energy from its surroundings during the process. In the air-from-tires case, this is measured in terms of the temperature of the gas, and the energy you need to worry about is heat energy from the surroundings. As a general matter, heat moves very slowly, so you want a fast process.

In the cold-atoms case, the quantity that matters is the microscopic distribution of the atom’s wavefunction among the possible states. To preserve the atom in the lowest-energy microscopic state of the trap, you need the expansion to be slow compared to the motion of the atom inside the trap.

The relevant time scales are very different, so these actually aren’t contradictory requirements– the speed of an atom inside the trap is very fast, so it’s not too hard to be slow compared to that, while still being fast compared to thermal flow. It’s really just a question of emphasis– if you work in a context where the microscopic distribution is what matters, then “adiabatic” means slow, and everything is over long before heat is an issue; if you work in a context where heat flow is important, then “adiabatic” means fast, but the individual gas molecules are moving so fast that you can’t keep track of them.

It does get you some odd looks when you move from one community to the other, though.

Comments

  1. #1 Nathan
    May 5, 2010

    I think that Feynman mentioned in one of the Feynman Lectures that Adiabatic has at least three different definitions in physics that mean completely different, but related things.

  2. #2 Jesse
    May 5, 2010

    Fascinating. I’ve actually been TAing for the engineering E&M and thermo class this semester, and we just finished up the thermo unit. That was my motivation for the question. However, even though I’ve had to use the quantum adiabatic approximation this semester, I hadn’t quite made the connection in my head as to how strange the different usages might seem to people.

  3. #3 tyler
    May 5, 2010

    Yes, it seems there are some nuanced diferences in the maning of “adiabatic” depending on the context. Can’t we generalize them though? Adiabatic most generally means that the system or process occurs without a change in heat, doesn’t it? In the case of the air escaping from the tire, it is an adiabatic process in the sense that it happens so fast that the kinetic energy of the air outside the tire does not have time to change the temperature of the air that has only just escaped from the tire; in the QM case above, the expansion of the well must be slow enough not to make it likely that the particle is found with a higher energy than before.

    Does this make sense?

  4. #4 Eric M
    May 6, 2010

    As tyler said above, adiabatic = fast is a coarse engineering approximation. The idea is that the cycle time is so short that heat transfer across the boundary is negligible.

  5. #5 perry
    May 6, 2010

    I think adiabatic as I use it refers to a separation of time scales, so that one set of variables becomes slaved to another. The fast variables track the slow ones, so you just need to solve for the slow ones, problem is easier.

  6. #6 Perry
    May 6, 2010

    Hit return too fast.

    Its a ratio of time scales (decay rates, pump rates, whatever) that matters. What makes something “fast”?????? Oh its faster than some other thing! All hail dimensionless units.

  7. #7 CCPhysicist
    May 9, 2010

    The word “adiabatic” means exactly what it the word says in Greek: a (not) diabatos (go across). It literally means heat can’t get across the system boundary, hence Q = 0. That means its best modern synonym is “insulated”. The other instances inherit the name because they also have Q=0. The reason to do it slowly (quasi-statically) is so it is reversible and can be drawn as a continuous curve on a PV diagram.

    I’m no Greek scholar, by the way. After years of teaching and wondering vaguely about why it sounds so much like “diabetes”, I finally looked in a dictionary. Shocking thought! I learned that diabetes was given its name by early Greek physicians from one of its symptoms, frequent urination. One dictionary says it simply means water “passes through” the patient, while another says it comes from two words that mean to walk with legs crossed. !!

    PS – Solenoid means “pipe”, a coil of wire wound around a cylindrical pipe.

  8. #8 lordaxil
    May 9, 2010

    CCPhysicist: If process is quasi-static (reversible) then don’t you mean dQ=0?

    In general, adiabatic processes don’t have to take place reversibly, but those that do are isentropic (since entropy and temperature are conjugate variables). The conservation of entropy in a reversible adiabatic process is directly analogous to the conversation of quantum number in a slowly-varying potential as described by Chad. Thus, there is really no conflict between the two usages of adiabatic if understood correctly.

  9. #9 csrster
    May 10, 2010

    So does that mean that, in the quantum case, if we move the walls rapidly the process is still adiabatic (in the strict thermodynamic sense) but not isentropic ?

  10. #10 lordaxil
    May 10, 2010

    That’s an interesting question. I guess it depends on whether or not the system makes a quantum jump from the ground state of old potential to an excited state of the new one. If I understood what Chad was saying correctly, then the only way to make certain there are no jumps to an excited state (thereby requiring energy input) is to do the process very slowly.

  11. #11 Matthew von der Ahe
    May 10, 2010

    Inconceivable!

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