# The notion of an adiabatic process in thermodynamics -vs- quantum mechanics

I'm confused about the terminology in the two contexts since I can't figure out if they have a similar motivation. Afaik, the definitions state that quantum processes should be very slow to be called adiabatic while adiabatic thermodynamic processes are supposed to be those that don't lose heat. Based on my current intuition, this would mean that the thermodynamic process is typically fast (not leaving enough time for heat transfer). What gives, why the apparent mismatch?

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Adiabatic means quasi-static and isoentropic - slow enough to create negligible amount of irreversible excitation. This is the common rationale of technically different definitions. E.g., Landau & Lifshic'es definition has two components - thermally isolated (to prevent entropy change by heat exchange) and slow (to prevent irreversible excitation). For a gaped quantum system adiabatic can by quite fast (just keep Planck constant times the characteristic driving rate below the value of the energy gap).

What is confusing indeed is that there can be an intermediate speed which you can be reasonably adiabatic - much faster than heat exchange with what you separate as the "reservior" but much slower than the equilibraton speed of the degrees of freedom being excited. That's why adiabatic can be fast and slow at the same time - there are two conditions to satisfy. These subtleties are often not made sufficiently clear,
but that's what we have physics.SE for :)

To sum up, don't make "waves" (entropy) and you'll adiabatic.

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This sentence made me quite scared of life for a moment: "Adiabatic means quasi-static and isoentropic" –  Magpie Apr 24 at 19:40
That's good - life is not adiabatic! :) –  Slaviks Apr 24 at 19:41
I see what you did there ;) –  Magpie Apr 24 at 19:50
If I understand correctly, the adiabatic time scale being slower than the equilibration time scale implies that the process will be quasi-static, and hence reversible. We need it to be slower than the heat-transfer ("thermalization"?) time scale. –  Siva Apr 25 at 2:24
In QM, all processes are reversible, so presumably we don't need the notion of quasi-static (which knocks off the smallest time scale). So I would naively think that the adiabatic condition you phrased in terms of the energy gap should give an upper limit on the time scale (to replace the notion of no heat transfer), but it seems to serve as a lower limit. That confuses me. –  Siva Apr 25 at 2:24

The terminological mismatch arises because different physicists use the terms differently in different contexts. For example, here is how Landau and Lifshitz define an adiabatic process in the context of thermodynamics:

Let us suppose that a body is thermally isolated, and is subject to external conditions which vary sufficiently slowly. Such a process is said to be adiabatic

As you can see, these authors combine the criterion of thermal isolation (no heat exchange with the environment) with a slowness assumption, to arrive at their definition of the term adiabatic. In contrast, consider Huang's definition of adiabatic in the context of thermodynamics;

Any transformation the system can undergo in thermal isolation is said to take place adiabatically.

In the context of quantum mechanics, Griffiths defines the term adiabatic as follows:

I would say, from personal experience, that the more widely held convention for the term adiabatic is not the one used by Landau and Lifshitz. In particular, most physicists I know use the term adiabatic in the context of thermodynamics to mean thermally isolated, while they use the term adiabatic in the context of quantum mechanics to mean sufficiently slow that certain approximations can be made.

Addendum. In the context of thermodynamics, the free expansion of a thermally isolated ideal gas is often referred to as an "adiabatic free expansion of a gas," see, for example here. Such a process is not isentropic. Using Slavik's definition would deem invalid the characterization of such a free expansion as adiabatic. However, all you need to do is google "adiabatic free expansion" to see how widespread such use of the terminology is.

People can make all of the unqualified, seemingly confident statements about what the term "adiabatic" means, but it's simply false that everyone uses the same definition, and I think its unproductive to call widely used conventions other than your own "not sensible."

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With all due respect, I disagree. Explosion in a thermally isolated box is not adiabatic by any sensible definition. So it is not only the isolation - I explain myself more in a separate answer. –  Slaviks Apr 24 at 19:38
@Slaviks What exactly are you disagreeing with? My statements pertain only to what I perceive to be a widely held convention about how a term is defined. Are you saying that you think using adiabatic as a synonym for thermally isolated is not a common convention? I'm certain that I can site multiple sources to show that there are a sizable number of physicists who use the term in that way. –  joshphysics Apr 24 at 19:49
In classical thermodynamics there is generally an assumption that the entire path is at or near equilibrium, which is why an abrupt event like an explosion probably shouldn't be called "adiabatic" in a thermodynamic sense even though it does not exchange heat with the environment. And yes, this makes the application of the theory to IC engines a little philosophically difficult, but it more or less works so people do it anyway. –  dmckee Apr 24 at 21:18
I'm not clear on how this fact about classical thermodynamics makes one convention for the term adiabatic more reasonable unless one wants to restrict the term to only be applicable in the domain of classical thermodynamics. I see nothing unreasonable about keeping adiabatic = $Q=0$ and simply adding the qualifier quasi-static when you also want to emphasize that the process being considered is slow in the sense you describe. I feel that making definitive statements about what adiabatic means is problematic given the varying usage of the term. –  joshphysics Apr 24 at 21:49
I agree with @joshphysics. In chemistry it is unambiguous that "adiabatic" means "thermally isolated". Thus an explosion, at least in its initial stages, can be and is treated as adiabatic because there is not enough time to exchange much heat with the surroundings. Further, adiabatic processes do NOT have to be at or near equilibrium. Those paths are called "reversible". Adiabatic paths can be reversible but are not mandated to be such. –  Paul J. Gans Apr 25 at 1:33
In quantum mechanics, the opposite of the adiabatic approximation is the sudden approximation. Take a system with initial Hamiltonian $H_0$, and change the Hamiltonian to $H_1$ over some time $T$. Then the adiabatic approximation is $T \rightarrow \infty$, and the sudden approximation is $T \rightarrow 0$. In the sudden approximation, the state of the system doesn't change (it "doesn't have time to change"), and it finds itself suddenly not in an eigenstate. In the adiabatic approximation, the state follows the perturbation, and is always in an eigenstate of the Hamiltonian.