2. Fast or slow?

What is an adiabatic process? Most of the readers would probably answer that this is a process with a gas which is so fast that there is no heat exchange with the surroundings. However, this is only a half of the truth, and actually the less important half. In fact, it is quite easy to understand that this is not entirely correct: consider a cylinder, which is divided by a thin wall into two halves; one half is filled with a gas at a pressure p, and the other one is empty. Now, let us remove momentarily the wall: the gas from one half fills the entire cylinder. Since no external work is done (the wall can be removed without performing a work), the energy of the gas is preserved, hence, the temperature remains the same as it was at the beginning. Meanwhile, for an adiabatic process we would expect a decrease of temperature by a factor of 2^γ-1: part of the internal energy is supposed to be spent on a mechanical work performed by the expanding gas. However, if the piston moves faster than the speed of sound, the gas will be unable to catch up and push the piston. So, the adiabatic law was not followed because the process was too fast!

It appears that the adiabatic law for thermodynamics has also a counterpart in classical mechanics – the conservation of the adiabatic invariant. For mechanical systems (oscillators) performing periodic motion, the adiabatic invariant is defined as the area of the closed curve drawn by the system in phase space (which is a graph where the momentum p is plotted as a function of the respective coordinate x), and is (approximately) conserved when the parameters of the system are changed adiabatically, ie. slowly as compared with the oscillation frequency. For typical applications, the accuracy of the conservation of the adiabatic invariant is exponentially good and can be estimated as e^–fτ, where f is the eigenfrequency of the oscillator, and τ is the characteristic period of the variation of the system parameters.

How are related to each other (a) adiabatic invariant and (b) adiabatic process with a gas? The easiest way to understand this is to consider a one-dimensional motion of a molecule between two walls, which depart slowly from each other (Figure 1). Let us use the system of reference where one of the walls is at rest, and the other moves with a velocity u << v, where v is the velocity of the molecule (the interaction of the molecule with the walls is assumed to be absolutely elastic). One can say that such a molecule represents an oscillator with a slowly changing potential:  the potential energy U(x) = 0 for 0<x< X (where X = a +ut) and otherwise, U(x) = ∞. The trajectory of the molecule in the phase space is a rectangle of side lengths X and 2mv. So, the adiabatic invariant is 2mvX; hence, vX = Const. For a one-dimensional gas, the distance X between the walls plays the role of the “volume” V, and mv²/2=kT/2, hence v ~ T^1/2 (here "~" means “is proportional to”). So, the adiabatic invariant can be written as V ²T = Const. On the other hand, from the adiabatic law for an ideal gas, we would expect TV ^γ‑1 = Const. For the one-dimensional gas, the number of the degrees of freedom i = 1, hence γ = c_p/c_V = (i+2)/i =3, and TV² = Const, ie. we can conclude that the adiabatic invariant and the adiabatic gas law give us exactly the same result!

How to prove that for an adiabatic forcing of an oscillator, the adiabatic invariant is conserved? Well, this is not a too simple mathematical task and thus we skip the proof here (it can be found in good textbooks of theoretical mechanics). However, for a simple particular case of an elastic ball between two walls (see above), it can be done more easily. Indeed, with each impact with the departing wall, the speed of the ball is decreased by 2u, and this happens once per time interval t = 2X/v. So, the ball decelerates with the rate of dv/dt = 2u/t =uv/X, hence dv/v= udt/X = –dX/X. Integrating this differential equation gives us directly Xv = Const.

Conservation laws play a central role both for the physical processes, and for the physics as a science (cf “Minimum of Maximum”), and adiabatic invariant is not an exception.  Perhaps the most important role of it is related to the quantum mechanics. Namely, during adiabatic processes, the system will not leave the stationary quantum state it has taken (as long as the state itself does not disappear). To motivate this claim, let us consider a biatomic molecule, which can be modelled as an oscillator. When treating the process classically, the trajectory of a harmonic oscillator in the phase space is an ellips of surface area J = πp₀x₀, where p₀ and x₀ are the amplitudes of the momentum and coordinate. Note that p₀ = mx₀ω₀, where ω₀ is the circular eigenfrequency of the oscillator; therefore, the full energy of the oscillator (calculated as the maximal kinetic energy) is E = p₀²/2m = p₀x₀ω₀/2 = J ω₀/2π = J f₀. Hence, the adiabatic invariant J = E/f₀: during adiabatic processes, the oscillation energy is proportional to the frequency. According to the quantum mechanics, the stationary energy levels of the oscillator are given by E_n=hf₀(n +1/2), where n is an integer representing the order number of the energy level. Comparing the classical and quantum-mechanical results leads us to the conclusion that during adiabatic processes, n = Const: the system will remain at the stationary state of the same order number where it was (Figure 2) . (While it is not always completely correct to combine classical and quantum-mechanical results, classical mechanics is a macroscopic limit of the quantum mechanics and hence, the conservation laws of both theories need to be compatible.)

Now, suppose our bi-atomic molecule is forced by an electromagnetic field in the form of an adiabatic pulse. In terms of classical mechanics we say that such a forcing is unable to pump energy into oscillations of the molecule, because the adiabatic invariant is conserved and hence, the energy of oscillations depends only on the current eigenfrequency. In terms of quantum mechanics we’ll say exactly the same, but the motivation will be different: the adiabatic pulse contains no photons which are resonant with the oscillator.

Another important role of the adiabatic invariant is protecting us from the cosmic radiation (in “collaboration” with the magnetic field of the Earth). It appears that the motion of a charged particle in a magnetic field can be represented as an Hamiltonian motion (we skip here the definition of the Hamiltonian motion as it would go too deeply into the subject of theoretical mechanics), with a re-defined momentum. It appears also that with this new momentum (the so-called generalized momentum), the adiabatic invariant of a gyrating (helicoidally moving) charged particle is its magnetic dipole moment (which is proportional to the magnetic flux embraced by the trajectory, hence this flux is also conserved). So, if a charged particle moves helicoidally along magnetic field lines towards a stronger magnetic field, due to the conservation of its magnetic moment, the perpendicular (to the magnetic field) component of its velocity will increase. Owing to the conservation of its kinetic energy, the parallel component of the velocity will decrease, and at a certain point, it becomes equal to zero: the particle is reflected back (Figure 3). This is exactly what happens with a majority of the charged particles approaching Earth along the field lines of its magnetic field.

Adiabatic invariant has simple every-day applications, too. Suppose you try to carry a cup of coffee – this will be quite simple even if the cup is completely full. Now try the same with a plate of soup – at least with full plate, this will be quite difficult! Finally, with a large full photographic tray, this will be nearly impossible! The reason is that when you try to keep your hands motionless, they still move slightly, but the feedback from your vision allows you to correct the mistakes. The characteristic time-scale of such a motion of hands is of the same order of magnitude as your reaction time, in the range of 0.2 – 0.4 s. This is to be compared with the reciprocal of the circular eigenfrequency ω₀^-1 of the water level oscillations. (ω₀^-1 differs from the full period T by 2π; ω₀^-1 serves as a better reference here, because the corrective motion of hands represents no more than a quarter of a full period of an oscillatory motion.) For a plate of depth h and length L, the smallest eigenfrequency can be estimated as the frequency of standing waves of wavelength 2L (see also problem No 2 of IPhO-1984). The speed of shallow water waves is (gh)^1/2, so that the eigenfrequency will be  f₀ = (gh)^1/2/2L. For a cup of coffee, the diameter and depth can be estimated as 7cm, hence the characteristic time scale of oscillations will be ω₀^-1 ≈ 0.03s; with respect to such oscillations, the hand motion is adiabatic – even if we apply our smallest estimate of 0.2s (note that counter-intuitively, here a slow reaction is better than a fast one). For a plate of H = 3cm and L = 25cm we get ω₀^-1 ≈ 0.15s – the hand motion is already not very adiabatic. Finally, for a photographic tray of H = 3cm and L = 60cm, we obtain ω₀^-1 ≈ 0.35s, which is really difficult to handle.

Finally, in the context of adiabaticity, it is interesting to analyse the IPhO problem about tides, which was posed in 1996 in Oslo (as Problem No 3). The problem is, indeed, very interesting:  you are given a simplified model of a complex and important phenomenon, which, regardless of simplicity, gives you reasonable estimate and teaches valuable physical concepts. Let us read its text and comment the model assumptions.

In this problem we consider some gross features of the magnitude of mid-ocean tides on earth. We simplify the problem by making the following assumptions:
(i) The earth and the moon are considered to be an isolated system,
/a very reasonable assumption: even the effect of the Sun is small in the reference frame of Moon-Earth centre of mass, where the inertial force and Sun gravity cancel each other out/
(ii) the distance between the moon and the earth is assumed to be constant,
/also reasonable: there are small variations, but nothing to worry about/
(iii) the earth is assumed to be completely covered by an ocean,
/this is definitely not the case, but at least the Pacific Ocean is very large; as a model, why not …/
(iv) the dynamic effects of the rotation of the earth around its axis are neglected, and
/Did you understand what they wanted to say? If not, you need to learn reading the problem texts! Well, it means that the forcing of the water by the Moon is to be assumed to be adiabatic (slow), so that the water level will take a quasi-equilibrium position (ie. equilibrium, where the equilibrium state changes slowly in time). The validity of this assumption will be discussed below./
(v) the gravitational attraction of the earth can be determined as if all mass were concentrated at the centre of the earth.
Again, a perfectly reasonable assumption: the gravitational field of a sphere (assuming that the mass density depends only on the distance from the centre) is outside the sphere the same as that of a point mass. The departure of the Earth's shape from a sphere is small, indeed.

And so, is the tide forcing really adiabatic? We need to compare the period of forcing with the eigenfrequency, or, the speed of the "piston" with the speed of waves. The speed of the "piston" is the Earth perimeter divided by 24 h, ie. v = 460 m/s. The relevant wave is, in effect, a tsunami with the estimated speed of (gH)^1/2 = 200 m/s (here, H = 4000 m is an estimate for the average ocean depth). So, the forcing is far from being adiabatic, we could say that the assumption (iv) is horribly wrong. On the other hand, if we solve the problem according to these assumptions, we obtain for the tide amplitude h = 27 cm, which has at least a correct order of magnitude; why? Well, because for a typical resonance response curve, the response amplitude at a double eigenfrequency (which we would need as the "piston" speed is ca twice the wave speed) is of the same order of magnitude as that of a zero frequency (which is obtained in this Problem). Further, since the tidal motion of the water is by no means quasi-stationary, the ocean boundaries will play an important role. What will happen is very similar to the motion of tea in a cup, when you push the tea by a spoon: basin boundaries reflect the moving water, creating vortices and complex pattern of tidal heights. To conclude, we learned that the above tide model fails for water tides (providing a very rough estimate of the tidal height); perhaps it can be used somewhere else with a better accuracy? The answer is "yes, for the tides of the Earth crust "! Indeed, the mantle thickness is of the order of few thousands km, which corresponds to almost ten-fold tsunami speed and makes the Moon as a "piston" reasonably adiabatic. The relative crust deformation due to tidal movements is so small that the elastic response of the crust is also negligible: the result h = 27 cm is indeed very close to reality.

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Jaan Kalda, Academic Committee of IPhO-2012