By Mark Levi
In the early days of quantum mechanics, physicists wondered why Planck’s constant is constant — why is the energy/frequency ratio of photons fixed? After all, the atom that emits photons is buffeted by a surrounding electromagnetic field; it is surprising that this buffeting does not change the ratio.
At a Solvay Conference in 1911, Einstein showed that a slightly analogous phenomenon occurs for the mathematical pendulum. If the string’s length is changed appreciably but slowly enough (taking a long time), then the ratio changes arbitrarily little. Quantities that behave in such a way are called adiabatic invariants.
Einstein derived the adiabatic invariance of the pendulum from the conservation of energy; the work of pulling the string is spent on lifting the bob and on changing the bob’s energy of oscillations. With some massaging, this statement yields the conclusion that \(E/ \omega\) is an adiabatic invariant.
In addition to Einstein’s physical explanation (which I believe can be made rigorous), another one exists that is based on the action-angle variables [1]; we can actually turn it into a geometrical proof that is almost free of formulas.
The coexistence of two seemingly unrelated explanations of the same effect suggests the need to rise above the maze to see the whole picture at a glance; as far as I know, this remains to be done. Instead, I would like to illustrate Einstein’s idea on a simple toy model of Ulam’s ping-pong: a particle bouncing between two walls — a baby model of ideal gas (see Figure 1).
Figure 1. Ulam’s ping-pong, wherein a particle collides with the walls without loss of energy. There is no gravity. The particle exerts average force \(F=mv ^2 /L\) on each wall: the analog of pressure in this single-atom gas.
Unlike in Figure 1, the wall in Figure 2 moves in slowly. The work spent on pushing the wall adds to the kinetic energy of the “molecule”:
\[FdL\approx d\biggl( \frac{mv ^2 }{2} \biggl).\tag1\]
Here, \(F\) is the averaged force of impacts on the wall in Figure 1 for the fixed wall; the "\(\approx\)" sign is due to the fact that I replaced the averaged force that the mover applied in Figure 2 with \(F\) — associated with the fixed wall in Figure 1. I claim that
\[F= \frac{m v ^2 }{L};\tag2\]
this is proven at the end.
Substitution of \((2)\) into \((1)\) gives
\[\biggl( - \frac{m v ^2 }{L} \biggl) dL\approx d(mv ^2 /2).\tag3\]
The minus sign is due to the fact that the mover in Figure 2 pushes left, i.e., in the negative direction. Rearranging \((3)\) yields
\[\frac{dL}{L} + \frac{dv}{v} \approx 0 \ \ \ \ \ \hbox{or} \ \ \ \ \ d\big( \underbrace{ \ln vL }_{\text{entropy}}\big) \approx 0 \]
so that \(vL \approx {\rm const}\).
Figure 2. Moving the wall in slowly.
Here are some remarks:
- \(vL\) is the area of the path in the phase plane when the walls are fixed (see Figure 3).
- \(\ln vL\), the log of the phase area, is the entropy of our “gas.” It is additive because phase volumes multiply when we add more dimensions. Just like in this simple case, the entropy of ideal gas is the log of the volume that is enclosed by the energy hypersurface in the phase space. It is also an adiabatic invariant of the gas.
- As with the pendulum, the energy/frequency interpretation for Ulam’s ping-pong holds: \(vL = \frac{E}{ \omega }\), where \(E = v ^2 /2\) is the energy of unit mass and \(\omega = 1/T = 1/(2L/v)\).
- The adiabatic invariance of \(vL\) is the one-dimensional version of the adiabatic equation for ideal gas: \[pV^\gamma = {\textrm{const.,}} \ \ \ \ \ \hbox{where} \ \ \ \ \ \gamma = \frac{f+2}{f}.\tag4\]Here, \(f\) is the number of degrees of freedom; \(f=1\) and \(\gamma=3\) for our “gas.” To see this connection, we note that pressure, volume, and temperature \((p, V, T)\) in ideal gas correspond to our \(F\), \(L\), and \(m v ^2 /2\) respectively. Since \((4)\) involves pressure and volume, we also wish to express our adiabatic invariant \(vL\) in terms of “pressure” \(F\) and “volume” \(L\). Substituting the \(v\) from \((2)\) into \(vL\) yields \[vL= \sqrt{ FL ^3 /m },\] so that \(FL^3 \approx {\rm const}\). This is the exact counterpart of \((4)\) for \(f=1\).
- \(p V ^ \gamma\), or rather its power, has a geometrical interpretation as the volume enclosed by the energy surface in the phase space.
Figure 3. Phase plane of Ulam’s ping-pong. The wall moves only during time \(0 < t < 1/ \varepsilon\) and is otherwise fixed. The area changes minimally: \(v_0L_0=v_1L_1+ O ( \varepsilon )\).
Proof of (2)
The momentum in each collision with the right wall in Figure 1 changes from \(mv\) to \(-mv\), i.e., by \(2mv\) at each impact. These impacts are spaced at times \(T=2L/v\) apart. The change of momentum per unit time (the force) is hence
\[\frac{2mv}{T} = \frac{m v ^2 }{L} ,\]
as claimed.
Interestingly, this is exactly the same force as the centripetal force upon a mass \(m\) that moves in a circle of radius \(L\) with speed \(v\); in fact, we can derive \((2)\) by considering motion with speed \(v\) on a circle, but one of radius \(L/2\); I omit this alternative derivation.
The figures in this article were provided by the author.
References
[1] Arnold, V.I. (1989). Mathematical methods of classical mechanics. New York, NY: Springer-Verlag.
Mark Levi ([email protected]) is a professor of mathematics at the Pennsylvania State University.