The wall that moves while the ball is in flight

Energy conservation is usually met as a fact about the world and it is not one. It is a consequence, and the thing it is a consequence of is that the laws do not change with time — the same symmetry argument that supplies momentum and angular momentum, applied to translations in time rather than in space.

So the way to break energy conservation is to make the rules depend on the time. Nothing exotic is needed. Move a wall.

The system whose energy is not conserved is the ball. The larger system — the ball together with whatever is holding and moving the wall — has rules that do not change with time, and its energy is conserved exactly — which is what the symmetry hands over and all it hands over. Nothing has broken. What has happened is that a system with a time-dependent boundary condition has been carved out of a larger one, and such a system has no conserved energy of its own.

That is not a loophole. It is the ordinary situation of every part of every machine.

Slowly, and the answer is an old one #

Move the wall slowly and the arithmetic is simple enough to do in a line.

While the wall retreats a small distance $|dL|$ at speed $u$, the ball makes about $v|dL|/2Lu$ round trips, losing $2u$ of speed at each collision with the retreating wall. So

$$dv = -2u \cdot \frac{v\,|dL|}{2Lu} = -\frac{v\,dL}{L},$$

and the wall’s speed cancels out. Integrating gives $vL = \text{constant}$, so $v \propto 1/L$ and the energy goes as $1/L^2$.

That constant is not an accident of this problem. It is the area of the ball’s path in phase space, which for a ball bouncing between walls is a rectangle two momenta tall and $L$ wide — and the statement that it is preserved under a slow change is the adiabatic theorem.

The theorem says an area in phase space is preserved when a parameter of the system is changed slowly compared with the motion the area belongs to. It is why the ratio of a pendulum’s energy to its frequency is constant when its string is slowly shortened; it is why a magnetic moment is conserved for a charged particle in a slowly changing field; and it is the classical ancestor of the rule that a quantum system in its ground state stays there if the change is slow enough, which is what makes an avoided crossing a crossing or not.

What “slowly” means is the whole content of it. Here the motion is a round trip taking $2L/v$, and the change is slow when the wall moves a small fraction of $L$ in that time — which is the condition $u \ll v$.

Quickly, and the answer is different in both directions #

The two directions fail differently and it is worth separating them, because between them they are the whole of why a fast process is irreversible.

Expanding fast, the ball keeps its energy. A wall retreating faster than the ball travels is never caught, so after the last collision nothing more happens: the box grows and the ball’s speed does not change. That is a free expansion — a gas let into a vacuum does no work and cools not at all, which is heat and temperature coming apart — and the figure draws it for one particle.

Compressing fast, the ball gains more than the adiabatic amount. The wall is moving toward the ball, so the ball meets it more often than the slow estimate says and each meeting adds $2u$ to a speed that is already rising. The extra energy is real work done by whatever is pushing the wall, and it ends up in the ball — work in the strict sense the strict sense the essay on pseudowork insists on, since the wall’s own point of application moves.

Put the two together and the asymmetry is exactly the second law in miniature. Compress quickly and then expand quickly and the ball does not come back to where it started: it is left with more energy than it began with, and the extra came from the agent that moved the wall. Do both slowly and it returns exactly. A cycle of fast processes is not a cycle, and the excess is the dissipation.

Nothing about that argument mentions many particles, entropy or temperature. It is one ball in a box, and the irreversibility is already there.

That is worth dwelling on, because irreversibility is usually introduced as a statistical fact — something that emerges from counting arrangements once there are enough particles for the counting to be overwhelming. Here there is one particle and no counting anywhere, and a cycle of fast processes still fails to return the system to its starting state.

The two accounts are not in competition. What the single ball shows is that the mechanical asymmetry between fast and slow is already present in the equations of motion: a fast process transfers more energy in one direction than a slow one does in the other, for reasons of timing rather than of statistics. What the counting adds is the reason the asymmetry cannot be undone by any clever arrangement, which does require many particles. The single ball can in principle be returned exactly, by moving the wall back through precisely the reverse history at precisely the reverse phases. A gas cannot, and the impossibility there is about how many phases would have to be got right at once.

So the one-particle picture is the mechanical half of the second law with the statistical half removed, and it is a useful thing to have seen separately — not least because the statistical half is usually presented as though it were all of it.

The invariant is a phase-space area, and that is not a metaphor #

The phase-space statement deserves a paragraph on its own, because it is the form in which the result generalises and because it looks like bookkeeping until it is used.

A ball bouncing between walls has, at any instant, a position between $0$ and $L$ and a momentum that is $+p$ or $-p$. Over one round trip it traces a closed rectangle in the plane of position and momentum, of area $2pL$. That number is the action of the motion, it is the same quantity that appears in the statement of mechanics as a rule about whole paths, and it is what the adiabatic theorem says is preserved.

Two things follow that the energy statement does not give.

It says which quantity to watch for any system. For a pendulum of slowly shortening string, the area is $E/\omega$, so a pendulum shortened to a quarter of its length doubles its frequency and doubles its energy — and the tension does exactly that much work, though following the work is far harder than following the area. For a charged particle spiralling in a slowly changing magnetic field, the area is the magnetic moment, and its constancy is the whole basis of magnetic confinement.

And it says what “slowly” is slow compared with. The area belongs to a particular periodic motion, and the change is slow when the parameter alters little during one period of that motion. A system with several periodic motions has several invariants, each with its own threshold, and a change can be slow for one and fast for another — which is how energy is moved from one degree of freedom to another in practice.

Where the theorem fails is where a period goes to infinity. Near a separatrix — the boundary between two kinds of motion, such as a pendulum on the point of going over the top — the period diverges, no change is slow compared with it, and the invariant jumps. That is not a small correction: it is a discontinuity, and it is how a slowly driven system crosses from one type of motion to another at all.

Kicks that line up, and kicks that do not #

The last case is the one that turned this arithmetic into an explanation of something.

If every collision is head-on, the speed rises by $2u$ each time and the energy grows as the square of the number of collisions. If the wall’s motion is as often one way as the other, the first-order term averages to zero — a head-on collision adds $4uv$ to $v^2$ and an overtaking one subtracts it — and what survives is the second-order term $4u^2$, which is always positive. So the energy grows in proportion to the number of collisions rather than as its square.

Both are heating. The random case gains energy on average even though the wall is as often helping as hindering, because the square is not linear. That is the Fermi mechanism, proposed in 1949 for the acceleration of cosmic rays: charged particles bouncing off moving magnetic clouds in the galaxy, gaining a little on average from an arrangement that has no preferred direction.

The trouble Fermi’s mechanism has is visible in the figure. Second-order growth is slow — the gain per collision goes as $(u/v)^2$, and the clouds move at a thousandth of the speed of light, so the gain is a part in a million per encounter. The time to reach the observed energies exceeds the time particles stay in the galaxy.

What works instead is a way of making every collision head-on. At a supernova shock front, the material ahead and behind is converging, so a particle crossing the front and scattering back across it always meets an approaching scatterer — whichever way it is going. The gain per crossing is first order in $u/v$, the growth is the upper curve rather than the lower, and it is fast enough. That is first-order Fermi acceleration, it is the standard account of cosmic rays below the knee of the spectrum, and the difference between it and the original is exactly the difference between the two lines drawn here.

And the same arrangement produces the power law. A particle in the shock region has a fixed chance per crossing of escaping and a fixed multiplication of its energy if it stays, and a process with a constant multiplication per step and a constant escape probability per step yields a power-law distribution — the same arithmetic that turns a fixed fraction per step into an exponential — of final energies with an exponent that is the ratio of the two rates. That exponent comes out near two for a strong shock, which is close to what is measured, and it does not depend on the details.

The same arithmetic, in four places that look unrelated #

The reason this is worth a place among the energy essays is that the pattern turns up wherever a boundary moves, and it is usually not recognised as the same thing.

A gas in an expanding universe. Photons travelling through space that is expanding lose energy in proportion to the expansion, and non-relativistic particles lose momentum the same way. The calculation is the one above with the wall replaced by the expansion itself, and the result — that a photon’s wavelength stretches with the scale factor — is usually derived quite differently and is the same statement. The universe’s expansion is slow compared with a light crossing time of anything small, so the adiabatic result applies and the energy is not conserved, exactly as here.

A pendulum on a shortening string. Pull the string of a swinging pendulum up through a hole and the swing gets faster and more energetic. The energy gained is the work done pulling, and it is larger than the weight of the bob times the distance pulled, because the string’s tension exceeds the weight by the centripetal term. The adiabatic invariant gives the answer in one step without computing the tension at all.

A particle in a magnetic bottle. A charged particle spiralling along a field line into a region of stronger field has a conserved magnetic moment, so its perpendicular energy rises as the field does, so its parallel energy falls — and at some point it turns round. That is magnetic mirroring, it is what traps particles in the Van Allen belts, and its invariant is the area of the particle’s own circular orbit in phase space.

And a musical string being tuned. Tighten a vibrating string and its energy rises; the ratio of its energy to its frequency does not. A player who plucks a string and then tightens it hears the pitch rise and the loudness rise together, in the fixed ratio the invariant sets.

Four systems with nothing in common except that a parameter of a periodic motion is being changed slowly, and in every one the quantity that survives is an area.

One ball is not a gas, and a fast piston makes a shock #

One ball is not a gas. The adiabatic law $E \propto L^{-2}$ for a single ball in one dimension becomes $TV^{\gamma-1} =$ constant for a gas, with $\gamma$ depending on how many ways a molecule can hold energy. The structure is the same — an invariant preserved by slow change — and the exponent is not.

A real fast compression makes a shock. In a gas, a piston moving faster than the sound speed does not compress uniformly: it drives a shock wave ahead of itself, and the gas between the piston and the shock is at a different state from the gas ahead. None of that exists for one particle, and the single-particle picture is right about the direction of the asymmetry and silent about the mechanism.

The random model here is a caricature. In the real second-order Fermi problem the collisions are not equally likely in both directions: a particle meets an approaching scatterer slightly more often than a retreating one, simply because it closes with the first faster. That imbalance is where the surviving second-order gain actually comes from, and the model drawn — an even coin toss between adding and subtracting $2u$ — gets the same scaling from a slightly different place.

And nothing here is relativistic. Cosmic rays are ultrarelativistic and the collisions have to be treated with the relativistic composition of velocities. The scalings survive, the arithmetic does not, and the energies involved are twenty orders of magnitude above anything a ball in a box would meet.

Whatever is holding the wall, which is where the energy comes from #

They cannot show the agent. Every figure here tracks the ball and none of them tracks whatever is holding the wall, which is where the energy comes from and goes to. The full system’s energy is conserved exactly; the figures draw the part of it that is not, and the missing half is off the page by construction.

Nor can they show why the adiabatic theorem is only approximate. An adiabatic invariant is conserved to all orders in the slowness parameter and not exactly: what is left over is exponentially small, of the form $e^{-c/u}$, which is smaller than any power of $u$ and is not zero. On a logarithmic axis at the resolution drawn, a quantity of that size is indistinguishable from nothing, and it is what makes an adiabatic invariant a very good approximation rather than a conservation law.

And they cannot show the phase. Whether a given collision helps or hinders depends on where the wall is in its motion at the moment the ball arrives, and for an oscillating wall the sequence of phases is itself a dynamical system — one of the first in which the transition between orderly and chaotic motion was studied. The figures average over that structure and draw the mean.

Still open: where the highest-energy particles are made #

The shock mechanism accounts for cosmic rays up to about a thousand million million electronvolts, above which the measured spectrum steepens. Above that the sources are not established: supernova remnants are too small and too short-lived to hold particles long enough, and the candidates are the jets of active galaxies, gamma-ray bursts, and processes nobody has proposed.

The difficulty is that the highest-energy particles arrive rarely — a few per square kilometre per century — and are deflected by galactic and intergalactic magnetic fields on the way, so their arrival directions do not point back to their sources. Correlations with catalogues of nearby galaxies have been reported and disputed for twenty years. What would settle it is either a source whose neutrinos or gamma rays can be seen at the same time, or enough events to see an anisotropy that survives the deflection.

The habit worth carrying away is about conservation laws and boundaries. A conservation law is a statement about a system whose rules do not change, and carving a subsystem out of a larger one usually destroys it. The energy of a ball in a moving box is not conserved, the energy of everything is, and the difference between those two statements is a wall that somebody is holding.