The sand that chooses a side

In 1871 James Clerk Maxwell imagined a vessel of gas divided in two by a wall with a small hole in it, and a being — he called it a “very observant and neat-fingered” one; later writers called it a demon — who watched the molecules approach the hole and opened a shutter only for fast ones going one way and slow ones going the other. Without doing any work it would make one side hot and the other cold, in defiance of the second law. The bit that has to be paid for explains why no demon can actually do it: knowing which molecule is fast is information, and erasing that information costs at least as much entropy as the sorting saves.

Replace the gas by sand and the demon is not needed. Divide a box of grains — sand, small steel balls, glass beads — into two compartments with a slot in the wall between them, a few centimetres above the floor, and shake the box up and down. Shake it hard and the grains fly about like a gas, crossing the wall in both directions, and the two compartments hold equal numbers. Reduce the shaking a little and the balance breaks: grains pile into one compartment, which becomes a nearly still heap, while the other is left with a few grains bouncing violently. Hans-Joachim Schlichting and Volkhard Nordmeier showed it as a teaching demonstration in 1996; Jens Eggers explained it in 1999, in a paper called Sand as Maxwell’s demon; and the group of Detlef Lohse and Ko van der Weele at Twente measured it carefully in the years that followed.

The explanation is the gas that cools itself into clumps put in a box with a wall across it. Inelastic grains that are crowded collide more often, lose more energy, and become colder than grains that are sparse; a cold gas presses less and throws fewer grains up to the slot. A compartment that holds slightly more grains than its share therefore sends slightly fewer across — and from there the imbalance grows by itself.

More grains can mean fewer leaving #

The whole phenomenon is in the shape of one curve. How many grains leave a compartment per second through the slot depends on how many are at the slot’s height and how fast they are moving there. Both depend on the number of grains in the compartment, $n$, in opposite directions. More grains means more grains available to leave, and the flux rises. But more grains also means more collisions per grain per second, each losing a fraction of the relative motion along the line of centres, so the gas is colder; a colder gas is more tightly bunched near the vibrating floor and fewer of its grains reach the slot. The flux model Eggers proposed, which the Twente experiments fitted with two constants, puts both effects in one line:

$$F(n) = A\,n^2\, e^{-Bn^2}.$$

The factor $n^2$ is the supply, the exponential is the cooling. The constant $B$ collects everything about the experiment: it grows with the total number of grains, with the energy each collision loses and with the height of the slot above the floor, and it falls as the shaking gets more vigorous. $B$ is therefore the knob, and turning the shaking down turns $B$ up.

The curve rises, peaks at $n = 1/\sqrt{B}$ and then falls. That peak is the whole story. A crowded compartment beyond the peak sends out fewer grains the more crowded it gets.

Where the flux comes from #

The model is not arbitrary, and taking it apart shows which physical facts the bifurcation depends on. A shaken compartment holds a granular atmosphere. Grains thrown up by the floor rise, slow, fall back and are thrown up again, and if their random speeds have a spread set by a granular temperature $T$, the density falls off with height the way an isothermal atmosphere’s does, exponentially, with a scale height $T/mg$ — the same arithmetic as the exponential that decides everything, with a shaking floor in place of a heat bath. The fraction of the grains that reach a slot at height $h$ is then about $e^{-mgh/T}$, and the flux through the slot is that fraction times the number of grains times their speed.

The temperature is where the grains’ number enters a second time. The floor feeds energy into each grain it strikes at a rate set by how hard it shakes. Collisions between grains take energy out, each removing a fraction $1-e^2$ of the relative kinetic energy along the line of centres, where $e$ is the coefficient of restitution, and the rate of collisions per grain rises with the number of grains in the compartment. Balancing the two gives a temperature that falls as the compartment fills. In Eggers’s estimate it falls as the inverse square of the number of grains, which puts $n^2$ into the exponent and gives the formula above; other ways of estimating the energy balance give other powers, and the experiments are fitted with the square.

What matters for everything that follows is only that the temperature falls fast enough with $n$ to overwhelm the growing supply of grains at some filling. That requires the collisions to be inelastic. With $e = 1$ nothing is lost when grains meet, the temperature does not depend on the crowding, the exponential is the same on both sides, and the flux rises for ever with $n$ — the grains then share themselves equally, as molecules do. The pitchfork below is the price of inelasticity and nothing else.

Shake more gently and the grains choose a side #

The number of grains in the first compartment changes at the rate grains arrive from the second minus the rate they leave: $\dot n = F(1-n) - F(n)$. Equal shares, $n = \tfrac12$, always balance. Whether they are stable depends on which way a small excess moves. Put a few extra grains in the first compartment: its outflow changes by $F'(\tfrac12)$ times the excess, and the other compartment’s outflow by the same amount with the opposite sign. If $F$ is still rising at one half, the fuller side sends out more and the excess is returned. If $F$ has already peaked, the fuller side sends out less and the excess grows.

So the equal state is stable exactly when the flux curve’s peak lies beyond one half — when $1/\sqrt{B} > \tfrac12$, that is $B < 4$. Past $B = 4$ it is unstable, and the grains run away to one of two new steady states where the flows balance again with one compartment crowded far past the peak and the other nearly empty. The two branches grow smoothly out of the equal state as $B$ passes 4, the shape a physicist recognises as a pitchfork bifurcation, and it is the same shape as a magnet cooled through its Curie point: above the critical point one symmetric state, below it two equivalent states that break the symmetry, with nothing in the rules to choose between them.

That last point is the unsettling part. The box is left–right symmetric, the grains are identical, the shaking is uniform, and the steady state the grains end up in is not symmetric. The symmetric state still exists — it is the dashed line — but it is the one state the grains will not stay in, because every fluctuation away from it grows.

Four hundred grains, one hop at a time #

The rate equation treats the grains as a continuous fluid. A real box holds a finite number, and they cross one at a time, at random. The simulation makes that explicit: at each step one grain crosses, from whichever side the flux model’s rates pick, at a random time drawn from those rates. The curves are therefore noisy, the way the count in a real experiment is.

On the stable side of the bifurcation the noise does nothing lasting. At $B = 3$ the share of grains in the first compartment wanders a few per cent either side of one half, and every wander is pulled back. On the unstable side the noise decides the outcome. At $B = 6$ the share wanders at first exactly as before; then a wander happens to go far enough that the compartment it favours has cooled appreciably, and from then on the drift is systematic. The first compartment empties to about eight per cent and stays there. Which compartment fills was decided by a random excess of perhaps a dozen grains in the first ten hopping times — rerun with another seed and the other compartment fills about half the time.

With few grains the fluctuations are large compared with the difference between the states, and a cluster can occasionally be kicked back out and re-form on the other side. With many grains that almost never happens. The time the grains wait before switching sides grows exponentially with their number, as the escape time from any well does when the barrier is proportional to the size of the system — the magnetisation that nothing keeps for ever is the same calculation — so a box of a few hundred grains is already, for practical purposes, locked in whichever state it chose.

The empty side is the hot side #

The temperature of a granular gas is borrowed language, and the drawing makes the borrowing precise: it is the mean kinetic energy of the grains’ random motion, set by a balance between the energy the vibrating floor pumps in and the energy the collisions throw away. In the crowded compartment the collisions win and the grains barely move; in the sparse compartment they rarely meet each other and fly about at the speed the floor gives them.

That is precisely the separation Maxwell’s demon was supposed to be unable to make — fast particles on one side and slow ones on the other, from an initial state in which both sides were the same — and it has been made by nobody. The escape is not subtle. The second law applies to a system that is isolated, or in contact with a single reservoir, and whose microscopic dynamics runs the same backwards; a shaken box of grains is none of those things. Energy flows in through the floor at one rate and out as heat in the grains at another, and a system with a steady flow of energy through it can hold a steady state far from equilibrium, with as much order in it as the flow will pay for. The ordering costs nothing thermodynamically because the dissipation in the collisions is far larger than the entropy the sorting removes.

What the granular demon shows instead is what an arrow of time looks like when it is built into the particles. The equation that only runs forwards is irreversible because of statistics, while each molecular collision is perfectly reversible. An inelastic collision is irreversible one collision at a time: film two grains colliding and running the film backwards shows them gaining energy from nothing. A gas made of such particles does not need to be driven towards uniformity; it is driven by its own dissipation towards whatever makes the dissipation largest, and a crowded cold heap dissipates more of the floor’s energy than two half-full compartments do.

With three compartments the grains remember #

Add a third compartment and the transition changes character. By the same argument as before, equal thirds are stable as long as the flux curve is still rising at one third, which holds until $B = 9$. But a clustered state — one compartment holding almost everything, two nearly empty — becomes available earlier, at $B = 6.55$, as a pair of states born together at a fold: the full branch, which is stable, and the dashed branch, which is not. Between 6.55 and 9 both the equal state and the clustered state are stable, and the dashed branch is the watershed between them.

That makes the box remember. Start with vigorous shaking and slowly reduce it: the grains stay in equal thirds all the way to $B = 9$, where that state disappears from under them and they collapse abruptly into a cluster. Now slowly increase the shaking again: the cluster persists all the way back to $B = 6.55$, where it in turn disappears and the grains spread out suddenly into thirds. The two jumps happen at different settings, and at any setting between them the state of the box says which way the knob was last turned. The Twente group saw exactly this hysteresis with three or more compartments, and saw the continuous pitchfork with two.

Near the fold the dynamics also slows dramatically. Just past $B = 6.55$ on the weak-shaking side, a cluster beginning to break up passes through the region where the fold used to be and lingers there, because the rate of change nearly vanishes where the two branches met — the ghost of a state that no longer exists, the same slowing described for the calm before a chaotic system settles. In an experiment it shows up as a cluster that seems stable for a long time and then gives way.

The granular clock and the sorting of sizes #

The model has been pushed in several directions. With two kinds of grain — large and small — in two compartments, the flux of each depends on both populations, and for some settings neither the mixed nor the segregated state is stable: the grains gather in one compartment, the small ones leak over, the large ones follow, and the cluster moves back and forth between the compartments with a regular period. That granular clock is a relaxation oscillation in a system with nothing oscillating in it, and it is a cousin of the big grain that comes to the top, where shaking sorts a mixture that shaking would mix if it were a gas.

With many compartments in a row, clustering can be used to transport grains: a slight asymmetry in the walls, repeated along the row, biases which neighbour a cluster forms in, and grains can be moved uphill along a conveyor with no net force pushing them, the granular version of a ratchet. Like the ratchet a fluctuation cannot run, it can do this only because it is driven and dissipating; switch off the shaking and nothing moves at all.

What the flux model leaves out #

One number for a compartment. The model describes each compartment by the fraction of grains in it and a single temperature. Real granular gases are strongly inhomogeneous — dense near the floor, dilute above — and at high densities they stop being gases at all, becoming a heap that the floor merely jiggles. The fitted flux function absorbs all of that into two constants.

A temperature that depends only on the number. The drawn temperatures assume the model’s inverse-square dependence on the share of grains. In a nearly empty compartment, where grains almost never meet one another, the temperature stops rising at whatever the floor gives a single bouncing grain, and the steep rise of the sparse compartment’s curve is the model running past its own range.

Point compartments. The slot is treated as a small hole joining two well-mixed gases. A wide slot, or a wall of finite thickness, adds grains that bounce on the wall and a flux that depends on the grains’ velocity distribution, and those details move the critical value of $B$.

A symmetric box. Any asymmetry in the compartments’ sizes or in the shaking tilts the pitchfork so that one side is always preferred, and the transition becomes a smooth, if steep, shift rather than a true symmetry breaking.

What the drawings cannot show #

The figures show fractions and temperatures and not the grains. What an observer sees is the part that is hardest to draw: in the cold compartment a flat, almost motionless bed with a skin of barely hopping grains on top, and in the hot compartment a few grains flying high enough to hit the lid, colliding mostly with the walls and floor. The difference between the two sides is as stark as a solid and a gas, and the model’s two numbers — a fraction and a temperature per compartment — are a very compressed description of it.

They also cannot show whether the flux model’s form is right in detail or only well fitted. Its exponent of two in both places comes from simple arguments about how the gas’s temperature depends on its density; other arguments give other exponents. The qualitative story — a flux that peaks, a symmetric state that loses stability when the peak passes the symmetric point — survives any flux with a maximum, and the numbers do not.

Still open: a statistical mechanics for things that are always losing energy #

The flux model works because it is fitted, and the question behind it is why systems like this should be describable by a few macroscopic variables at all. For a gas in equilibrium, statistical mechanics says which states are likely: the ones with the most microscopic arrangements. For a driven dissipative system there is no general principle of that kind. Proposals exist — that such systems settle into states maximising or minimising the rate of entropy production, or that effective temperatures can be defined that play the equilibrium role — and each works in some cases and fails in others. Whether the clustered state of the shaken compartments, or the patterns of a vibrated granular layer, can be predicted from a principle rather than from a fitted flux is one of the open questions in the physics of matter held away from equilibrium.

The habit worth carrying away is to ask of any conserved quantity what the flux of it depends on. When the flow out of a region falls as the region fills, a uniform state is unstable, and the system will pile up without anything pulling the pieces together — traffic, which slows as it gets denser; grains, which cool as they crowd; and any gas whose particles lose energy when they meet.