The gas that cools itself into clumps

A layer of steel balls on a plate, shaken up and down violently enough, stops behaving like a heap and starts behaving like a gas. The balls fly in every direction, collide with each other, fill whatever volume they are given, and exert a pressure on the walls. It is the regime at the far end of the shaking, beyond the sorting and convection that milder shaking produces, and the vocabulary of the kinetic theory of gases applies to it almost word for word — density, pressure, a temperature measured as the mean kinetic energy of a grain. What the vocabulary hides is the one difference that decides everything. A molecule’s collisions keep the energy. A grain’s collisions lose some of it, and a gas whose collisions lose energy does not stay a gas.

A temperature that only falls #

In a molecular gas a collision exchanges energy between two molecules and destroys none of it, so the total kinetic energy is fixed and a gas left alone in an insulated box stays at the temperature it started at. The distribution of speeds settles into Maxwell’s, and nothing further happens.

Two grains colliding do something different. Their relative velocity along the line joining their centres is reversed and reduced: after the collision it is $e$ times what it was before, where the coefficient of restitution $e$ is less than one for every real material. Steel on steel is about 0.9, glass beads a little higher, sand much lower. The kinetic energy of the relative motion along that line falls by a factor $e^2$, and the difference goes into heating the grains’ interiors — into the thermal motion of their atoms, which is a different temperature altogether and plays no further part. A granular gas’s “temperature”, the mean kinetic energy of a grain’s own motion, only falls.

The word temperature is being used loosely, and it is worth saying how loosely before leaning on it. In a molecular gas the temperature is the thing two systems share when they stop exchanging energy, and the mean kinetic energy per molecule is half a $kT$ for every way of moving. A granular gas has a mean kinetic energy per grain and nothing else of that list. Two granular gases in contact do not come to a common value of it, rough grains share their energy unequally between translation and spin, and the distribution of grain speeds is not Maxwell’s — a cooling granular gas has more fast grains in its tail than a molecular gas of the same mean energy, because the slow grains in dense regions have lost their energy and the fast ones in the sparse regions have not. The granular temperature is a useful measure of how agitated the gas is. It is not a thermodynamic temperature, and the laws that make a molecular gas relax to a uniform state have no grip on it.

What does carry over is the mechanics. The pressure a granular gas exerts on a wall is a rate of arrival of momentum, exactly as for molecules — density times the mean square speed, with a correction for the grains’ own size that is large when they fill a quarter of the space. The distance a grain travels between collisions is a mean free path set by the density and the grain’s cross-section. And every collision, however inelastic, conserves momentum, as every collision does: the energy lost is lost from the relative motion only, and the motion of the pair’s centre of mass is untouched.

How fast the temperature falls follows from two lines of kinetic theory. Each grain loses a fixed fraction of its energy per collision, of order $1 - e^2$, and it collides at a rate proportional to its speed, which goes as $\sqrt{T}$. So the energy per grain obeys

$$\frac{dT}{dt} \propto -(1-e^2)\,n\,T^{3/2},$$

whose solution is Haff’s law,

$$T(t) = \frac{T_0}{(1 + t/t_0)^2}.$$

The gas cools as the inverse square of time: quickly while it is hot and collisions are frequent, and ever more slowly as it slows down and the collisions that cool it become rare. It never reaches zero in a finite time, in this uniform picture, because a slower gas cools more slowly.

The simulated gas follows the law closely at first. That is not a fit. The law’s one constant is fixed by the first fifth of the cooling, and the agreement over the following decade of time is the law being tested and passing. Then the gas leaves it, and cools more slowly than a uniform gas would. Something has changed in the way energy is being destroyed, and the first figure shows what: the gas is no longer uniform.

Why the denser patch loses #

The mechanism is an instability with no attraction in it anywhere, and it can be followed with nothing more than Haff’s law applied twice.

Take two neighbouring patches of the same gas at the same temperature, one slightly denser than the other. In an ordinary gas the denser patch has the higher pressure — pressure is density times temperature — and it pushes grains out into its neighbour until the densities even up. That is why a molecular gas stays uniform: any clump disperses. In a granular gas the denser patch also collides more often, because each grain has more neighbours, so it cools faster. Its pressure is its density times a temperature that is falling faster than its neighbour’s.

The crossover takes under two collision times for a patch 20 per cent denser. After it, the denser patch has the lower pressure and grains are pushed into it from the hotter surroundings. More grains make it denser, more density makes it cool faster, faster cooling lowers its pressure further, and the runaway has begun. The whole sequence — a region that loses pressure because it is dense, and is compressed further because it has lost pressure — is the same shape as the collapse that gravity drives in a cloud of gas, where a denser region is pulled harder by its own weight and grows. There the instability needs attraction and has a critical size; here it needs dissipation and has one too. In both, the uniform state is an equilibrium that a small disturbance cannot leave alone.

The critical size is the reason the effect is easy to miss. A patch smaller than a certain length is smoothed out by grains flying in and out of it faster than the dissipation can act, so only disturbances larger than that length grow. That length is several mean free paths divided by $\sqrt{1 - e^2}$, so for nearly elastic grains it is enormous and the gas has to be very large to cluster at all. A small box of nearly elastic grains stays uniform and obeys Haff’s law all the way down. The box in the simulation is several of those lengths across, which is why the clusters appear in it.

Why a cluster is cold #

The departure from Haff’s law in the second figure — the gas cooling more slowly once it has clumped — looks at first like a contradiction. A cluster is dense, dense regions cool fastest, and a gas full of clusters ought to cool faster than a uniform one.

The resolution is in what a collision can destroy. It removes energy from the relative motion of the two grains along the line of their centres and from nothing else. Inside a cluster the grains have spent many collisions destroying their relative motion, and what remains is mostly the cluster’s motion as a whole — a crowd of grains drifting together, each moving nearly parallel to its neighbours. The kinetic energy of that shared drift counts towards the granular temperature, because the temperature is simply the mean kinetic energy per grain, but no collision within the cluster can touch it, because the motion of a centre of mass is untouched by internal forces. Only collisions between clusters, which are rare, can reduce it.

So a clustered gas stores much of its remaining energy in a form its own collisions cannot reach, and its measured temperature falls more slowly than a uniform gas’s would. The departure from the law in the cooling figure is a measurement of that: the energy has moved from random motion, which the collisions destroy at Haff’s rate, into collective motion, which they barely touch. A cluster is cold inside and moving as a body, and the temperature of the whole gas is a mixture of two quite different kinds of motion that the one number cannot distinguish.

At high enough density the grains in a cluster stop moving relative to each other altogether. Contacts become lasting rather than momentary, forces are carried along chains of touching grains, and the cluster has become a little solid — the transition by which a heap becomes rigid, reached from the gas side rather than by pouring. A cooling granular gas therefore visits every state that grains can be in, gas, liquid-like flow and jammed solid, without any change of temperature in the thermodynamic sense and without any external agent: only its own collisions.

Counting the grains #

A picture can suggest clumps where there are none — the eye finds patterns in scattered points — so the clustering is worth measuring rather than seeing.

Points scattered independently fall into cells with a count whose variance equals its mean. The starting gas is more even than that, because discs that occupy a quarter of the area cannot overlap and are therefore spread out more regularly than points would be — a molecular liquid is the same, and the ratio is what the compressibility of a dense gas measures. The cooled gas is seventeen times more variable than random points, which is not a statistical fluctuation of a uniform gas at any temperature. The grains have moved.

What the counts cannot say is whether the clusters are permanent. In the simulation they grow, merge and move, and over longer times they coarsen into fewer and larger structures — bands that wrap round the box, in a box with periodic walls — whose size is set by the box rather than by the physics. A real granular gas is always in a container, and its walls and the gravity pulling it to one side are part of what it becomes.

The limit where collisions never end #

Inside a dense cluster the grains move together with small relative speeds, and each collision takes away a fraction of an already small speed. In the extreme this becomes something that sounds impossible: a finite number of grains colliding an infinite number of times in a finite time.

The threshold for three grains is $7 - 4\sqrt 3$, about 0.072. Below it, each pair of collisions multiplies the relative speeds by a factor less than one, the times between collisions shrink in proportion, and a geometric series of ever-shorter intervals adds up to a finite time. At that time the three grains are in contact and moving together; the sequence of collisions has an accumulation point, and the simple model of instantaneous binary collisions says nothing about what happens after it. The phenomenon is called inelastic collapse. It is the many-grain relative of a ball bouncing on a floor, whose bounces also form a geometric series and stop after a finite time, and it is exactly why a simulation of a granular gas has to be told what to do when relative speeds become tiny. The one behind the figures here treats collisions below a ten-thousandth of the starting speed as elastic, which stops collapse without affecting anything larger.

In a long chain of grains the threshold rises towards one — collapse needs only a slightly inelastic material when enough grains are lined up — and inside the clusters of a cooling gas, where many grains move nearly together, it happens readily. In a real material it is cut off by something the model leaves out: grains are not perfectly rigid, a collision lasts a finite time, and at low enough impact speeds real contacts become nearly elastic or stick. Where exactly the cut-off falls is a property of the material’s surface rather than of the gas.

A restitution that depends on the impact #

The coefficient of restitution is not a constant. It depends on the impact speed, rising towards one for gentle impacts and falling for violent ones as the contact deforms plastically. A constant $e$ is a first approximation that gives Haff’s law exactly; a speed-dependent one changes the late-time cooling and softens the clustering.

Rotation and friction are ignored. Real grains are rough, and a glancing collision exchanges energy between the translation and the spin of the grains. A gas of rough grains has two temperatures — one for translation and one for rotation — which settle into a fixed ratio, and the cooling rate depends on both.

There is no gravity and no container. The simulation lives in a box whose walls wrap round, which is the cleanest way to study the gas and the least like any real apparatus. A granular gas under gravity forms a dense layer at the bottom and a thin atmosphere above it; a driven one, fed energy through vibrating walls, reaches a steady state in which clusters form where the energy input is weakest.

Two dimensions are not three. The discs are a model of a monolayer on a plate, which is how many experiments are done. In three dimensions the instability works the same way and its details differ.

Clusters that move, merge and shear #

The three panels of the first figure are three moments of a continuous motion, and what they cannot show is the motion itself: the clusters are not static. Each is a crowd of grains moving roughly together with a small spread of velocities about their common motion, and clusters collide, merge and shear past each other. A still frame shows where the grains are and not that the dense regions are also the cold ones — the local temperature inside a cluster is far below that in the space between, which is where the few fast grains live.

Nor can a simulation of two thousand discs establish what a gas of a billion grains would do. It shows the instability, the departure from the law and the growth of clusters, and it is small enough that the clusters soon reach the size of the box. Every conclusion here about the long-time state is a conclusion about this box.

Still open: what a cooling granular gas settles into #

The early stages — Haff’s law, the instability, the critical size — are well described by a kinetic theory of inelastic grains that extends the theory of ordinary gases. What happens at long times is not settled. Simulations in large boxes find the clusters coarsening without end, with a characteristic size growing as a power of time, and some find the late-time dynamics resembling the sticky collisions of particles that merge on contact — a model from cosmology for the formation of structure from a smooth start — while others find differences. Whether a freely cooling granular gas in an unbounded space has a universal late state, and what law its cooling follows once the clusters dominate, is still an active question in the theory of dissipative gases.

Past that lies the gas that is kept hot rather than left to cool — shaken from below, sheared between plates, poured down a slope — where the dissipation in every collision is balanced by an input of energy and the gas reaches a steady state that is not an equilibrium. The habit worth carrying from here is to ask of any uniform state whether the process that maintains it can also be undermined by it. A gas stays uniform because a denser region has the higher pressure; make denser regions lose energy faster and that single sign flips — and the uniform state becomes the one arrangement a disturbance will not leave alone.