Illustrated Physics
physics, drawn
Concept

Kinetic theory — where it appears

The account of a gas as a large number of moving particles, from which pressure, temperature and transport are derived by averaging. It gives pressure as a rate of arrival of momentum, temperature as a mean kinetic energy, and viscosity as a transport of momentum across an imaginary surface.

Named by 13 essays across 3 fields — each of them below, with the objects they name alongside it.

Molecular speeds at 4 temperatures. The distribution of molecular speeds in a gas, with each curve enclosing the same area. Raising the temperature moves the peak right and lowers it: the same molecules, spread over a wider range of speeds.

The speeds in a still room

The air in a quiet room is not still. Every molecule in it is moving at hundreds of metres per second, and temperature is a single number summarising an entire distribution.

thermodynamics · Kinetic theory A path through a crowd. A point crossing a field of 90 scatterers, rebounding off each. The mean length of 4000 such segments is 0.2983 box widths, against the textbook form 1/2nr = 0.3086 — a departure of -3.3 per cent, from a measurement that knows nothing of the formula. It does not agree exactly and should not: the closed form is derived for a vanishingly dilute field and these discs cover 9.2 per cent of the plane. Two finite-density effects pull opposite ways — crowding shortens the path, and discs shadowing one another lengthen it — so which side of the formula a given field lands on is not something the formula can tell.

How far a molecule gets

A molecule of air travels about sixty-eight nanometres between collisions — some two hundred times its own size, and a ten-millionth of the width of a room. That ratio is the reason a gas can be treated as a continuous substance at all, and the reason it sometimes cannot.

thermodynamics · Kinetic theory The same law, across twenty-eight decades. The mean free path 1/nσ against cross-section, for a target density of 6.83·10³⁰ targets per cubic metre — solid lead. It is a straight line of slope minus one, because there is only one thing in the law. At 10⁻²⁸ m² the path is 1.46 mm; at 10⁻⁴⁷ m² the path is 1.55 light-years. Nothing about the physics changes between those ends. Only the area does.

How far a neutrino gets

A mean free path is one over the number density times the cross-section, and nothing else. Change only the cross-section — by twenty-eight powers of ten — and the same arithmetic that gives a molecule seventy nanometres in air gives a neutrino a light-year of solid lead.

astrophysics · Kinetic theory The staircase equipartition cannot climb. The heat capacity of hydrogen at constant volume, in units of R, against temperature on a logarithmic axis. The three translational directions contribute 3/2 at every temperature. The two rotations switch on near 85.4 K — the temperature at which kT matches the first rotational step — taking the total to 5/2, computed here by summing the rigid rotor's partition function over two hundred levels rather than by assuming the plateau. The vibration switches on near 6332 K, taking it to 7/2. At 50 K the value is 2.469; at 300 K the value is 2.502; at 5000 K the value is 3.376. Classical equipartition predicts 7/2 at every temperature, including at four kelvin, and the size of the steps is set by ħ — which is how a heat capacity measures a quantum constant.

Half a kT for every way of moving

A heat capacity ought to be a count. Every quadratic term in a system's energy carries half a kT of it, so warming a gas is a matter of enumerating the ways its molecules can move — and the fact that the count comes out wrong for hydrogen is how a thermometer measured Planck's constant.

thermodynamics · Equipartition The viscosity of a gas, over 8 decades of pressure. The viscosity of 3 gases at 300 K against pressure, on a logarithmic pressure axis and a linear viscosity one. The lines are flat, and that is the whole figure. Viscosity is the rate at which momentum is carried across a shear, which is the density of carriers times the distance each one carries it: ⅓ρv̄λ. Doubling the pressure doubles the density and halves the free path, and the two cancel exactly, so the same gas at a hundredth of an atmosphere is exactly as viscous as at one — which is not what anybody expects of a thinner gas and is what is measured. nitrogen comes out at 17.9 μPa·s against a measured 17.9, helium comes out at 19.3 μPa·s against a measured 19.9, argon comes out at 21.7 μPa·s against a measured 22.7. The flatness ends when the free path reaches the apparatus rather than the next layer of gas: at a vessel 10 mm across that is around 0.68 Pa for nitrogen, 1.94 Pa for helium, 0.69 Pa for argon, below which there is no gas-to-gas hand-off left to make.

The viscosity that does not care how much gas there is

Pump most of the air out of a vessel and the air that is left is exactly as viscous as it was. Maxwell derived that in 1860, did not believe it, and spent six years building an apparatus to measure it — which is a better description of how a prediction becomes knowledge than any amount of agreement would have been.

thermodynamics · Kinetic theory The line that slopes the wrong way. Mean temperature against total energy, for a self-gravitating sphere at nine radii. The points fall on a straight line of negative slope: taking energy away makes the body hotter. The heat capacity read off the drawing is -4.10e+34 J/K against the -4.10e+34 J/K the theorem gives, and the sign is the whole content. The dashed line is an ordinary gas in a rigid box, whose temperature rises when energy is added, as everything one can put a thermometer in does.

The ball of gas that heats up as it cools

Take energy away from a self-gravitating cloud and its temperature rises. Its heat capacity is negative, which nothing else stable does, and the consequence is that a star radiating into cold space is not cooling down — it is running up, and its whole life is a slow fall it cannot stop.

astrophysics · Self-gravity What a coordinate is worth, by the shape of its energy. The mean energy stored in one coordinate, in units of kT, against the power with which that coordinate enters the energy. The curve is 1/n and the points are the same average obtained by integrating the Boltzmann weight numerically, agreeing to 3.3e-7 per cent at worst. an energy going as |x|^1 holds 1.0000 kT; an energy going as |x|^1.5 holds 0.6667 kT; an energy going as |x|^2 holds 0.5000 kT; an energy going as |x|^3 holds 0.3333 kT; an energy going as |x|^4 holds 0.2500 kT; an energy going as |x|^6 holds 0.1667 kT. The familiar half a kT is the case n = 2 and nothing more general than that: a coordinate whose energy is linear in it — the momentum of an ultrarelativistic particle, or a field with no restoring force but a constant tension — carries a whole kT, and one confined by a very steep wall carries almost nothing. Equipartition is a theorem about quadratic terms, and calling it a theorem about degrees of freedom is the substitution that makes it fail.

The share that is not half a kT

Equipartition is quoted as half a kT for every degree of freedom, and it is nothing of the kind. It is half a kT for every *quadratic* term. A coordinate whose energy is linear in it carries a whole kT, and a gas hot enough that its particles' energy is pc rather than p²/2m therefore holds twice what the counting says — which drops its ratio of specific heats to four thirds and puts a star on the edge of being able to hold itself up.

thermodynamics · Equipartition One word, two mechanisms, opposite signs. Viscosity on a logarithmic axis against temperature over the range 280 to 360 kelvin, where gases and liquids can both be measured. The gases rise and the liquids fall, and the two families are separated by three decades of magnitude as well as by sign. The logarithmic slopes at the middle of the range are 0.76 for air, 0.69 for helium, -6.13 for water, -5.41 for ethanol, -23.00 for glycerol, so the steepest liquid responds 30 times more strongly than the gas and in the other direction. Nothing about the word viscosity requires this: what is being measured in both cases is the ratio of a shear stress to a shear rate, and that definition says nothing about what carries the momentum. In a gas it is molecules in free flight, so heating speeds up the carriers; in a liquid the molecules are permanently in contact and what has to happen is one of them getting past its neighbours, so heating removes an obstacle rather than adding a carrier. The obvious question this raises is what a dense gas near its critical point does, where neither picture holds, and the honest answer is that neither formula on this chart applies there at all.

The thickness that goes both ways

Heat a liquid and it thins; heat a gas and it thickens. The two are not a strong effect and a weak one but opposite signs, differing by a factor of thirty in size as well — and the word viscosity names one measurement made on two mechanisms that have almost nothing in common.

fluids · Viscosity The pair potential, and the two things it does to a gas. The Lennard-Jones potential between two molecules, in units of its own depth and range, with the Mayer function it produces at 1, 3, 8 times the well depth in temperature. The virial coefficient is minus the integral of that function over volume, so the two parts of the potential contribute with opposite signs: the steep repulsive core makes the function minus one there, giving a positive contribution — molecules take up room — and the attractive well makes it positive, giving a negative one. At low temperature the attraction dominates and a gas is easier to compress than an ideal one; at high temperature the core dominates and it is harder. Between them is one temperature at which they cancel.

The first correction to the gas law

An ideal gas has no forces between its molecules. The first correction to what it does is computable from those forces alone — one integral over the pair potential — and its sign flips at a temperature where a real gas obeys the ideal law without being ideal at all.

thermodynamics · Kinetic theory 14 decades of fluid, and a floor none of them reaches. The ratio of viscosity to entropy density for 8 substances, in units of the proposed lower bound, on a logarithmic axis. The quantity is a viscosity divided by how much entropy a cubic metre of the substance holds, and it has the dimensions of Planck's constant over Boltzmann's — so a bound on it is a statement with no material properties in it at all. The span here is a factor of 1.6e+14, from pitch at 20 °C at the top to the quark–gluon plasma at the bottom, which sits a factor of 1.6 above the floor. The values marked as computed are worked out from a viscosity and a tabulated entropy; the rest are quoted from the compilations, because a minimum along an isobar is an inference from many measurements rather than a single one.

Whether a fluid can be made arbitrarily thin

Viscosity has no units anybody would call fundamental, and nothing obviously stops it being as small as you like. Divide it by the entropy in a cubic metre and the units become Planck's constant over Boltzmann's — and a conjecture from 2005 says that ratio has a floor. Across fourteen decades of ordinary fluid nothing has been measured below it, the closest thing to it is the hottest matter ever made, and a kinetic-theory argument reaches the same number from the opposite direction.

fluids · Viscosity A gas of grains that lose 51 per cent of their energy per collision, cooling into clumps. 2000 discs in a square box with periodic walls, covering 25 per cent of its area, started with random velocities and left alone. Each collision keeps a fraction e = 0.7 of the relative velocity along the line of centres. The panels are the same gas at three temperatures, where temperature means the mean kinetic energy of a grain: 0.85 T₀, 0.019 T₀, 0.00049 T₀. It starts uniform — the grain counts in a 10 × 10 grid have a variance 0.3 times their mean, more even than scattered points because discs cannot overlap — and ends in dense bands and clumps with empty space between, the variance 17 times the mean. Nothing attracts anything. The clumps form because a dense patch collides more often, cools faster, loses pressure and is squeezed denser by the hotter gas around it.

The gas that cools itself into clumps

Shake a box of grains hard enough and they fly about like the molecules of a gas. Stop shaking and the gas cools, because every collision destroys a little of the motion — and it does not cool evenly. A denser patch collides more often, cools faster, loses pressure, and is squeezed denser by the hotter gas around it. With nothing attracting anything to anything, the gas gathers itself into clumps and bands, and in the limit three grains can collide infinitely many times in a finite time.

fluids · Granular matter How well a diamond-like crystal conducts heat, from 1 K to 1000 K. The thermal conductivity of a model crystal with diamond's sound speed and Debye temperature, from kinetic theory applied to its phonons, against temperature on logarithmic axes: natural carbon in a sample 1 mm across, isotopically enriched carbon, and natural carbon in a sample 0.1 mm across; copper's room-temperature 400 W/m K is marked for comparison. The model is calibrated to natural diamond's 2,200 W/m K at room temperature. At low temperature the conductivity rises as the cube of the temperature, limited by the sample's walls; at high temperature it falls as umklapp collisions shorten the phonons' paths. Between, it peaks: 3.7·10⁴ W/m K at 100 K for the natural crystal; 7.8·10⁴ W/m K at 94 K for the enriched crystal; 1.2·10⁴ W/m K at 126 K for the thin natural crystal. The model gets the shape — the rise as the cube of the temperature, a peak near 100 K, the fall — but overestimates the heights, because it leaves out the phonon collisions that conserve momentum and every defect but isotopes; measured enriched diamond peaks near 4 × 10⁴ W/m K. An insulator with no free electrons still conducts heat five times better than copper at room temperature.

The gas of sound that carries a diamond's heat

Diamond has no free electrons and conducts heat five times better than copper. What carries the heat is a gas — not of molecules but of quantised sound waves, phonons — and the kinetic theory that gives the conductivity of air gives the conductivity of a crystal with the same formula: a third of heat capacity times speed times mean free path. The gas is strange in two ways. At low temperature its particles cross the whole crystal without meeting anything, so the conductivity depends on how big the sample is. And at room temperature one atom in a hundred being a neutron heavier is enough to scatter its fastest-oscillating members.

thermodynamics · Kinetic theory A sound that survives only in a hot-electron plasma. The damping of an ion-acoustic wave at kλD = 0.3, as the fraction of its amplitude lost per radian of oscillation, −γ/ω, against the ratio of electron to ion temperature, for hydrogen and argon plasmas, on logarithmic axes: from the kinetic dispersion relation (solid) and from the weak-damping formula (dashed), which is valid only where the damping is small. With equal temperatures the wave loses 0.45 of its amplitude per radian in hydrogen — it is gone within a cycle. At Tₑ/Tᵢ = 10 the loss is 0.031; at 100 it is 0.0126, set by the electrons alone. Argon's floor is lower, 0.0020, because its ions are heavier and its electrons a smaller fraction of a sound wave's speed.

The sound a plasma carries on its electrons' heat

Sound in a gas needs two things: something heavy to be pushed, and a pressure to push it. In an ordinary gas the same molecules supply both. In a plasma they can be split — the ions carry the inertia and the electrons, a thousand times lighter and often a hundred times hotter, carry the pressure — and the result is a sound that travels at a speed set by one particle's temperature and the other's mass. It travels only while the two temperatures stay far apart. Warm the ions to match the electrons and the sound dies within a cycle, with nothing to rub against.

astrophysics · Plasma oscillation

Named alongside it

The objects these essays reach for when they reach for this one.

Mean free pathEquipartitionHeat capacityViscosityPressureTemperatureThe Boltzmann factorContinuumCross-sectionDegrees of freedomDissipationEquation of state

All concepts