Illustrated Physics
physics, drawn
Concept

Maxwell–Boltzmann distribution — where it appears

The spread of molecular speeds in a gas at one temperature, with a peak, a long tail, and no molecule at the average. Three different speeds characterise it — the most probable, the mean and the root mean square — and which one belongs in a formula depends on what is being averaged.

Named by 5 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 The isothermal atmosphere against the real one. Pressure as a fraction of its sea-level value, against altitude. The curves are the isothermal barometric formula at 220, 288, 400 kelvin, whose scale heights are 6.4, 8.4, 11.7 kilometres. The points are the measured standard atmosphere. At 20 kilometres the 288 kelvin model is 71 per cent out, because the air up there is not at 288 kelvin.

Why the air thins with height, and why that is the same law as the speeds

The pressure of the atmosphere falls exponentially with altitude, and the distribution of molecular speeds falls exponentially with energy. These are not two results that happen to look alike. They are one statement read on two axes.

thermodynamics · Kinetic theory 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 A delay that grows as the square root of the length. The differential group delay between the fastest and slowest polarisation states of a fibre built from sections 100 m long, each a slightly birefringent waveplate with its axis at a random angle, against length up to 400 km. Three individual fibres are drawn faint, and the root-mean-square delay over 300 of them heavy. The ensemble grows as a power 0.50 of the length — the square root, because each section rotates the polarisation it receives before adding its own delay, so the delays add like the steps of a random walk in three dimensions rather than like lengths laid end to end. At 100 km the mean delay is 5.18 ps against 5.00 ps for a coefficient of 0.5 ps/√km. Had the axes all been aligned, the same sections would have added to 172 ps at that length, growing in proportion, and the drawn dashed line leaves the frame within a few kilometres. The randomness is what keeps the delay small, and it is also what makes it impossible to compensate with a fixed device.

The delay that is a random variable

A fibre's core is very slightly elliptical, so its two polarisations travel at very slightly different speeds — and the ellipse turns, at random, every hundred metres or so. The delays of the pieces do not add. They random-walk, so the total grows as the square root of the length, follows the same distribution as the speeds of gas molecules, differs from one wavelength to the next, and on any particular day may be three times its average.

optics · Dispersion How much slower a warm atom ticks. The fraction by which a free atom's clock runs slow on average because of its thermal motion, (3/2)kT/mc², against temperature on logarithmic axes, for five atoms used in clocks or tests of relativity: hydrogen 4.1 × 10⁻¹¹ at 300 K, aluminium ion 1.5 × 10⁻¹² at 300 K, strontium 4.8 × 10⁻¹³ at 300 K, caesium 3.1 × 10⁻¹³ at 300 K, mercury ion 2.1 × 10⁻¹³ at 300 K. Every line falls in proportion to the temperature and to one over the mass. A clock aiming at one part in 10¹⁸ must hold strontium below 0.63 mK and hydrogen below 7.3 µK — or know their temperature well enough to correct for it. Pound and Rebka's gravitational shift across 22.5 m, 2.46 × 10⁻¹⁵, is marked for scale.

The clock that runs slow because it is warm

Every atom in a gas is moving, and every moving clock runs slow. The first-order Doppler shifts of a hot gas cancel on average; the time dilation does not, because it has only one sign. A hydrogen maser at 40 °C runs slow by four parts in a hundred billion from this alone, a nucleus in a crystal runs slow even at absolute zero, and the first test of gravitational time dilation nearly drowned in a one-kelvin temperature difference.

relativity · Time dilation

Named alongside it

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

Mean free pathKinetic theoryTemperatureAtomic clockBirefringenceThe Boltzmann factorContinuumDebye modelDifferential group delayDiffusionDispersionDissipation

All concepts