Illustrated Physics
physics, drawn
Concept

Fluctuations — where it appears

The departures of a quantity from its average, arising from the finite number of things being averaged over. Their mean square is fixed by a susceptibility — for a fluid's density it is ρkTκ_T, which is exactly one for an ideal gas and 0.062 for water — and it is the fluctuations rather than the molecules that scatter light.

Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.

Two straight lines that are not the same line. Half the difference between the liquid and vapour densities, in units of the critical density, against the distance from the critical temperature — both logarithmic, over 5 decades. The van der Waals curve is solved for the coexisting pair at each temperature and its slope in the last decade is 0.500, which is the mean-field ½. Real fluids give 0.326. The two differ by 11 per cent at t = 0.1 and by a factor of 7.9 at the bottom of the axis, which is why an equation with the wrong exponent in it looked right for eighty years.

The point at which the two become one

Heat a sealed tube of carbon dioxide and the meniscus inside it does not boil away — it fades, the two densities converging until there is nothing to separate. Twenty millikelvin before that happens the fluid turns milky, and the exponent describing the last approach is a number van der Waals got wrong and could not have got right.

thermodynamics · Phase change N when the phases are random, N² when they are not. Scattered intensity against the number of scatterers, both logarithmic, for two ways of adding the same amplitudes. The lower curve averages 400 draws of N unit amplitudes with independent random phases and grows as N^1.000; the upper one adds them in phase and grows as N². At 3000 scatterers the two differ by a factor of 2921. Nothing about the scatterers is different between the two — same number, same strength, same wavelength. Only the arrangement is, and it is worth three decades here.

Why a litre of water is not blue for the reason the sky is

The same molecules that make the sky blue also make the refractive index of air, and the two numbers agree because the sideways sum has random phases and the forward one does not. Condense those molecules into a liquid and the sideways sum collapses by a factor of sixteen — and what is left is thirty-four times smaller than the absorption that actually colours the water.

optics · Scattering A magnetisation curve, and the same curve magnified. On the left, the position of a domain wall against the applied field, over the whole of a crystal. It looks like a curve. On the right, a window 2.0% of its width, taken 42% of the way across, at the magnification a sensitive measurement reaches: the wall does not move at all while the field rises, then jumps, then stops again. The largest jump in the window covers 0.67 units of wall position in no field change at all. There is nothing smooth underneath this — the smooth curve on the left is a staircase with 1400 steps in it, drawn small.

The curve that is really a staircase

A magnetisation curve is drawn as a smooth line and is nothing of the sort. Measured finely enough it is a sequence of jumps of every size, audible as a crackle in a coil, with no typical jump and no smooth motion underneath.

electromagnetism · Magnetisation A ratio in the exponent. The Boltzmann factor against the energy of a state measured in units of kT, with the logarithm up the axis so that the straight line is the whole content. Every kT of energy costs a factor of e, so 5, 12, 20 times kT are factors of 10^-2.2, 10^-5.2, 10^-8.7. At 295 K, kT is 25.4 meV, so a barrier of 0.4 eV is 15.7 kT and a factor of 1.5e-7. That is the sense in which a third of an electronvolt is not a small energy: it is small compared with a chemical bond and enormous compared with kT, and it is the second comparison that decides whether anything happens.

The temperature a molecule does not have

Temperature fixes a system's average energy and nothing more. The actual energy wanders, by an amount tied to the heat capacity, and the relative size of the wandering falls as one over the square root of the number of degrees of freedom — so a mole has a temperature and a molecule does not.

thermodynamics · Equipartition Runs that break the second law, and how often. The work done in a process repeated many times, and the same for the process run in reverse with its work reflected, for a free-energy change of 4 kT and a dissipation of 3 kT. The average work exceeds the free-energy change, which is the second law, and individual runs do not have to: the shaded tail is the fraction of runs that do less work than the free energy — trajectories in which the entropy of the universe went down — and it is 11.03% here. The two curves cross exactly at the free-energy change, whatever the dissipation, which is what makes an irreversible measurement able to report an equilibrium quantity.

The second law, with a probability attached

Entropy increases, on average. For a small system pulled quickly, individual runs go the other way — and how often is not a matter of taste but an exact number, fixed by a relation with no adjustable constant in it and no requirement that anything be near equilibrium.

thermodynamics · Entropy How full a surface is, against the pressure of the gas above it. The fraction of adsorption sites occupied on a surface that binds a molecule with 0.35 eV, against the gas pressure on a logarithmic axis, at 250, 300, 350 K. Each curve is θ = p/(p + p½) — Langmuir's isotherm — and each is the same S-shape moved along the axis: on a logarithmic pressure scale the coverage is a function of ln p alone, shifted by the binding energy over kT. The half-full pressure is 5.58 Pa at 250 K, 132 Pa at 300 K, 1341 Pa at 350 K. Warming the surface does not change the shape; it slides the curve to higher pressure by a factor that is a Boltzmann factor, e^(ε/kT), which is how a binding energy is read off a family of isotherms.

The site that fills like an electron level

A patch of catalyst holding a gas molecule, a haem in a muscle cell holding oxygen, an energy level in a metal holding an electron: each is a place that can hold one particle or none, in contact with a reservoir it trades particles with. Each fills according to the same function of one variable — how far the reservoir's chemical potential lies above the energy of the site. Langmuir's isotherm and the Fermi function are not analogous. They are the same law, and haemoglobin works because its sites break it.

thermodynamics · Chemical potential The price of precision never falls below two. For a clock ticking forwards and backwards with a free-energy drop a per step, the product of its squared relative uncertainty and the entropy it has produced, in units of k: exactly a·coth(a/2), whatever the time elapsed. It approaches the bound of 2 only as the driving vanishes and grows as a once the driving is strong; the three clocks compared elsewhere sit at 2.04 (0.5 kT), 2.63 (2.0 kT), 8.01 (8.0 kT). No steady-state process of any design lies below the line at 2 — the thermodynamic uncertainty relation.

The precision a clock pays for in heat

Anything that counts — a molecular motor stepping along a filament, a chemical oscillator keeping time in a bacterium, a clock ticking on a wall — has to be driven, because a count that is not driven runs backwards as often as forwards. The heat it gives off buys it regularity, and there is an exact floor under the price: the squared relative uncertainty of the count, times the entropy produced, is never less than two in units of Boltzmann's constant.

thermodynamics · Entropy London's attraction, and the distance where it gives out. The dispersion energy between two identical atoms times the sixth power of their separation, relative to London's constant, against the separation in units of c/ω₀ — the reduced wavelength of each atom's strongest transition — on logarithmic axes, from the Casimir–Polder integral with a one-resonance polarisability. Close in, the product is one: London's inverse sixth power. It has fallen to a half by 3.82 c/ω₀, and far out it falls as one over the distance, so the energy goes as the inverse seventh power, reaching 23.2 per cent of London's value at ten reduced wavelengths and 2.44 per cent at a hundred. For two helium atoms, whose strongest line is at 58 nm, c/ω₀ is about 9 nm; for two sodium atoms, 94 nm.

The attraction that weakens when light is too slow

Two neutral atoms attract because each one's fluctuating electrons polarise the other, and the two fluctuations fall into step. London's calculation of the force, as the inverse sixth power of the distance, assumes the field from one atom reaches the other instantly. It does not. Beyond a distance set by the wavelength of the atoms' own strongest transition — nine nanometres for helium, about a hundred for sodium — the field returns too late to find the fluctuation that made it. The attraction then falls one power faster. Casimir and Polder worked this out in 1948, and it was the starting point for the Casimir force between plates.

electromagnetism · Retardation The Casimir pressure between warm plates, from vacuum to heat. The attractive pressure between two parallel, perfectly reflecting plates at 300 K against their separation, on logarithmic axes, from the Lifshitz sum over Matsubara frequencies. Dashed: the zero-temperature law π²ħc/240a⁴. Dotted: the classical law ζ(3)kT/4πa³, which contains no Planck constant. At 1 μm the full pressure is 0.0013 Pa, within 0.2 per cent of the vacuum value; at 10 μm it is 3.96·10⁻⁷ Pa, 3.05 times the vacuum value and within 0.002 per cent of the classical law. The two laws cross at 3.28 μm; the thermal length ħc/kT is 7.63 μm. The second solid curve removes one term — the static magnetic fluctuation, which a metal of finite conductivity does not reflect — and sits 15 per cent lower at 1 μm and a factor of two lower far out. The plates are otherwise ideal mirrors: a model, chosen to isolate that one term.

The force that forgets Planck's constant

Two mirrors in a vacuum attract, with a pressure that contains Planck's constant and nothing else about the world but the speed of light. Warm the mirrors and pull them apart, and past a few micrometres at room temperature the Planck constant drops out of the answer. The force is still there and is now larger than the vacuum alone would give, but it has become classical: a count of thermal fluctuations of the static field, with kT of energy each. Half of that classical force comes from one fluctuation — a static magnetic field — and whether a real metal mirror reflects it is a question that has divided the people who measure the force for twenty years.

electromagnetism · Retardation Two kinds of flicker in one heat bath. The variance of the number of photons in one mode of black-body radiation, against hν/kT, both on logarithmic axes, split into its two parts: n̄, the variance a shower of independent particles would have, and n̄², the variance of the intensity of randomly phased waves; their sum is what thermodynamics demands of Planck's law. Where the photons are rare, at hν much more than kT, the particle term wins; where they are crowded, the wave term wins; they are equal at hν/kT = ln 2 = 0.693, where each mode holds one photon on average. At hν = 5kT the wave term is 0.68 per cent of the particle term; at hν = kT/10 the particle term is 10.5 per cent of the wave term.

The flicker that is half wave and half particle

In 1909, sixteen years before quantum mechanics, Einstein asked how much the energy in a small corner of a hot oven should fluctuate, and answered with nothing but thermodynamics and Planck's law. The answer has two terms. One is exactly what a spray of independent particles would give; the other is exactly what randomly overlapping waves would give. Neither theory of light could produce the sum, and the sum is what the oven does. The two terms are equal where each mode of the radiation holds one photon on average — a dividing line that runs through every hot thing, from the microwave sky to the Sun.

quantum · Photon A quiet resonator growing louder before it sings. The pressure in a resonator's 200 Hz mode over two and a half seconds, kicked continually by the same random noise, at three settings approaching the onset of self-sustained oscillation: the mode's natural decay rate falling from 20 to 6 to 2 per second. Each panel is on the same scale. Far from onset the noise is a faint, ragged hiss, each burst dying almost as soon as it starts. Closer, the bursts last longer and grow larger, because the mode forgets each kick more slowly and the kicks pile up; at the last setting the resonator hums in long swells of a nearly steady tone. Nothing has started oscillating on its own — every setting is below onset — but the noise has already begun to sound like the oscillation it is approaching.

The noise that warns a tube is about to sing

A resonator that is about to break into self-sustained oscillation — a combustor about to hum, a heated tube about to howl — is silent on average right up to the moment it starts. Its noise is not. As the margin of stability shrinks, the random hiss in the mode grows louder, remembers each kick for longer and narrows towards the coming tone, in proportions fixed by the margin itself. The instability can be heard before it exists, except in the case engines most often present, where noise starts the song before the warning has properly begun.

waves · Resonance

Named alongside it

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

The Boltzmann factorCoherenceCompressibilityCritical pointDetailed balanceEquipartitionIrreversibilityPolarisabilityScatteringThe second lawStatistical mechanicsUniversality

All concepts