Illustrated Physics
physics, drawn
Concept

Fluctuation dissipation — where it appears

The relation between the random fluctuations a system shows at equilibrium and the way it dissipates energy when pushed. The same friction that damps a motion sets the size of its thermal jiggling, so measuring one determines the other.

Named by 7 essays across 5 fields — each of them below, with the objects they name alongside it.

Seven walks from one point. 7 random walks of 400 steps each, all starting at the same place. None of them goes anywhere in particular and none of them stays put; the typical distance reached after n steps is the square root of n, so quadrupling the time doubles the spread. The tracks are seeded, so this is a property of the figure rather than of any one run.

The jiggle that proved atoms

A pollen grain in still water never stops moving. For eighty years that was a curiosity with no explanation; then it became the measurement that settled whether matter is made of particles, by turning a microscope and a stopwatch into a count of how many molecules are in a mole.

thermodynamics · Diffusion Two pushes that add up to a friction. The force on a sodium-23 atom from each of two counter-propagating laser beams tuned 0.5 linewidths below resonance, at 0.1 of saturation each, and their sum, against the atom's velocity in units of the linewidth over the wavenumber. Each beam pushes along its own direction and is heard loudest by an atom moving towards it, because the Doppler shift brings the red-detuned light up into resonance. At rest the two pushes cancel exactly; moving, the atom scatters more from the beam ahead of it than from the one behind, and the difference points against the motion. Near zero velocity the sum is a straight line through the origin — a friction, with slope −0.0907 ħk² — and it is largest at 3.13 m/s, beyond which the atom has been Doppler-shifted out of resonance with both beams and the grip weakens. The damping time it implies for sodium-23's mass is 17.5 microseconds.

The friction made of light

Two laser beams pointed at each other push an atom both ways at once, and at rest the pushes cancel. Moving, the atom hears the beam ahead of it louder than the one behind, and the difference is a friction. The photons that supply the friction arrive one at a time, so they also kick — and the temperature where the two balance contains the width of a spectral line and nothing else.

astrophysics · Radiation pressure The modes of a wire, counted. Nyquist's argument of 1928, with its count performed. Two resistors joined by a lossless line are in equilibrium, and the line is a one-dimensional cavity whose standing waves are spaced c/2L apart in frequency. Each mode has an electric and a magnetic energy, both quadratic, so equipartition gives it kT — the same half a kT per quadratic term that a heat capacity counts. The upper points are how many modes fall in a band of a hundred and thirty-seven megahertz, for five line lengths, from 17 on a thirteen-metre line to 8,558 on one of six kilometres. The lower points are the power each end therefore receives, as a fraction of kTΔf, and they converge on one: a longer line has proportionally more modes and takes proportionally longer to deliver them, so the length cancels. The longest line lands within 0.005 per cent and the shortest is 4.5 per cent low, because thirteen metres holds only seventeen whole modes in the band and the remainder is a real granularity rather than an error. What is left in the limit is kTΔf — a noise power with no resistance in it at all, and none of the line's properties either.

Half a kT in a piece of wire

Count the quadratic terms in a molecule's energy and equipartition gives a heat capacity. Count them on a transmission line instead and the same theorem gives a resistor's noise voltage — 4kTRΔf, with nothing in it about what the resistor is made of. A fifty-ohm input at room temperature says 0.91 nanovolts in every root hertz, and no design removes it.

thermodynamics · Equipartition The same thermal motion, spread three ways. How the thermal jiggling of a spring-mounted mass is distributed over frequency, for quality factors 3, 30, 300, with frequency in units of the resonance and the spectrum in units that make the area under each curve its mean-square displacement. All three curves enclose exactly the same area, kT/k — 1.000, 1.000, 1.000 of it in the units drawn — because equipartition fixes the energy in the spring and not how fast it changes. What the damping chooses is where that motion goes: a high Q concentrates it in a peak 100 times taller than the low-Q one, and leaves the rest of the spectrum 100 times quieter at 10 times the resonance frequency.

The noise a high Q moves out of the way

Anything held by a spring at a temperature jiggles, and equipartition says exactly how much: the mean square of its displacement is kT divided by the spring's stiffness, whatever the spring is made of and however it is damped. That looks like a floor no measurement can go below. It is a floor on the total, and the total can be moved. An oscillator with little damping puts almost all of its thermal motion into a narrow peak at its own frequency and leaves every other frequency quiet — which is why the mirrors of gravitational-wave detectors hang from fibres of glass that ring for minutes.

thermodynamics · Equipartition The heat two surfaces exchange when they almost touch. The radiative heat-transfer coefficient between two parallel half-spaces at 300 K, in W/m²K on a logarithmic axis, against the width of the vacuum gap between them, also logarithmic, from fluctuational electrodynamics: silicon carbide (solid) and gold (dashed), with two perfect black bodies, 4σT³ = 6.12 W/m²K, for comparison (dotted). Far apart, silicon carbide exchanges 3.3 W/m²K — less than black — and gold, a good mirror, only 0.024. Closer than the thermal wavelength, about 17 µm, waves that cannot leave either surface begin to tunnel across the gap. At 100 nm silicon carbide exchanges 137 W/m²K, at 10 nm 9338, at 1 nm 9.3·10⁵ — rising as the inverse square of the gap, 1.5·10⁵ times the black-body value at 1 nm. Gold exceeds black only below a few hundred nanometres and then grows much more slowly: 1284 at 10 nm, 2561 at 1 nm.

The heat that crosses a gap too narrow for light

Stefan and Boltzmann's law is usually read as a ceiling: no surface can radiate more heat than a perfect black body at the same temperature, 6.1 watts per square metre for every degree of difference at room temperature. It is a ceiling only for surfaces that are far apart. Bring two of them closer than the wavelength of their thermal glow and they begin to exchange waves that never leave either surface — waves that cling to it and die away within a wavelength — and those waves tunnel across the gap. Two plates of silicon carbide ten nanometres apart exchange fifteen hundred times what two black bodies can, almost all of it in a single colour, and the colour is the one in which silicon carbide is too shiny to glow at all.

quantum · Blackbody 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 The same damped motion in two phase planes. A cloud of damped oscillators that start on a small circle of positions and velocities, drawn at times 0, 2, 4, 6 (outlines) with the centre's path. Left: position against the physical momentum mẋ. The cloud spirals inward and shrinks: its area falls as e^(−γt), to 0.30 of its start by t = 6, as friction squeezes every motion towards rest. Right: position against the canonical momentum of the Caldirola–Kanai Lagrangian, p = e^(γt)mẋ. The same cloud keeps exactly its starting area while it is stretched out of shape, as Liouville's theorem requires of any Hamiltonian flow. The friction has been hidden in the definition of momentum: the canonical momentum grows exactly as fast as the real one decays.

The friction an action can hold only by changing its clock

The principle of least action generates every conservative law of motion from a single integral, and it has no place for friction: there is no ordinary Lagrangian whose equation of motion is a damped oscillator's. Multiply the Lagrangian by an exponential of time and friction appears — but only by redefining momentum so that it grows as fast as the real momentum decays. The phase-space area the principle conserves is then no longer the one that counts motions, and quantised, the damped oscillator becomes more sharply known than the uncertainty principle allows. Friction belongs to a system and the surroundings it loses energy to, and an action for the system alone can only fake it.

mechanics · Least action

Named alongside it

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

Brownian motionEquipartitionBlackbodyDampingLinewidthAvogadro numberCombustion instabilityCritical slowing downDegrees of freedomDiffusionDoppler effectDoppler limit

All concepts