20 metres per second: heat that travels like sound, and a hot stripe that went colder than the cold around it

Part IHeat that forgets

Chapter 1Twenty metres per second

Cool liquid helium below 2.17 kelvin and give it a short pulse of heat at one end. The heat does not seep outwards and fade, as heat does everywhere else in daily life. It sets off through the liquid as a wave, at about twenty metres per second. At 1.80 K the recommended value is 19.90 m/s; between 1.30 and 1.85 K it stays between 19.0 and 20.4 m/s, with a maximum of 20.37 m/s at 1.65 K.¹ That is 72 km/h, the speed of a car on a country road (this page's arithmetic).

The wave is slow by the standards of the liquid it travels in. Ordinary sound in helium at 1.8 K runs at 231.3 m/s, so a pressure pulse crosses a metre in 4.3 milliseconds and the heat pulse needs 50. The heat wave is more than eleven times slower (11.6 times at 1.8 K, this page's arithmetic from the same tables).¹

What travels is not the liquid. There is no net flow of liquid from one end of the tube to the other; what moves is a pattern of slightly warmer and slightly colder liquid, a wave of temperature and entropy with almost no change in pressure. Landau gave it the name second sound,^2,3 and in helium it exists only in the superfluid phase. At the lambda point, 2.1768 K, its speed falls to zero; above it, heat only spreads.

For decades this was a curiosity of liquid helium, the one substance that stays liquid down to absolute zero at ordinary pressure. Then the same physics turned up in crystals, at first only within 20 degrees of absolute zero. On 3 April 2026 a team at Sun Yat-sen University and its partners reported it at room temperature, in flakes of graphite made of 99.93% carbon-12: a stripe of heat painted by a laser turned into a stripe of cold before it faded.⁴

This page follows the idea in four parts. Part I explains why heat normally spreads and forgets where it came from, and how a short delay turns spreading into a wave. Part II is superfluid helium: the two fluids, the wave, the argument it settled and the tools it became. Part III is crystals: the collisions that let heat flow like a wind, and graphite's climb to room temperature. Part IV asks what it means for the second law and the arrow of time, and where it might lead.

Chapter 2Why heat spreads

In a gas, or in an insulating crystal, heat is carried by particles that collide often: molecules in a gas, quanta of vibration called phonons in a crystal. After each collision a carrier's direction is close to random. A packet of heat is therefore carried along a random walk, and a random walk spreads without ever going anywhere in particular.

Averaged over many carriers, the random walk gives Fourier's law of 1822: the heat flux q points down the temperature gradient.⁵ Combined with conservation of energy it gives the diffusion equation:

q = −k∇T, ∂T/∂t = α∇²T, α = k/(ρc)k: thermal conductivity; ρ: density; c: specific heat; α: thermal diffusivity, in m²/s.

Diffusion has two signatures. The first is the square-root law: the distance heat reaches grows as √(αt), so ten times farther takes a hundred times longer. In the planes of graphite at room temperature, where α is about 11 cm²/s,⁶ heat diffuses about 1 µm in a nanosecond and about 10 µm in a hundred nanoseconds (this page's arithmetic).

The second signature matters more for this page. Under diffusion a peak can only sink and spread; it never overshoots. Mathematically this is the maximum principle: the temperature never rises above its starting maximum or falls below its starting minimum. Paint a pattern of hot and cold stripes and let it go, and the stripes fade in place. The hot stripes stay the hot ones until everything is even, with the contrast decaying as exp(−αq²t), where q = 2π/λ for stripes a distance λ apart. The decay time λ²/(4π²α) is about 32 ns for 37.5 µm stripes in room-temperature graphite (this page's arithmetic), and the contrast never changes sign.⁶

Fourier's law has one embarrassing feature: it is infinitely fast. Light a match here and, according to the diffusion equation, the temperature rises at once everywhere, exponentially little far away. Nobody minds, because the error is invisible at everyday times and distances. Fourier's law is an excellent approximation whenever collisions scramble the heat flow quickly compared with the times and distances you look at. It fails only at very short times, at very small scales, and in special materials. Those are where the rest of this page lives.

Chapter 3A memory turns spreading into a wave

The repair is a delay. In a paper usually dated 1948, Cattaneo let the heat flux take a short time τ to respond to a temperature gradient, and keep going for about that long once started; Vernotte proposed the same in 1958.^7,8 With energy conservation the delay turns the diffusion equation into the telegraph equation:

τ ∂q/∂t + q = −k∇T
⇒ τ ∂²T/∂t² + ∂T/∂t = α∇²TFor times much shorter than τ it is a wave equation with speed v = √(α/τ); for times much longer than τ it is the diffusion equation again.

Two natural units come with it, and the figures on this page use them throughout: the memory time τ, how long the heat flux remembers its direction, and the memory length vτ, how far heat runs in that time.

Now paint the stripes again. A pattern T = A(t) cos(2πx/λ), started with no heat flux, obeys an equation for a damped oscillator:

A″ + A′/τ + v²q²A = 0, A(0) = 1, A′(0) = 0q = 2π/λ. The oscillator is underdamped, and the stripe contrast changes sign, if and only if vqτ > ½, that is λ < 4πvτ ≈ 12.6 vτ.

So the stripes swap, hot becoming cold, if and only if they are closer together than about 12.6 memory lengths. Below that threshold the contrast falls through zero and comes out the other side. With stripes 2 memory lengths apart it reaches −60% near one memory time; at 3 memory lengths, −46% at 1.54 memory times, after crossing zero at 0.89; at 10 memory lengths, only −1.6%, late, near 8.3 memory times. At 12.6 memory lengths and above, the contrast never changes sign, however long you wait (all this page's arithmetic). Figure 1 shows the stripes and the contrast over time. Make the stripes finer or wider, press Play, and switch to "No memory" to compare with Fourier's law.

Figure 1🦓 Try it

Does the hot stripe stay hot?

Finer stripes, then Play

Stripe spacing

flip threshold

Stripe contrast now

First flip at

Coldest the hot stripes get

A pattern of warmer and cooler stripes, as painted by the crossed laser pulses of a transient-grating experiment, evolving under Cattaneo's law τ∂q/∂t + q = −k∇T, which with energy conservation gives τ∂²T/∂t² + ∂T/∂t = α∇²T. The stripe contrast obeys A″ + A′/τ + v²q²A = 0 with no initial heat flux; time is in memory times τ and the spacing λ in memory lengths vτ, with v = √(α/τ). The stripes swap only if λ < 4πvτ ≈ 12.6 vτ. Grey: Fourier's law with the same diffusivity α (the limit τ → 0), A = exp(−αq²t), which never changes sign. Upper band: the stripes at the start; lower band: at the marked time, with thin lines through the stripes that started hot. Colour strength grows with the size of the deviation. This page's model and arithmetic: a cartoon of phonon hydrodynamics, not the Boltzmann-equation calculations used for graphite; values are small deviations from the ambient temperature.Two rows of stripes: how they started, and how they look at the moment you pick. Orange is warmer than average, teal colder. Press Play on the fine stripes: the hot ones fade, then turn cold. Make them wide and they only fade, like ink in water. Drag along the curve to move through time.

The thing to notice is the threshold. Below it the curve crosses zero; above it the curve only sinks towards zero, like Fourier's grey curve, which never crosses at any spacing. When the trace dips below the line, the stripes that started hot are colder than average and the ones that started cold are warmer. For that moment heat has moved from colder places to hotter ones.

One correction to a tempting picture. In this model a single hot pulse on a line splits into two and spreads, but it never leaves a dip below its surroundings behind (this page's arithmetic, for pulse widths from a quarter of a memory length to four). The cold dip needs geometry that brings heat back together: a periodic pattern of stripes, a narrow spot in two dimensions, or the middle of a heated ring (Chapter 10).

There is a pleasing picture of why a memory makes waves. A walker who keeps her direction for a while and reverses at random spreads out along a line according to exactly the telegraph equation, as Goldstein showed in 1951 and Mark Kac in 1974; in two or three dimensions the match is only approximate.^9,10 Take it as an analogy for the equation, not as the mechanism: in real second sound individual carriers collide constantly, and what persists is the drift of the whole gas (Chapter 8).

The telegraph equation is itself a cartoon. Pushed hard, with large amplitudes or sharp sources, it can predict negative amounts of heat, which is unphysical,^11,12 so every figure here plots small deviations from the surrounding temperature. Fed with graphite numbers (the room-temperature α = 11 cm²/s and a wave speed of 3,200 to 4,000 m/s, near the value measured at 85 K), it would let stripes flip in ordinary graphite at periods up to 3.5 to 4.3 µm, where experiments see only plain decay (this page's arithmetic). The graphite analyses therefore solve the full Boltzmann equation for the phonons, which reduces to equations of this damped-wave kind only as a limit; Simoncelli, Marzari and Cepellotti call the generalisation "viscous heat equations".¹³ Experimenters still use the damped-wave form to read a speed and a damping time off their data.⁴

Part IITwo fluids, two sounds

Chapter 4A liquid that is two liquids

Helium stays liquid at ordinary pressure however cold it gets: its atoms are light and attract each other only weakly, so their quantum zero-point motion keeps them from freezing unless the liquid is squeezed to about 25 times atmospheric pressure. Below 2.17 K liquid helium-4 changes into something else, called helium II. In January 1938 two letters in the same issue of Nature reported that helium II flows through narrow channels with almost no resistance: Kapitza's from Moscow, and Allen and Misener's from Cambridge.^14,15

Laszlo Tisza, inspired by Fritz London's idea that helium II was a kind of Bose–Einstein condensate, proposed in Nature on 21 May 1938 that the liquid behaves as two interpenetrating fluids.^16,2 The superfluid has no viscosity and carries no entropy. The normal fluid has viscosity and carries all the entropy. Tisza also pointed out that such a liquid should carry waves of temperature.

Landau put the idea on a different footing in 1941. In his theory the normal fluid is the gas of thermal excitations of the liquid: phonons, the quanta of sound, and rotons, excitations of higher momentum, whose spectrum he revised in 1947.^3,2 At absolute zero there are no excitations and the whole liquid is superfluid. As the temperature rises the gas of excitations grows: the normal fraction is 0.7% at 1.0 K, 31% at 1.8 K, and 100% at the lambda point.¹

One liquid, two ways of moving. It is tempting to picture warm "normal" atoms swimming through cold "superfluid" ones. That is wrong. Every atom belongs to the same quantum liquid; the liquid flows as if it were a mixture of two fluids, and the two "fluids" are two ways it can move.²

Chapter 5Second sound

A two-fluid liquid can carry two kinds of wave. In first sound the two parts move together: it is ordinary sound, a wave of density and pressure, at 218 to 238 m/s in helium II, slowest near the lambda point.¹ In second sound the two parts move in opposite directions, so that the net flow of mass almost cancels:

ρ_sv_s + ρ_nv_n ≈ 0ρ_s, ρ_n: densities of the superfluid and normal parts; v_s, v_n: their velocities.

Where the normal fluid gathers, the liquid holds more entropy and is warmer; where the superfluid gathers, it is colder. The density hardly changes. The speed follows from the two-fluid equations, which Landau first wrote down:^3,2

u₂² = (ρ_s/ρ_n) T s²/cs: entropy per unit mass; c: specific heat per unit mass. Valid when the two specific heats (at constant pressure and volume) are nearly equal, as in helium. From the 1998 tables of entropy, specific heat and normal fraction it gives 19.95 m/s at 1.8 K against the measured 19.90, and agrees within 3.5% from 1.5 to 2.0 K (this page's arithmetic).

The formula explains the end of the wave. As the lambda point approaches, the superfluid fraction ρ_s/ρ vanishes: there is less and less to slosh against, and u₂ falls to zero at 2.1768 K. Figure 2 shows the measured speed against temperature, with the two famous predictions dashed. Drag the temperature, and press Play to see the two parts of the liquid slosh past each other.

Figure 2🌡️ Try it

The speed of a heat wave in helium

Drag the temperature

Temperature of the helium

Predictions:

Heat wave (second sound)

Ordinary sound

Ordinary sound is faster by

Normal fluid (carries the heat)

Speed of second sound in liquid helium-4 at saturated vapour pressure: recommended values of Donnelly and Barenghi (1998, Table 4.3), dots at tabulated temperatures. Dashed: Tisza's 1938 formula as printed by Donnelly (2009), u₂ = 26[(T/T_λ)(1 − (T/T_λ)^5.5)]^½ m/s, and Landau's two-fluid theory evaluated by this page for an ideal gas of excitations on the measured spectrum (Table 13.3), which ignores interactions between excitations and so overshoots near the lambda point. The pale zone marks temperatures not yet reached in 1948; other groups measured below 1 K in the early 1950s, and Peshkov reached about 0.5 K in 1960. The arrow at the top points to 137.5 m/s, the limit u₁/√3 for a pure gas of phonons (Landau's theory; Ward and Wilks 1951), far above this scale. The strip above illustrates a travelling second-sound wave, slowed down: normal fluid and superfluid move in opposite directions with ρ_sv_s + ρ_nv_n ≈ 0, so the liquid as a whole stays put while the temperature oscillates; the bar above it is the normal fraction. Readouts: ordinary sound from Table 3.3, normal fraction from Table 2.4.The line is how fast a heat wave travels in helium at each temperature. Drag the temperature: a long flat stretch near 20 metres per second, then a plunge to zero at 2.17 kelvin. Press Play to see the normal part and the superfluid part slosh past each other while the liquid stays put. Then drag below 1 kelvin and see which of the two predictions the real data follow.

Notice how flat the curve is, close to 20 m/s from 1.3 to 1.9 K, and how sharply it falls to zero at the lambda point. The pale zone on the left is what nobody had measured by 1948. The next chapter is about what was found there.

Making the wave was harder than predicting it. In Moscow the first attempts used piezoelectric crystals to drive pressure waves, and failed; Evgeny Lifshitz then showed that periodic heating would work much better.² In 1944, during the war, Peshkov did exactly that, with an alternating current in a heater and a resistance thermometer to feel the passing wave.^17,2 After the war, at Yale, Cecil Lane and Henry and William Fairbank, at first unaware of Peshkov's work, used an idea of Lars Onsager's: at the free surface, the temperature wave makes the liquid evaporate and condense in step, and that makes ordinary sound in the vapour. A heater at the bottom of a tube ran at 1 kHz, a microphone above the liquid listened at 2 kHz, and resonant peaks appeared as the level fell.^18,2 You cannot hear second sound in the liquid; a thermometer feels it, and Onsager's trick converted it into something a microphone could hear.

Helium II has more than two sounds. In 1959 Kenneth Atkins predicted two more: third sound, a ripple on films of helium a few atoms thick, in which the wall holds the normal fluid still and only the superfluid moves, and fourth sound, which runs through powders packed so tightly that, again, only the superfluid can move.^19,2

Helium-4, not helium-3. Two recent graphite papers describe Peshkov's discovery as made in superfluid helium-3.^20,4 It was helium-4, helium II, as the title of his paper says. Helium-3 becomes superfluid only below about a thousandth of a kelvin at ordinary pressure, or about two and a half thousandths when squeezed to its melting pressure, where it was discovered at Cornell in 1972.²¹

Chapter 6The race below one kelvin

Tisza and Landau agreed that second sound should exist. They disagreed about its speed at low temperature. Tisza's formula, as printed by Donnelly in 2009, is

u₂ = 26 [(T/T_λ)(1 − (T/T_λ)^5.5)]^½ m/sT_λ: the lambda temperature. It falls to zero as T → 0.

Landau's theory, worked out by Lifshitz, predicted the opposite: below about 1 K the rotons die out, the phonons take over, and the speed rises. Above 1.3 K the two predictions nearly coincide, so the low-temperature speed became the prize.²

The first round went to Tisza, on paper. Unable to cool below 1.4 K, the Yale group wrote in 1947 that "Tisza's theory seems to be somewhat favored over Landau's". They had also attributed Landau's formula for the speed to Tisza, and Landau protested; the group printed an erratum acknowledging his priority. In 1949 the two theorists published adjacent letters, both titled "On the Theory of Superfluidity". Landau paid tribute to Tisza's two-fluid idea but called his entire quantitative theory "entirely incorrect".^2,22,23

Experiment settled it. In 1948 Peshkov reached about 1.2 K and saw the speed flatten off; by the early 1950s other groups, cooling the liquid magnetically, had pushed below 1 K and seen the speed rise;⁵⁸ and in 1960 Peshkov reached about 0.5 K, "in very good agreement with Lifshitz".^24,2 The recommended values today rise from 18.35 m/s at 1.10 K to 29.28 m/s at 0.80 K and 83.85 m/s at 0.60 K.¹ A wave that gets faster as the liquid gets colder.

Drag Figure 2 below 1 kelvin, or press its "Below 1 K" chip, and the measured points appear. Tisza's formula gives 13.6 m/s at 0.6 K, low by a factor of six. Landau's recipe, recomputed by this page as an ideal gas of phonons and rotons on the measured excitation spectrum, gives 82.6 m/s, within 2%, and stays within 4% of the data up to 1.6 K. Closer to the lambda point it runs too high (by 8% at 1.8 K and 21% at 2.0 K), as it should: there the excitations are crowded enough to interact, and the ideal gas ignores that (all this page's arithmetic from the 1998 tables). Tisza's formula fits above 1.3 K partly by accident: the roton contributions to the normal density and to the entropy happen to depend on temperature in similar ways.²

Where does the rise end? In a pure gas of phonons, second sound runs at the speed of ordinary sound divided by √3: it is the low-temperature limit of Landau's theory, and Ward and Wilks showed in 1951 that it is the sound of the phonon gas itself.^3,25 With u₁ = 238.2 m/s near absolute zero, the limit is 137.5 m/s (this page's arithmetic). It cannot be reached: Peshkov found that below about 0.5 K second sound "does not exist in any meaningful way", presumably because there are too few excitations left to reach equilibrium among themselves.² The same √3 returns in Chapter 8, in crystals.

The √3 reaches beyond helium. Any gas whose pressure is a third of its energy density, a gas of phonons or a gas of light, carries waves at its particles' speed divided by √3. Until the universe was about 380,000 years old, light and ordinary matter were locked together in a hot plasma that carried sound at about half the speed of light: c/√3 while light dominated, a little less as the share of ordinary matter grew (dark matter took no part). The frozen pattern of those sound waves is the series of acoustic peaks in the cosmic microwave background.²⁶

Chapter 7Heat that echoes

Peshkov's 1948 apparatus made heat echo, literally. A glass tube closed at the bottom served as a resonator. A heater sat on a glass disc that could be moved up and down; a thermometer, an open winding of phosphor-bronze wire, sat on a second movable frame. A current at frequency f heats at 2f, and moving the thermometer traced out the standing waves of temperature in the tube.² Modern experiments generate and detect second sound with superleak transducers: a plastic film with micrometre holes that the superfluid can pass and the normal fluid cannot, vibrating at typically 20 to 50 kHz.²

The most practical echo is a sonar. Superconducting niobium cavities, the accelerating structures of many modern accelerators, are tested in baths of superfluid helium, and a small defect on the inner surface can make a cavity quench: lose superconductivity, dump its stored energy as heat, and launch a second-sound wave into the helium. In 2008 Z. A. Conway and colleagues at Cornell described a method that uses "a few (e.g. 8)" oscillating superleak transducers around a nine-cell cavity to time the wave's arrival; three or more arrival times locate the source.²⁷ Figure 3 is this page's model of that measurement. Drag the hot spot along the cavity, change the timing noise of the sensors, and send the wave.

Figure 3📡 Try it

A sonar made of heat

Drag the hot spot, send the wave

Helium temperature

Timing noise of each sensor

Heat-wave speed

First sensor hears it after

Typical miss

This page's model of second-sound quench location on a 9-cell superconducting cavity in He II (cells 115 mm long, 103 mm in equator radius, drawn in cross-section): a hot spot emits a second-sound front that travels in straight lines at u₂(T) (Donnelly and Barenghi 1998); eight sensors, like the oscillating superleak transducers of Conway et al. (LINAC08) and Plouin et al. (2019), record arrival times with Gaussian timing noise; the spot and the emission time are fitted by least squares, repeated 150 times to show the spread (inset, magnified; the cross is the true spot). The miss scales as u₂ times the timing noise times a geometry factor; with eight sensors around the cavity the factor is a little below one. Shadowing by the cavity, the transducers' response and the processes in the niobium that make real signals arrive early (Plouin et al. 2019) are ignored, so real misses can be larger. Play is slowed down 50 times. Conway et al. recorded their quench at 1.94 K.Drag the hot spot anywhere on the cavity, then send the heat wave. Each sensor notes when the wave arrives; the dots in the magnified circle are where the sensors think the spot is. Make the sensors' clocks sloppier and the dots spread out.

In this model the spread of the dots is roughly the wave speed times the timing noise, times a geometry factor: at 20 m/s, 0.25 ms of noise is 5 mm, and with eight sensors around the cavity the typical miss is a little smaller, about 4 mm; 0.05 ms of noise gives about 0.8 mm (this page's arithmetic). Spots near the end cells, seen by sensors mostly on one side, give a longer cloud.

On Cornell's cavity the method found the quench site in a single cold test, where the usual approach needs two or more. Thermometers pressed onto the outside agreed with the location, and an inspection of the inner surface with a long-distance microscope then found, where the second sound pointed, a pit a fraction of a millimetre across on the equator weld (0.12 mm by 0.06 mm in the paper's figure caption; its text gives these as the radii).²⁷ A team at CEA in France studied the technique systematically and found a catch: in most cavity tests the signals arrived earlier than straight-line second sound predicts. A heater test gave no sign of faster second sound; a model that adds processes in the niobium itself predicted the arrival times much better.²⁸

Second sound is also a counter. In 1956 Henry Hall and William Vinen found that it is attenuated in rotating helium II and deduced that the attenuation comes from the array of quantised vortex lines that rotation creates.^29,2 Measuring that attenuation later let Donnelly's group follow turbulence over five orders of magnitude in vortex density.²

The best-known engineering use of helium II is not second sound itself. The main magnets of CERN's Large Hadron Collider run at 1.9 K, in superfluid helium, because its very high thermal conductivity carries heat away; the machine holds 120 tonnes of helium.³⁰ The same counterflow of normal fluid and superfluid that makes second sound is what carries that heat, but the LHC does not use second sound.

Part IIIHeat as a wind

Chapter 8A gas of sound inside a crystal

In an electrically insulating crystal, heat is carried by phonons, the quanta of lattice vibration. The crystal is filled with a gas of them, and the gas has its own collisions, because the forces between atoms are not perfectly springlike. Helium's normal fluid at low temperature is also a phonon gas, which is why Ward and Wilks's result was later seen as the bridge from helium to crystals.^31,25

Not all collisions are equal. Rudolf Peierls showed in 1929 that collisions which conserve the phonons' total crystal momentum cannot, on their own, stop a heat current. Only collisions in which the momentum "folds over" the crystal's periodic structure, which he called Umklapp processes, can.^32,31 Call the first kind normal collisions and the second, together with scattering by isotopes, defects and boundaries, resistive collisions. Normal collisions alone bring the gas to a drifting equilibrium, a "displaced" distribution that keeps flowing like a wind.⁶ Resistive collisions make it forget which way it was going.

Second sound lives in a window: normal collisions happening many times within the time you watch, resistive ones far less often than once, τ_N ≪ t ≪ τ_R; in a stripe experiment, the stripe spacing must lie between the two mean free paths.^6,4 Too many resistive collisions and heat diffuses. Too few normal ones and phonons fly straight across, ballistically; in a grating experiment that gives a signal that wiggles but does not make a clean negative dip.^20,4 Figure 4 is a toy phonon gas with three velocities and the two kinds of collision. Drag the dot across the map of collision rates and watch the stripes.

Figure 4👣 Try it

When does heat remember?

Drag the dot

Bumps that keep the flow

Bumps that stop the flow

Heat

Coldest the hot stripes get

First flip at

A toy phonon gas (this page's construction): carriers with velocities −c, 0, +c (weights 1/6, 2/3, 1/6). Normal collisions (time τ_N) relax them towards a drifting equilibrium that keeps energy and heat flux; resistive collisions (time τ_R: umklapp, isotopes, boundaries) towards one that keeps only energy. The trace is the amplitude of a thermal grating of spacing L started with no heat flux, solved exactly; axes count collisions per crossing time L/c, and the map's shading is the deepest dip within six crossing times. Frequent normal and rare resistive collisions give second sound at c/√3 (first minimum at 0.87 L/c); frequent resistive collisions give diffusion with D = c²τ_R/3; without collisions the signal oscillates but, in this toy, never falls below +1/3. A cartoon of the Callaway picture, not the full calculations used for graphite (Huberman et al. 2019; Ding et al. 2022; Xie et al. 2026).Drag the dot. To the right, heat carriers bump into each other more, in ways that keep the crowd flowing. Downwards, they hit more pillars that send them back. The dot starts in the corner where the stripes slosh: lots of friendly bumps, almost no pillars. Drag it out and watch the flip vanish. Time is counted in crossing times: how long a carrier takes to fly from one hot stripe to the next. Orange is warmer than average, teal colder.

The sloshing corner needs both: many collisions that keep the flow and few that stop it. There the stripes flip to −82% at 0.87 crossing times, which is the time a wave at c/√3 takes to cover half a stripe spacing, the same √3 as in helium's phonon gas. With no collisions at all the signal oscillates but never falls below a third of its starting value; that floor belongs to this toy, in which two thirds of the carriers stand still. Real ballistic phonons can dip slightly below zero, but not deeply, because phonons with many different speeds and directions blur their wiggles together.²⁰ With frequent resistive collisions it only fades, as diffusion with D = c²τ_R/3 (this page's model). Removing isotopes cuts resistive collisions (the figure's "pillars") and so moves the dot up.

The theory of phonon hydrodynamics grew from there. Guyer and Krumhansl solved the linearised phonon Boltzmann equation in 1966 and set out the conditions for second sound and for a viscous, Poiseuille-like flow of heat.³³ Robert Hardy distinguished in 1970 between "drifting" second sound, sustained by normal collisions, and a "driftless" kind at high frequencies.³⁴ The distinction returns with germanium in Chapter 11.

Why did crystals need such cold? Umklapp collisions need phonons of high momentum, which freeze out only at low temperature, and real crystals contain isotopes and defects that also destroy momentum. The window opened only in very pure crystals, below 20 K: in solid helium near half a kelvin (1966), in sodium fluoride at about 11 to 14.5 K (1970; other reviews give 10 to 18 K) and in bismuth at 1.2 to 4 K (1972).^35,36,37,6 Reports of second sound in strontium titanate at 30 to 40 K, in 1995 and 2007, were disputed by other authors; a 2022 preprint led by the author of the 2007 report claims a direct demonstration.^6,38 Before 2019, second sound had been seen only in isotopically pure solids, apart from those disputed reports.⁶ A common misconception says that in second sound the phonons stop colliding. The opposite is true: it needs many collisions, of the right kind.

Chapter 9Graphite opens the window

Graphite is generous because of its layers. They are held together only weakly, so one family of vibrations is unusually soft: the slow transverse acoustic mode, the counterpart of the "flexural" rippling of a single sheet of graphene. Its very large anharmonicity and density of states make normal collisions intense. The other half of the story is that umklapp collisions stay weak: the in-plane bonds are stiff and the atoms light, so most of the phonons that carry heat have wavevectors far smaller than umklapp needs, and between about 80 and 200 K umklapp scattering is suppressed.⁴ Hydrodynamic heat flow was predicted in graphene and other two-dimensional materials in 2015, and in graphite in 2018.^39,40,41,6

The tool is the transient thermal grating. Two crossed laser pulses interfere on the surface and heat it in a sinusoidal pattern of stripes, with a period λ set by the angle between them. The heated stripes expand and make a tiny surface ripple; a probe beam diffracts off the ripple, and a reference beam makes the detection sensitive to sign, so the signal says which stripes are the hot ones.⁶ Under diffusion the signal decays exponentially, with time constant λ²/(4π²α), and never changes sign. Under second sound it goes negative, meaning that "the local maxima" are now "where the local minima used to be"; the first dip comes after the wave has crossed half a period, so v = λ/(2t_dip).^6,4 Figure 1 is a cartoon of exactly this experiment.

In 2019 a team at MIT led by Keith Nelson and Gang Chen reported second sound in graphite.⁶ With 10 µm stripes in ordinary graphite (natural carbon, 1.1% carbon-13), the signal at 85 K fell below zero and oscillated; it was still oscillatory at 104 and 125 K, and gone at 150 K and, at the cold end, at 50 K. The measured speed at 85 K was about 3,200 m/s, against 3,650 m/s computed, faster than graphite's slow transverse sound (about 1,480 m/s) and slower than its fast one (14,700 m/s). In earlier materials second sound had been slower than every sound wave. At room temperature the same experiment showed plain exponential decay, with a diffusivity of 11 cm²/s at 37.5 µm. That speed belongs to 85 K; it is not a room-temperature value.

Chapter 10A ring with a cold heart

The most striking signature came from a different geometry. In 2021 a team led from the University of Texas at Austin heated graphite with a picosecond laser pulse shaped into a ring 3 µm wide, 15 µm in radius according to the paper's figure captions (its abstract says diameter), and watched the temperature at the unheated centre.⁴² In large-grained graphite the centre warmed, then fell below its starting temperature, near 6 ns after the pulse. The cooling grew clearer as the sample was cooled to 80 K and was seen from 80 to 120 K. It was absent in small-grained graphite (tested with rings of three sizes), and the temperature rise stayed positive when a plain spot of laser light heated the surface beside the probe. Heating a ring made the middle colder. The team also recalled an older echo of the effect: in sodium fluoride, a bolometer at the far, insulated end of a crystal saw a negative temperature swing when a second-sound pulse reflected there.^42,36

Figure 5 runs a heated ring through the telegraph equation of Figure 1. Press Play and watch the centre; then make the ring wider, and switch to "No memory" for diffusion.

Figure 5⭕ Try it

Heat a ring, watch the middle

Press Play

Ring radius

Centre now

Hottest the centre gets

Coldest the centre gets

Ring-shaped heating of a plane (Gaussian ring of width 0.2 vτ), evolved with the two-dimensional telegraph equation of Figure 1 on a radial grid; the disc is a top view of the plane and the trace is the temperature at the centre relative to the surroundings, as a fraction of the ring's starting peak. A ring of radius vτ focuses to 1.49 times that peak and then dips to −0.30; radius 3vτ dips to −0.09; 6vτ does not dip. Grey: diffusion with the same diffusivity (the exact solution for a thin ring holding the same heat), which can neither exceed the starting maximum nor fall below the surroundings. This page's model and arithmetic (grid step 0.02 vτ). Jeong et al. (2021) heated a 15 µm ring on graphite with picosecond pulses and saw the centre fall below its starting temperature near 6 ns, at 80–120 K.A laser heats a thin ring. Press Play and watch the middle. With memory, the heat rushes inward, piles up, overshoots and leaves the middle colder than everything around it. Switch to "No memory" and the middle only warms a little. Orange is warmer than the surroundings, teal colder. In 2021 a team saw this cold heart in real graphite.

With memory, the heat rushes inward, piles up and overshoots. For a ring one memory length in radius, the centre reaches 1.49 times the ring's starting peak at 0.90 memory times, something diffusion forbids, and then dips to −0.30, colder than the surroundings, at 1.36 memory times. A ring of radius three memory lengths dips only to −0.09; at six memory lengths there is no dip at all, because the heat has forgotten its direction by the time it meets in the middle (this page's arithmetic). Under diffusion the centre warms gently, never above the ring's starting peak and never below the surroundings. The model is a cartoon; the 6 ns dip in a 15 µm ring is the measurement.

Chapter 11Room temperature, April 2026

In 2022 the MIT group pushed to stripes of about 2 µm with sub-picosecond time resolution and saw second sound at 200 and 225 K.²⁰ At short periods the apparent speed rose, because ballistic phonons mixed into the signal. The paper also made a prediction: in graphite purified of carbon-13, second sound should survive at room temperature, at periods of 1.0 to 1.5 µm.

The prediction was tested by Zhikun Xie and colleagues in a team led by Ke Chen at Sun Yat-sen University and Masahiro Nomura at the University of Tokyo, with partners at MIT (Zhiwei Ding), at Japan's National Institute for Materials Science and at Huazhong University of Science and Technology. Their paper was received on 7 November 2025 and published on 3 April 2026.⁴ Their samples were millimetre flakes of graphite about 150 µm thick, grown at high pressure and high temperature from carbon that is 99.93% carbon-12, and they compared them with ordinary highly oriented pyrolytic graphite containing 1.10% carbon-13, using a transient grating with periods from 0.95 to 54 µm and about 100 fs resolution.

At 300 K and a period of 0.95 µm, the signal from the purified graphite crossed into the negative and slowly recovered: the hot stripes had turned cold. The ordinary graphite only decayed. At 258 K the dip was deeper. At room temperature there was only one dip, with no second positive swing, because damping by umklapp collisions is strong there.⁴

The difference between the two samples is one carbon-13 atom in 91 against one in 1,429 (this page's arithmetic). A carbon-13 atom is 8% heavier than a carbon-12 atom and scatters short-wavelength phonons, destroying their momentum. The mass disorder that drives this scattering is 15.5 times smaller in the purified sample (this page's arithmetic, from the mass variance of the two isotope mixtures), and the authors' first-principles calculations find the isotope scattering rate suppressed by more than an order of magnitude.⁴ That was enough to open the window at room temperature for stripes about a micrometre apart.

The wave also helps heat along. The effective thermal conductivity read from the decay rises a little above the bulk value, with a bump near a 3 µm period at 300 K and near 5 µm at 258 K; the authors report enhancements of up to 8.6%, which their abstract rounds to "nearly 10%". Ordinary size effects can only lower the effective conductivity at small scales.⁴

How settled is it? The result is peer reviewed, it confirms the 2022 prediction, and it matches the authors' first-principles simulations. The agreement with the prediction is not an outside check: the 2022 MIT paper's co-first authors, Zhiwei Ding and Ke Chen, both helped conceive the 2026 experiment, and Ke Chen co-led it. It is also, so far, one group's measurement: as of 24 October 2026 we found no independent replication and no published comment. This page quotes no room-temperature speed: the measured speed depends on the grating period, because ballistic phonons mix in at short periods, and the value is given only in a figure of the paper. Figure 6 places the main sightings of second sound on one chart, by year and by temperature. Tap a bar to read about it, and switch to "Graphite only".

Figure 6🗺️ Try it

From 1 kelvin to room temperature

Tap a sighting

seenpaler: reached after 1948 (helium); the wider range in other reviews (NaF)seen in pure carbon-12 graphite (1 carbon-13 atom in 1,429; natural carbon has 1 in 91)claimed, disputed×looked for, not seen

Sighting

When

Temperature

Status

Reported observations of second sound, one row per report in order of publication (later is higher; rows are evenly spaced, so the gaps between years are not to scale), against the temperature range in which it was seen (logarithmic scale). Filled: observed; hollow: claimed and disputed; crosses: looked for and not seen. Superfluid helium-4 from Peshkov (1944), measured to about 1.2 K by 1948, below 1 K by the early 1950s (de Klerk et al. 1954) and to about 0.5 K by 1960 (Donnelly 2009; the later range paler); solid helium at 0.42–0.58 K (Ackerman et al. 1966, as summarised by Huberman et al. 2019); NaF (Jackson et al. 1970; 11–14.5 K in Huberman et al. 2019, 10–18 K in other reviews, shown faint); bismuth (Narayanamurti and Dynes 1972); SrTiO₃ (reports in 1995 and 2007, disputed, as summarised by Huberman et al. 2019; a 2022 preprint claims a direct demonstration); ultracold atomic gases at billionths of a kelvin, off this scale (Sidorenkov et al. 2013; Christodoulou et al. 2021; Hilker et al. 2022; Yan et al. 2024); natural graphite (Huberman et al. 2019; Jeong et al. 2021, ring geometry; Ding et al. 2022); germanium (Beardo et al. 2021; disputed by Huberman et al. 2025); 99.93% carbon-12 graphite, with natural graphite as the control (Xie et al. 2026). Grating periods and sample scales differ between experiments; tap a bar for its note. As of 24 October 2026.Each bar is a sighting of heat moving as a wave, in the order it was reported, placed at the temperature it happened at. Tap one to read about it. Confirmed heat waves in solids stayed below 20 kelvin for decades, then graphite climbed all the way to room temperature.

Notice the long wait: from the 1960s to 2019, the established sightings of second sound in solids stayed below 20 K. Then graphite climbed in seven years from about 100 K to room temperature, and the last step needed both the right crystal and the right isotopes. The pair of points at 300 K, pure graphite seen and ordinary graphite not, is the whole isotope story in one picture.

One other room-temperature claim exists, and it is disputed. In 2021 Albert Beardo and colleagues reported "high-frequency second sound" in natural germanium from 7 K up to room temperature, inferred from the phase lag of the material's response to rapidly varying heating: a driftless kind, in Hardy's terms.⁴³ Samuel Huberman, Chuang Zhang and Jamal Abou Haibeh argued, in a critique posted in 2022 and published on 21 October 2025, that one of Hardy's conditions for driftless second sound is not met, and that a phase lag alone is "not a suitable experimental observable" for second sound.⁴⁴ As of 24 October 2026 we found no resolution. The germanium result is a claim under dispute, not an established room-temperature observation.

Part IVWhat it means

Chapter 12Uphill, for a moment

Diffusion is the textbook example of an irreversible process: run the film backwards and heat gathers itself into a hot spot, which never happens. The ideal wave equation, by contrast, looks the same with time reversed. Second sound in helium II is close to that ideal, a weakly damped oscillation of entropy from place to place. Heat waves in crystals sit in between.

In a stripe experiment the sign flip is exactly the forbidden-looking event. Under diffusion, as Huberman and colleagues put it, the hot and cold stripes "cannot switch places because heat can only move from hotter to colder regions".⁶ Under second sound they do switch, so for a moment heat moves from colder regions to hotter ones.

No law is broken. The local temperature is not the whole state of the gas: a drifting phonon gas also carries momentum (crystal momentum, in a crystal), and that momentum carries the heat on past the even state, the way a pendulum's momentum carries it through the bottom of its swing. Resistive collisions damp the swing and produce entropy, and the entropy of the whole system still rises. The bookkeeping needs care: with the ordinary local-equilibrium entropy, the Cattaneo equation can appear to violate the second law, and modified equations and generalised entropies have been proposed to repair it.¹² The honest one-line version: second sound is heat that remembers, for a while, which way it was going. The arrow of time is delayed, not reversed. Why heat runs downhill at all, and why running it backwards is so improbable, is Boltzmann's statistical argument; Nº 3 follows it to its strangest conclusion.

How much inertia does heat have? For a pure phonon gas, Landau's theory gives the density of the normal fluid as^3,45

ρ_n = 2π²(k_BT)⁴ / (45ħ³u⁵) = (E + P)/u²u: the speed of sound; E = π²(k_BT)⁴/(30ħ³u³): the energy density of the phonon gas; P = E/3: its pressure. The second equality is this page's algebra.

The right-hand side is a formula relativity knows well: a drifting gas of light carries momentum (E + P)/c² per unit volume and unit drift speed. Einstein showed in 1905 that energy has inertia, E/c² (Nº 5). In a phonon gas, heat has inertia too, with the speed of sound in place of the speed of light, so for the same energy it is (c/u)² times larger: about 1.6 × 10¹² for helium's phonons, with u = 238.2 m/s (this page's arithmetic). That inertia is what carries the heat past the even state.

Two repairs to finish. A stripe that goes "cold" drops below the stripes beside it, not below the temperature it had before the pulse: the pump light also warms the whole patch evenly, and the grating signal records only the difference between hot and cold stripes, which is what changes sign.⁴ A ring's centre, which the laser leaves unheated, can dip a little below its starting temperature.⁴² Neither goes below absolute zero, and neither has anything to do with negative absolute temperature (Nº 17), which is a statement about the populations of energy levels in equilibrium, not about transport. And the telegraph cartoon's ability to produce negative heat content (Chapter 3) is a defect of the cartoon, not a feature of the experiments.

Chapter 13The same wave at 63 nanokelvin

Second sound is not special to helium or to crystals. Wherever a superfluid coexists with a gas of thermal excitations, the two can slosh against each other. In 2013 a team in Innsbruck and Trento observed second sound in a trapped gas of lithium-6 atoms with resonant interactions, and used it to measure the gas's superfluid fraction.⁴⁶ Bose gases followed: a two-dimensional superfluid in 2021 and a three-dimensional one in 2022, both in Cambridge.^47,48

In February 2024 a group at MIT turned the idea into thermography. The radio-frequency spectrum of a strongly interacting gas of lithium-6 atoms in a box gave them a thermometer with sub-nanokelvin resolution at every point, so they could watch heat move: above the superfluid transition a hot spot diffuses; below it, heat propagates as second sound. The transition showed up directly as the sudden change from one to the other, with a peak, right at the transition, in the second-sound diffusivity, which measures how fast the wave is damped. In the run they show first, the gas was at 63 nK, three quarters of its transition temperature.⁴⁹

The speeds span a remarkable range. In that gas the atoms' Fermi speed is only about 4 cm/s (this page's arithmetic, from a Fermi temperature near 500 nK), and first- and second-sound speeds in such a gas, calculated and measured, are fractions of the Fermi speed.^2,45,49 In helium, second sound runs at 20 m/s; in graphite at 85 K, at about 3,200 m/s. The same physics spans more than five orders of magnitude in speed (this page's estimate).

Chapter 14Will it cool a chip?

Heat flow at micrometre and nanometre scales is a practical problem in electronics, and hydrodynamic heat flow in graphite could matter for it. The demonstrations so far are laboratory experiments, and several work only at cryogenic temperatures:

a thermal conductivity of 4,300 W/(m·K) at room temperature in graphite 8.5 µm thick, well above that of diamond, linked by its authors to phonon hydrodynamics;⁵⁰
phonon Poiseuille flow, heat flowing like a viscous fluid in a pipe, in a 5.5 µm wide ribbon of isotopically purified graphite, up to 90 K;⁵¹
a graphite "thermal Tesla valve", 90 nm of 99.98% carbon-12 graphite that conducts 15.2% better in one direction than the other, at 45 K;⁵²
a small rise of effective conductivity above the bulk value at micrometre periods, seen at room temperature in carbon-12 graphite (the authors report up to 8.6%).⁴

Theory is running ahead. Simulations predict strongly oscillatory heat transport in graphene at 300 K over lengths of about 68 nm and below,⁵³ and similar behaviour in single-layer hexagonal boron nitride;⁵⁴ a July 2026 preprint connects such predictions to grating experiments through the Boltzmann equation.⁵⁵ First-principles simulations of phonon Tesla valves in graphite report rectification ratios of up to about 8 for a single stage and about 16 for several stages,⁵⁶ although a June 2026 comment, not yet peer reviewed, argues that calculations of this kind cannot give true two-terminal rectification.⁵⁷ None of this is a product. It is fair to say that wave-like heat flow could matter for heat management at small scales; it is not fair to say that it will cool your phone.

The open questions, as of 24 October 2026:

Will an independent group reproduce room-temperature second sound in carbon-12 graphite, and what speed will it report?
Is there second sound in germanium? A direct, spatially resolved experiment could settle the dispute.
Which other crystals can reach the window at room temperature, by removing isotopes or by working at smaller scales? Simulations point to graphene below about 68 nm, and to single-layer hexagonal boron nitride.
Can heat valves, lenses or echoes in graphite work at room temperature, at scales that matter for devices?
How far does the simple damped-wave picture hold, now that the field prefers the full Boltzmann equation and "viscous heat equations"?
In cold atoms, does the peak of second-sound diffusion at the superfluid transition follow the same critical law as in liquid helium, as the MIT team suggests?

Chapter 15Words and sources

Diffusion: Spreading by random walk: heat reaches a distance growing as the square root of time, and peaks only sink and spread.
Thermal diffusivity, α: Conductivity divided by heat capacity per volume; it sets how fast heat diffuses, in m²/s.
Memory time, τ: How long the heat flux keeps its direction after the temperature pattern changes; also called the relaxation time.
Memory length, vτ: How far heat runs in one memory time.
Telegraph (Cattaneo) equation: The diffusion equation with a memory term; a damped wave equation for temperature.
Superfluid: The part of helium II, or of a cold atomic gas, that flows without viscosity and carries no entropy.
Normal fluid: The gas of thermal excitations (phonons and rotons in helium II), which has viscosity and carries all the entropy.
Roton: An excitation of superfluid helium with higher momentum than a phonon, introduced by Landau; with the phonons it makes up the normal fluid, and it dies out below about 1 K.
Lambda point: The temperature at which helium-4 becomes superfluid: 2.1768 K at saturated vapour pressure.
First sound: Ordinary sound: a wave of pressure and density.
Second sound: A wave of temperature and entropy; in helium II, the superfluid and normal fluid moving in opposite directions.
Phonon: A quantum of vibration of a crystal lattice (or of sound in helium); the carrier of heat in an insulating crystal.
Normal collision: A collision between phonons that keeps their total crystal momentum, and so keeps the heat flowing.
Umklapp collision: A collision in which the total crystal momentum folds over the lattice's periodicity and is lost: it stops the flow.
Isotope scattering: Scattering of phonons by atoms of a different mass, such as carbon-13 in carbon-12; it also destroys momentum.
Transient thermal grating: A laser method that paints stripes of heat on a surface and follows how they evolve.
Ballistic heat: Heat carried by phonons that fly across without colliding.

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20 metres per second

Heat forgets

Heat with a memory

Sound you can't hear

A fight settled below 1 kelvin