The heat that arrives before it could

The arguments about thermodynamics and relativity so far have been about equilibrium. The body that has no temperature when it moves found that a moving body has no single temperature; the count that no observer can disagree about found the entropy invariant; the column that is hotter at the bottom found that equilibrium in gravity means a temperature that varies with height; the heat that turns into matter found particles created from heat and held in equilibrium. None of them asked how heat gets from one place to another on the way to equilibrium.

That question has an awkward answer. The standard law of heat flow, which Fourier wrote down in 1822 and which every engineer uses, is incompatible with relativity, and not subtly. It predicts that heat released at a point raises the temperature everywhere, instantly. The prediction is wrong in principle for every material and harmless in practice for almost all of them. But the reason it is harmless tells when it is not, and in the hot, dense fluids that relativistic heavy-ion collisions and neutron-star mergers produce, the diffusion equation had to be replaced. The replacement has consequences that can be seen in ordinary laboratories too.

A point of heat, spread two ways #

Fourier’s law says heat flows down the temperature gradient, $\mathbf{q} = -\kappa\nabla T$, at every instant. Combined with conservation of energy it gives the diffusion equation, $\partial T/\partial t = D\nabla^2 T$. Released at a point at time zero, heat spreads as a Gaussian whose width grows as $\sqrt{Dt}$, as the summer that reaches the cellar in December found for the seasonal temperature wave in the ground. A Gaussian has no edge. At any time after zero, however short, the temperature has risen by some amount at every distance.

The figure compares it with the repaired law. In 1948 Cattaneo, and independently Vernotte, proposed that the heat flux takes a short time $\tau$ to respond to a change in the gradient:

$$\tau\frac{\partial\mathbf{q}}{\partial t} + \mathbf{q} = -\kappa\nabla T.$$

Combined with energy conservation this gives the telegraph equation, $\tau\,\partial^2T/\partial t^2 + \partial T/\partial t = D\nabla^2T$, which has a signal speed $v = \sqrt{D/\tau}$. Its solution for a point of heat has an edge: nothing at all beyond a distance $vt$.

The figure draws both at three times, in units where the relaxation time and the signal speed are one. At half a relaxation time, the diffusion equation has put 62 per cent of the heat farther away than a signal travelling at $v$ could have reached. The telegraph equation has put none there. Most of its heat, 78 per cent, is still concentrated in two sharp fronts racing outward at $v$, drawn as vertical marks because they are infinitely narrow. At two relaxation times, 32 per cent of the diffused heat is out of reach, and the telegraph’s fronts have faded to 37 per cent, with the rest filling in behind them. At eight, the two solutions nearly agree in the middle, the diffused heat beyond the edge has fallen to 5 per cent, and the fronts carry only 1.8 per cent. After many relaxation times the telegraph solution becomes the diffusion equation’s Gaussian.

How much heat outruns every signal #

The acausal part of the diffusion equation is large at short times and vanishes quickly at long ones.

The figure plots the fraction of the diffused heat that lies beyond the signal’s reach, as a function of time. At a tenth of a relaxation time, 82 per cent of it is where no signal could have taken it. At one relaxation time, 48 per cent. At ten, 2.5 per cent. At a hundred, a part in a million million. The failure is severe on timescales shorter than $\tau$ and negligible on timescales much longer, because the Gaussian’s tail beyond the front falls off as $e^{-t/4\tau}$.

Relativity sets the scale. The fastest a signal can travel is $c$, so the relaxation time must satisfy $\tau \ge D/c^2$, and the diffusion equation is safe for any process slower than that. For copper, whose thermal diffusivity is about $10^{-4}$ square metres per second, $D/c^2$ is about $10^{-21}$ seconds; for water it is $10^{-24}$. No heat-flow experiment on ordinary materials has ever been sensitive to times that short, which is why Fourier’s law has survived two centuries of use with an infinite signal speed built in.

A thermometer that should not have moved yet #

The acausality is clearest from the point of view of a distant thermometer.

The figure places a thermometer four signal-lengths from the source. By the diffusion equation it begins to register the moment the heat is released: by one relaxation time it reads 0.0052 of the peak scale, although no signal could reach it before four relaxation times. It peaks at about eight. By the telegraph equation it reads exactly zero until four relaxation times, when the front arrives as a sharp spike carrying 6.8 per cent of the released heat, and then it relaxes towards the diffusion curve.

For any purpose that looks only at the late part of that record, the two agree. The diffusion equation is a good approximation after the front has passed and a false description before it. The trouble arises when a theory needs to respect causality structurally — when it is written for a relativistic fluid, where different observers must agree about which events can affect which. An equation that lets heat outrun light can be transformed into a frame in which heat flows backwards in time, and in relativistic fluid dynamics that shows up as instabilities that blow up without limit. The first relativistic generalisations of Fourier’s law, by Eckart in 1940 and by Landau and Lifshitz, inherited the diffusion equation’s structure and turned out to be unstable for exactly this reason, a result established by Hiscock and Lindblom in 1985.

Where diffusion becomes a wave #

The relaxation time does more than cap the signal speed. It changes what heat does at short wavelengths.

The figure takes a sinusoidal temperature ripple and asks how it evolves. Under diffusion it decays at a rate $Dk^2$ and never oscillates: short ripples die fastest, and none travels. Under the telegraph equation long ripples behave the same way, but above a wavenumber of $1/2v\tau$ the ripples oscillate. They travel as waves, with a phase speed approaching $v$, and their decay rate stops growing and levels off at $1/2\tau$.

Heat moving as a damped wave is called second sound. It is not a curiosity. In superfluid helium it is the dominant way heat moves, because the superfluid and normal components can oscillate against each other; there it is a genuine wave with a well-defined speed of about 20 metres a second. In very pure crystals at low temperature, where phonons collide mostly with each other in ways that conserve their momentum, heat pulses have been seen travelling as second sound: in solid helium in 1966, in sodium fluoride and bismuth in the early 1970s, and in graphite in 2019, above a hundred kelvin, where collisions that conserve phonon momentum dominate. In each case the relaxation time is set by the phonons’ collisions and is long enough to be measured, and a telegraph-like equation describes the result.

Why ordinary matter never notices #

The relativistic bound $\tau \ge D/c^2$ is easily satisfied by every ordinary material, and the reason is simple enough to state in one line.

In a gas of heat carriers moving at speed $u$ and colliding every mean free path $\ell$ — how far a molecule gets — the diffusivity is about $u\ell/3$ and the time between collisions is $\ell/u$. The physical relaxation time is roughly the collision time, and its ratio to the relativistic minimum is $3c^2/u^2$. The figure plots that ratio against the carriers’ speed. For air molecules, moving at 500 metres a second, the real relaxation time is a million million times longer than relativity requires. For the conduction electrons in copper, moving at 1,600 kilometres a second at the Fermi surface, a hundred thousand times. The diffusion equation’s faster-than-light tail sits in a range of times nothing physical in these materials reaches.

The bound comes within a small factor only when the carriers move at nearly the speed of light. Photons inside the Sun diffuse outward, bouncing from electron to electron, and their collision time is only three times the causal minimum; that is the regime of the light that takes a hundred thousand years to leave, where the diffusion picture of radiative transfer needs flux limiters near the Sun’s surface for exactly this reason. The other regime is the quark–gluon plasma made in collisions of heavy nuclei, where the particles are relativistic, the collision time is comparable with the time it takes light to cross a proton, and the fluid is so nearly perfect that its viscosity comes close to the smallest ever proposed. There no approximation that ignores the relaxation time is acceptable.

The fluid that needed a second-order theory #

The relativistic replacement for Fourier’s and Navier and Stokes’s laws is second-order hydrodynamics, developed by Israel and Stewart in 1979 and since refined. It treats the heat flux and the viscous stresses as quantities that relax towards their equilibrium values over finite times, exactly as Cattaneo’s equation does for heat, and it adds terms coupling them to the fluid’s motion. The resulting equations have finite signal speeds and are stable, provided the relaxation times are long enough, which translates into conditions on the fluid’s transport coefficients.

That theory is what is now used to model the fireballs of heavy-ion collisions at the Relativistic Heavy Ion Collider and the Large Hadron Collider. The fluid there lives for a few times $10^{-23}$ seconds, expands at nearly the speed of light, and its measured flow patterns have been used to extract its shear viscosity. Without relaxation times the simulations would be acausal and unstable; with them, they reproduce the measured angular distributions of emerging particles closely. The same equations are used for the hot matter in merging neutron stars, where the timescales are milliseconds but the fluid is relativistic, and where viscosity and heat conduction may affect the gravitational waves emitted after the merger.

The relaxation times themselves are measured quantities there, not adjustable fixes. Fits to heavy-ion data constrain the ratio of the shear viscosity to the entropy density to within a factor of about two of the value $\hbar/4\pi k_B$ that calculations in certain strongly coupled theories give, and the relaxation time of the shear stress is found to be a few times $\eta/(sT)$, a few tenths of the time light takes to cross a proton. A fluid with a relaxation time much shorter than that would be acausal, and one with a much longer one would not flow like a fluid at all; the quark–gluon plasma sits where both requirements are only just met.

In the telegraph equation the relaxation time sets both the signal speed and the timescale over which the heat’s memory of its initial arrangement fades. That echoes the answer that cannot come first, where causality in a material’s response to a field ties together its behaviour at different frequencies. Here causality ties the diffusivity to a relaxation time, and requires the second to exist if the first is finite.

Where non-Fourier heat has been seen, and where it has been claimed #

Outside the relativistic regime, departures from Fourier’s law appear whenever heat is followed on timescales shorter than its carriers’ collision time or over distances shorter than their mean free path. Both have been reached in the laboratory.

When a metal film is struck by a laser pulse lasting a few femtoseconds, the pulse heats the conduction electrons first, and for a picosecond or so they are far hotter than the lattice. Over that time the electrons carry heat into the film ballistically, at speeds near the Fermi velocity, a million metres a second, rather than diffusively. Measurements of how quickly the back surface of a gold film tens of nanometres thick responds show the heat arriving sooner than diffusion allows and later than free flight, the intermediate regime the telegraph equation was designed for, and models that give the electrons a finite relaxation time describe it. In semiconductors, heating a spot smaller than the phonons’ mean free path — hundreds of nanometres in silicon at room temperature — makes the hot spot cool more slowly than Fourier’s law predicts, because the phonons fly out ballistically and do not yet behave as a diffusing gas. Both effects matter for the thermal design of transistors, whose hottest regions are now smaller than those mean free paths.

A different set of claims has not held up. In the 1990s several experiments reported heat travelling as waves, with relaxation times of seconds, in processed meat, sand and other heterogeneous materials, and hyperbolic heat conduction was proposed for modelling laser surgery on tissue. Later experiments designed to reproduce the effect did not find it, and the apparent waves have been attributed to the materials’ structure — heat moving through several components at different rates — rather than to a genuine relaxation of the heat flux. A relaxation time of seconds would imply a signal speed of millimetres per second and second-sound waves in steak, and the evidence for that does not currently stand.

The contrast is instructive. The genuine departures from Fourier’s law all occur where a microscopic time or length has become comparable with the scale of the experiment, and each can be traced to a particular kind of carrier and its collisions. The claims that failed were in materials where no carrier with a long relaxation time was identified. The random walk that underlies diffusion — the same walk the jiggle that proved atoms traced for pollen — is an excellent description once many steps have been taken, and every correction to it lives in the first few steps.

What the telegraph equation leaves out #

Cattaneo’s equation is the simplest causal repair, and it is not the whole story.

It is phenomenological. The relaxation time is put in by hand. Kinetic theory, which follows the distribution of the heat carriers, derives such an equation as the first correction beyond Fourier’s law, with the relaxation time set by the collision rate, and it also shows that the telegraph equation is not exact: at wavelengths comparable with the mean free path, heat transport becomes ballistic and no local equation describes it.

It has a sharp front no real material shows. The delta-function fronts in the figures are an artefact of a single relaxation time. Real carriers have a distribution of speeds and relaxation times, which smears the front into a finite pulse, as the second-sound experiments see.

It can violate the second law transiently. For some initial conditions the telegraph equation lets heat flow briefly from cold to hot, a known embarrassment that the fuller theories resolve by modifying the entropy to include the heat flux as a variable, as extended irreversible thermodynamics does.

Still open: how far hydrodynamics reaches #

The success of relativistic hydrodynamics for the quark–gluon plasma raised a question that is still being answered. Hydrodynamics assumes the fluid is close to local equilibrium, with gradients gentle compared with the mean free path. In heavy-ion collisions, and even more in collisions of small systems such as a proton with a lead nucleus, the gradients are steep, and yet hydrodynamic simulations still describe the data. Explanations include the idea of a hydrodynamic “attractor”, a universal evolution that systems reach before they are near equilibrium, and calculations in strongly coupled theories where hydrodynamics works surprisingly early. How small and how far from equilibrium a system can be while still flowing as a fluid, and whether the relaxation-time equations are describing genuine fluid behaviour or merely fitting it, is an active question.

The habit worth carrying away is to ask of any transport law how fast its signals travel. A law in which a flux responds instantly to a gradient carries its influence at infinite speed, and the repair is a relaxation time, which turns a diffusing cloud into a wave with a front at short times and recovers diffusion at long ones. For ordinary matter the relaxation time is so much longer than relativity requires that the repair is invisible. For a fluid made of particles moving near the speed of light it is the difference between a stable theory and one that predicts heat flowing into the past.