The second experiment that cannot disagree

Heat one end of a bar of metal and a voltage appears between the ends. Send a current through the same bar with both ends at the same temperature and it carries heat from one end to the other, warming one junction and cooling the other.

Those are two different experiments. The first needs a heater and a voltmeter; the second needs a power supply and a calorimeter. There is no obvious reason why they should have anything to do with one another, and the number they measure is the same to within the precision of the computation.

What is actually being compared #

Both experiments are about a flow driven by something other than its own gradient, and there are two of them.

A current can be driven by a voltage — that is Ohm’s law — or by a temperature difference, which is the Seebeck effect and is what a thermocouple is.

A heat flow can be driven by a temperature difference — that is Fourier’s law — or by a current, which is the Peltier effect and is what a thermoelectric cooler is.

The direct effects are unrelated: a material’s electrical conductivity and its thermal conductivity are separate numbers. The two cross effects are not. Writing the two flows as a linear response to the two driving forces gives a two-by-two matrix, and the claim is that the two off-diagonal entries are equal.

For the thermoelectric case that comes out as $\Pi = ST$, the Peltier coefficient being the Seebeck coefficient times the absolute temperature.

Measuring it twice without using either answer #

The figure computes both for a model conductor, and the two computations share no step.

The Seebeck coefficient is obtained by imposing a temperature difference across the conductor, computing the current that results, and searching for the voltage that brings it back to zero. That is a root-find on one integral.

The Peltier coefficient is obtained by imposing a voltage at uniform temperature and taking the ratio of the heat flow to the charge flow. That is a ratio of two other integrals.

Sweeping the conductor’s transmission resonance across the chemical potential moves both coefficients through a wide range, through zero, and out the other side with the opposite sign — and the two curves agree to $3.8\times10^{-7}$ of the sweep’s own scale at every point.

The sign change is the part worth pausing on. A conductor whose transmission rises with energy carries heat one way and one whose transmission falls carries it the other, and the thermopower changes sign with it. Both coefficients change sign at the same place, which is what the diagonal in the second figure is showing.

The device the relation makes possible #

A cross-effect that runs both ways is a machine that runs both ways, and the thermoelectric pair is the clearest example of a heat engine and a refrigerator being the same object.

A heat pump’s performance rises steeply as the temperature it is pumping from approaches the temperature it is pumping to, and a thermoelectric module is such a machine with no moving parts at all. Run it one way and it moves heat against a gradient for electrical work; run it backwards and it delivers electrical work from a gradient. That the same device does both, with coefficients tied together, is the relation this essay is about wearing an engineering coat.

Pass a current through a junction between two materials with different thermopowers and one side cools while the other warms — a solid-state refrigerator with no compressor, no working fluid and no moving parts, which is what cools an infrared detector or a laboratory sample stage. Reverse the roles: hold the two sides at different temperatures and the same module delivers a current, which is what powers a spacecraft from a lump of decaying plutonium.

The two operations are governed by the two coefficients this essay is about, and because they are one coefficient, a material that is good at one is exactly as good at the other. That is a strong statement: a search for a better thermoelectric cooler and a search for a better thermoelectric generator are the same search.

What limits both is not the relation but the accompanying conductivities. The useful quantity is a dimensionless figure of merit combining the thermopower with the electrical conductivity, which should be large, and the thermal conductivity, which should be small — and the mechanism that carries charge in a metal also carries heat, so the two cannot be adjusted independently. The best materials reach a figure of merit near one after seventy years of work, which corresponds to about a sixth of Carnot’s ceiling.

There is one more thing the numbers in the figure say that is easy to miss. The two coefficients are large where the transmission function is steep at the chemical potential and vanish where it is flat, because what drives a thermoelectric current is an asymmetry between the electrons above the chemical potential and the holes below it. A material with a symmetric band structure has no thermopower at all, however good a conductor it is. That is why metals — whose transmission is nearly flat over the few tens of millielectronvolts that matter — have thermopowers of a few microvolts per kelvin, and why the useful thermoelectrics are heavily doped semiconductors, where the chemical potential sits on the edge of a band and the asymmetry is enormous.

Where the equality comes from #

Kelvin derived the relation in 1854 and did not think he had proved it.

His argument runs a thermoelectric circuit as a reversible engine and applies the second law, which gives the answer — but the circuit is not reversible. Ordinary heat conduction is going on at the same time, and ordinary Joule heating, and there is no justification within thermodynamics for setting those aside and treating the thermoelectric part as though it were on its own. Kelvin said so.

Every process here is irreversible in the plainest way: a concentration profile spreads, entropy rises, and nothing runs backwards. That is exactly what made the reversible argument for the relation unsatisfactory for nearly eighty years — a symmetry derived from time-reversal invariance, asserted about processes that visibly do not reverse. The resolution is that the invariance is used at the level of the fluctuations, which do reverse, and not at the level of the average flow, which does not.

The relation was checked experimentally over the following decades and held, and the reason was not found until 1931. Onsager’s argument does not come from thermodynamics at all; it comes from underneath it.

The microscopic equations of motion are the same run forwards and backwards. That symmetry says something about the fluctuations of a system in equilibrium — the ones a suspended particle makes visible: the correlation between one quantity now and another a moment later equals the correlation between the second now and the first a moment later. Onsager’s regression hypothesis — since made a theorem — says that a spontaneous fluctuation decays on average by the same macroscopic law that governs a deliberately imposed disturbance. Put the two together and the matrix of coefficients relating flows to forces must be symmetric.

So a statement about a laboratory measurement over centimetres has been derived from the reversibility of collisions between individual particles.

It is worth noticing what kind of statement has been established. Thermodynamics tells what is forbidden — no process may decrease the entropy — and says nothing about the coefficients of the processes that are allowed. Onsager’s relations are a statement about those coefficients, and they cannot be derived from the two laws alone; they need the microscopic dynamics. In that sense they sit between statistical mechanics and thermodynamics rather than inside either, which is roughly what “irreversible thermodynamics” means as a subject.

The experimental history is worth a line, because the relation was tested long before it was understood. Measurements of the Thomson coefficient — the third member of the thermoelectric family, which describes heat released when a current flows along a temperature gradient — provide an independent check, since Kelvin’s second relation ties it to the derivative of the thermopower. Both relations were confirmed to within experimental error in a dozen materials before 1900, which left the situation that a result everybody used had a derivation nobody accepted. Onsager’s paper closed a gap that had been open for three quarters of a century, and the citation for his Nobel prize thirty-seven years later names those relations and nothing else.

What has to be chosen correctly for it to work #

The relation is symmetric only when the flows and forces are paired in the right way, and getting that wrong is the commonest way to find a false violation.

What pairs the flows with their forces is the rate at which entropy is produced: written as a sum of products of flows and forces, that rate names the pairing uniquely. Use any other pairing — a temperature gradient against a heat flow rather than the gradient of $1/T$, say — and the matrix that results is not symmetric. The relations are true of one particular choice of variables, and the choice is not a convention but a consequence of what the entropy production is a sum of.

The requirement is that the rate of entropy production be the sum of each flow multiplied by its own force. For the thermoelectric case that means pairing the heat flow with the gradient of $1/T$ rather than of $T$, and the charge flow with the gradient of the electrochemical potential over $T$. Choose those and the matrix is symmetric; choose the more obvious pairings and it is not, by exactly the factor of $T$ that turns $\Pi = ST$ into $\Pi = S$.

That is worth stating plainly because it is where the content sits. The relations are not a claim that any two cross-coefficients are equal; they are a claim that the matrix is symmetric in the basis where entropy production is a simple sum, and finding that basis is the work.

The same requirement generalises directly. In a mixture with several species, several temperatures and several chemical reactions there are many cross-effects, and the relations halve the number of independent coefficients — which is the practical reason they matter in the modelling of anything from a fuel cell to a membrane that sorts by counting.

Where it fails, and why the failure is useful #

The derivation used time-reversal symmetry, so anything that breaks it breaks the symmetric form.

A magnetic field is where the symmetry acquires a qualification. A field makes the microscopic motion look different run backwards, because a charge deflected one way by a field is deflected the same way when its velocity is reversed — so time reversal has to reverse the field too. The reciprocal relations therefore hold between the coefficient at $\mathbf{B}$ and the coefficient at $-\mathbf{B}$, not at the same field, and that qualified form is what a Hall measurement tests.

A magnetic field is the standard case. Reverse time and the currents that make the field reverse too, so the field reverses with them; the microscopic symmetry is therefore between motion in a field and motion in the opposite field. The relation becomes

$$L_{ij}(\mathbf{B}) = L_{ji}(-\mathbf{B}),$$

which is a real restriction and is not symmetry of the matrix at any single field. The antisymmetric part that survives is exactly where the Hall effect lives, and with it the whole family of thermomagnetic cross-effects — Nernst, Ettingshausen, Righi–Leduc — each of which is a flow at right angles to a force.

Rotation does the same thing, for the same reason, which is why Coriolis-dominated transport in a rotating fluid has the same structure.

So the failure is not a limitation of the theory. It is a classification: the symmetric part of the response comes from reversible microscopic dynamics, the antisymmetric part requires something that distinguishes the two directions of time, and knowing which is which says what a material can and cannot do.

The same symmetry, in places that are not thermal #

Once the source of the equality is identified as time-reversal symmetry of the microscopic motion, the pattern turns up wherever two effects are each other’s inverse.

The same symmetry appears where nothing is thermal at all. Two circuits each induce an electromotive force in the other, and the two mutual inductances are equal — $M_{12} = M_{21}$ — however different the two coils are in size, shape or number of turns. That equality is a symmetry statement of the same family, provable the same way, and it is old enough that nobody finds it surprising, which is the best evidence that the thermal version should not be surprising either.

Two coils have one mutual inductance. The voltage induced in the second by a changing current in the first, per unit rate of change, equals the voltage induced in the first by the second — however different the two circuits are in size and shape. That is a statement about a symmetric matrix of coefficients, provable from the field equations and traceable to the same reversibility.

A transmitting antenna and a receiving one have the same pattern. An antenna’s directional response as a receiver is identical to its pattern as a transmitter, which is why one measurement suffices and why antenna ranges are built to do whichever is easier.

Diffusion and conduction cross in a mixture. A temperature gradient drives a separation of species and a concentration gradient drives a heat flow, and the two coefficients are again one.

The common form is: two channels, each capable of driving the other, and one number for the pair. Where the pattern fails, something has broken the symmetry of the underlying dynamics, and — as with the magnetic field above — that failure is usually the effect somebody wanted.

The coefficient that cannot be measured directly #

A quantity that two experiments agree on is a quantity worth having a value for, and there is an awkwardness in getting one that the relations themselves eventually solve.

Connect a voltmeter across a bar of material whose ends are at different temperatures and the reading is not the bar’s thermopower. The voltmeter’s leads run from the hot end and the cold end back to the instrument, so they too span the temperature difference and generate their own thermoelectric voltage, and what is measured is the difference between the bar’s coefficient and the leads’. Every thermocouple works on exactly that difference, which is why one is specified as a pair of materials and never as one.

So the absolute thermopower of a single substance is not directly measurable by any arrangement of wires. Two results rescue it.

The first is that a superconductor has zero thermopower. Below its transition the current is carried by pairs that transport no entropy, so a thermocouple made of any material against a superconducting one reads that material’s absolute coefficient outright. That anchors the scale, over the few kelvin where a convenient superconductor is available.

The second is Kelvin’s other relation. The Thomson coefficient — the heat released or absorbed when a current flows along a temperature gradient in a single homogeneous conductor — needs no second material and is therefore measurable on its own, and it equals $T\,\mathrm{d}S/\mathrm{d}T$. Measure it up a temperature range, divide by $T$, integrate, and the absolute thermopower is carried upward from the superconducting anchor to wherever the measurements stop.

That is the whole construction of the standard absolute scale, and every quoted thermopower is referred to it. It is a good illustration of what the relations are for: not to save an experiment, but to make a quantity accessible that no direct experiment reaches.

The same pairing, with no heat in it #

The relation’s reach is easiest to appreciate in a case with nothing thermal in it at all, and there is one that was measured before Onsager was born.

Push liquid through a porous plug or a fine capillary whose walls carry a surface charge. The liquid drags some of the mobile counter-ions along with it, so a pressure difference produces an electric current, and if the circuit is open, a voltage — the streaming potential.

Now do the other experiment. Apply a voltage across the same plug at equal pressures. The ions move, drag the liquid with them, and a flow results — electro-osmosis.

Two effects, two apparatus, one coefficient. Saxén’s relation says the streaming potential per unit applied pressure equals the electro-osmotic volume flow per unit current, and he verified it experimentally in 1892 — four decades before anybody could say why it had to be true. It is the thermoelectric pair with pressure in place of temperature and volume flow in place of heat, and it is symmetric for the same reason.

The pairing has a use underground. Water moving through rock generates streaming potentials, so a survey that measures voltages at the surface can locate subsurface flow — a leaking dam, a geothermal upwelling, water moving into a volcano — with electrodes and no drilling. What makes the interpretation possible is that the coupling coefficient can be calibrated in the laboratory by the other experiment, on a core sample, driving current and measuring flow. The relation is what licenses carrying a number measured one way into a measurement made the other.

Where the model runs out #

Everything is linear. The relations are statements about the first derivative of a flow with respect to a force, and they say nothing about what happens when the driving is strong. Far from equilibrium there is no general symmetry, and the search for a replacement — of which the fluctuation theorems are the most successful part — has been going on since.

Local equilibrium is assumed. Writing a temperature and a chemical potential at each point requires that each little region be internally equilibrated, which fails when the gradients vary over a mean free path. That is the ordinary situation in a nanoscale device, and the transport there is described by the Landauer picture the figures use rather than by local coefficients at all — which is why the model here computes currents from a transmission function and only afterwards extracts the two coefficients.

The conductor is a model. A single resonance of a chosen width at a chosen position is a fair description of a quantum dot and a caricature of a metal. What the figure demonstrates is that the relation holds identically in a model with no symmetry built into it; a real material’s coefficients depend on band structure, phonon drag and impurity scattering, none of which is here.

And nothing in the relation says either coefficient is large. Thermoelectric devices are limited by a figure of merit combining the thermopower with the two conductivities, and the difficulty of building a good one is that the three quantities are not independent in any real material — a good electrical conductor is usually a good thermal one, for the reason kinetic theory gives, and the useful combination is stubbornly small.

The ladder from here #

Later rungs on this anchor: the fluctuation–dissipation theorem, which relates a system’s response to a force to the spontaneous fluctuations it has anyway and is the same idea taken one step further; the thermomagnetic effects and how the antisymmetric part is measured; entropy production as a variational principle, and where the claim that it is minimised is true and where it is not; and the fluctuation theorems, which say something exact about arbitrarily strong driving where the linear relations say nothing.

The neighbouring ladders are the equation that only runs forwards, which is where an irreversible macroscopic law is got out of reversible microscopic ones, the coupling that is the same both ways, where two mutual inductances are equal for a related reason of symmetry, and entropy is a count, where the quantity whose production rate does the pairing is defined.