Concept

Cross effect — where it appears

A flow driven by a force of a different kind, such as a flow of matter driven by a temperature gradient or of heat driven by an electric current. The coefficients linking cross-effects come in equal pairs, which is Onsager's reciprocity.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as entropy production, onsager relations — the same set of essays touches all of them, so they are one junction rather than several.

Two experiments, two computations, one coefficient. The Seebeck coefficient of a resonant conductor against where its resonance sits relative to the chemical potential, together with the Peltier coefficient divided by the temperature. The first is obtained by applying a temperature difference and finding the voltage that stops the current; the second by applying a voltage at uniform temperature and taking the ratio of the heat flow to the current. Different driving, different measurement, different integral — and the two curves agree to 3.8e-7 of the sweep's own scale across the whole of where they pass through zero and change sign. That equality is Kelvin's relation Π = ST, guessed in 1854 from an argument its author knew was not sound and proved by Onsager in 1931 from microscopic reversibility. It is not a property of this conductor; it holds for every one.

The second experiment that cannot disagree

Heat one end of a wire and a voltage appears across it. Pass a current through the same wire at uniform temperature and it carries heat. Those are two different experiments with two different apparatus, and the coefficient in front is the same number in both — not approximately, and not for some materials. The reason is that the equations of motion underneath look the same run backwards.

thermodynamics · Diffusion $Nitrogen that runs the wrong way. The nitrogen mole fraction in each of two bulbs joined by a capillary, as in Duncan and Toor's experiment: one bulb starts with 0.50086 nitrogen and the rest carbon dioxide, the other with 0.49879 nitrogen and the rest hydrogen, at 35 °C. Solid curves: the capillary solved at each instant from the Maxwell–Stefan equations with the three pairs' diffusivities, 83.8, 68.0 and 16.8 mm²/s. Dashed: Fick's law for nitrogen alone, which can only let the two start values relax together. At the start the nitrogen gradient is 0.00207, yet nitrogen flows at 309 times the rate that gradient would drive. From 0.1 to 6.5 hours it flows from the bulb with less nitrogen into the bulb with more, opening a difference of 0.1468 at 6.5 hours; at 6.6 hours its flux passes through zero with a difference of 0.1468 still in place. Each gas is conserved to a part in 10⁹. The carbon dioxide moving out of the first bulb drags nitrogen with it, because the nitrogen–carbon dioxide pair has by far the smallest diffusivity and so the strongest friction.$

The gas that flows towards more of itself

Fick's law says a substance diffuses from where there is more of it to where there is less. In a mixture of three gases, nitrogen can do the opposite for hours — flowing into the bulb that already holds more nitrogen, and then stopping while a difference remains — and in a welded bar of steel, carbon crosses into the side that is already richer. Nothing is wrong with the second law. Diffusion flattens chemical potential, and with more than two components, or a second element changing it, that is not the same as flattening concentration.

thermodynamics · Diffusion

Named alongside it

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

Entropy production Irreversibility Onsager relations Transport coefficient Detailed balance Diffusion Equilibrium Kelvin relation Linear response Microscopic reversibility Peltier effect Reciprocity

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