Physics 1931

Reciprocal Relations in Irreversible Processes

Lars Onsager

When one set of forces drives many flows at once, each cross-effect mirrors the other — exactly.

Choose your version

In depth · the introduction

Some materials turn a temperature difference into electricity — and run backwards, the very same material uses electricity to pump heat. Onsager proved the two tricks are locked together by exactly one number.

The big idea

Most of nature is a one-way street. Heat flows from hot to cold, ink spreads through water, a current warms a wire — and none of it spontaneously reverses. These are the 'irreversible' processes, and for a long time they had no exact law of their own, only the rough statement that disorder tends to grow.

Lars Onsager found the missing rule. When several of these one-way flows happen at once and push on each other — heat dragging charge along, a voltage dragging heat — the strength with which the first effect feeds the second is exactly equal to the strength with which the second feeds the first. He proved this 'reciprocal relation' from a deep fact: the laws of motion of atoms look the same run forwards or backwards in time.

How it came about

Onsager was a Norwegian-born chemist working in the United States, famous for thinking in silence and writing almost nothing down. In 1931, at Brown University, he published the result in two short, severe papers in the Physical Review — so compressed that few colleagues grasped what he had done.

There was a clue waiting in the textbooks. Back in 1854 Lord Kelvin had noticed that the two thermoelectric effects — making electricity from heat, and pumping heat with electricity — seemed to share a single coefficient, but he could not really prove it. Onsager's reciprocity supplied the proof, and showed Kelvin's lucky guess was one case of a universal symmetry. The work was so far ahead of its time that the Nobel Prize did not arrive until 1968, nearly forty years later.

Why it mattered

It gave the messy, irreversible half of the world its own law — sometimes called the fourth law of thermodynamics. Once you know that coupled flows are symmetric, you can predict one cross-effect from its partner, design thermoelectric coolers and generators, and understand how cells pump ions and water across membranes. A whole field, non-equilibrium thermodynamics, grew from this single symmetry.

A way to picture it

Think of a revolving door between two rooms. If pushing on the door from room A nudges someone in room B forward, then pushing from room B nudges someone in room A forward by exactly the same amount — the coupling can't be a one-way bargain. Onsager showed that nature's coupled flows are like that door: the give and the take are always equal, because the underlying motions don't care which way time runs.

Where it sits

Carnot, Clausius and Boltzmann (all in this Library) built the thermodynamics of equilibrium — of systems that have already settled down. Onsager took the next step, into systems that are still flowing and changing but only gently disturbed. From his relations the trail runs onward to Ilya Prigogine's work on systems driven far from equilibrium, and to the fluctuation theorems that describe the jittery thermodynamics of tiny machines.

The original document

Original source text

L. Onsager · Physical Review 37 (1931): 405–426 · Physical Review 38 (1931): 2265–2279

Part I — the reciprocal relations

The first paper sets up the linear laws of irreversible transport: when a system is driven a little out of equilibrium, each flow (of heat, of electric charge, of matter) is taken proportional to the thermodynamic forces (gradients of temperature, voltage, concentration), through a matrix of coefficients. Onsager's central claim is that this matrix is symmetric — the coefficient coupling force k to flow i equals the one coupling force i to flow k.

He derives the symmetry not from thermodynamics alone but from a deeper principle: microscopic reversibility, the fact that the underlying equations of motion run the same forwards and backwards in time. Joined to the hypothesis that, on average, a spontaneous fluctuation decays back to equilibrium by the same laws that govern an imposed flow, this fixes the cross-coefficients to be equal.

Onsager points to the long-standing thermoelectric example: William Thomson (Lord Kelvin) had written down the reciprocal relation between the Seebeck and Peltier effects in 1854, but by an argument he himself admitted was not rigorous. The reciprocal relations supply the missing proof — and extend it to conduction in anisotropic crystals, coupled diffusion, and beyond.

[ … ]

Part II — least dissipation of energy

The second paper recasts the result as a variational principle: among the possible flows, the actual ones are those that, in a precise sense, dissipate energy least for a given rate of entropy production. The reciprocity of the coefficients is what makes this principle consistent. The full development, with the fluctuation theory behind it, is in the original papers at the source.