"Reciprocal relations" occur between different pairs of forces and flows in a variety of physical systems.
For example, consider fluid systems described in terms of temperature, matter density, and pressure.
In this class of systems, it is known that temperature differences lead to heat flows from the warmer to the colder parts of the system; similarly, pressure differences will lead to matter flow from high-pressure to low-pressure regions.
The theory developed by Onsager is much more general than this example and capable of treating more than two thermodynamic forces at once, with the limitation that "the principle of dynamical reversibility does not apply when (external) magnetic fields or Coriolis forces are present", in which case "the reciprocal relations break down".
[1] Though the fluid system is perhaps described most intuitively, the high precision of electrical measurements makes experimental realisations of Onsager's reciprocity easier in systems involving electrical phenomena.
In fact, Onsager's 1931 paper[1] refers to thermoelectricity and transport phenomena in electrolytes as well known from the 19th century, including "quasi-thermodynamic" theories by Thomson and Helmholtz respectively.
Onsager's reciprocity in the thermoelectric effect manifests itself in the equality of the Peltier (heat flow caused by a voltage difference) and Seebeck (electric current caused by a temperature difference) coefficients of a thermoelectric material.
Similarly, the so-called "direct piezoelectric" (electric current produced by mechanical stress) and "reverse piezoelectric" (deformation produced by a voltage difference) coefficients are equal.
Experimental verifications of the Onsager reciprocal relations were collected and analyzed by D. G. Miller[2] for many classes of irreversible processes, namely for thermoelectricity, electrokinetics, transference in electrolytic solutions, diffusion, conduction of heat and electricity in anisotropic solids, thermomagnetism and galvanomagnetism.
In this classical review, chemical reactions are considered as "cases with meager" and inconclusive evidence.
Further theoretical analysis and experiments support the reciprocal relations for chemical kinetics with transport.
[3] Kirchhoff's law of thermal radiation is another special case of the Onsager reciprocal relations applied to the wavelength-specific radiative emission and absorption by a material body in thermodynamic equilibrium.
For his discovery of these reciprocal relations, Lars Onsager was awarded the 1968 Nobel Prize in Chemistry.
The presentation speech referred to the three laws of thermodynamics and then added "It can be said that Onsager's reciprocal relations represent a further law making a thermodynamic study of irreversible processes possible.
"[4] Some authors have even described Onsager's relations as the "Fourth law of thermodynamics".
In a simple fluid system, neglecting the effects of viscosity, the fundamental thermodynamic equation is written:
For non-fluid or more complex systems there will be a different collection of variables describing the work term, but the principle is the same.
The above expression of the first law in terms of entropy change defines the entropic conjugate variables of
and are intensive quantities analogous to potential energies; their gradients are called thermodynamic forces as they cause flows of the corresponding extensive variables as expressed in the following equations.
However, if we assume that the macroscopic velocity of the fluid is negligible, we obtain energy conservation in the following form:
is the rate of increase in entropy density due to the irreversible processes of equilibration occurring in the fluid and
, with the thermal conductivity possibly being a function of the thermodynamic state variables, but not their gradients or time rate of change.
In the absence of heat flows, Fick's law of diffusion is usually written:
Since this is also a linear approximation and since the chemical potential is monotonically increasing with density at a fixed temperature, Fick's law may just as well be written:
is a function of thermodynamic state parameters, but not their gradients or time rate of change.
For the general case in which there are both mass and energy fluxes, the phenomenological equations may be written as:
It can be seen that, since the entropy production must be non-negative, the Onsager matrix of phenomenological coefficients
The fact that they are at least proportional is suggested by simple dimensional analysis (i.e., both coefficients are measured in the same units of temperature times mass density).
The rate of entropy production for the above simple example uses only two entropic forces, and a 2×2 Onsager phenomenological matrix.
Assuming the fluctuations are small, the probability distribution function can be expressed through the second differential of the entropy[6]