In
thermodynamicsThermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
, the
Onsager reciprocal relations express the equality of certain ratios between
flowIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
s and
forceIn physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s in
thermodynamic systemA thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....
s out of equilibrium, but where a notion of local equilibrium exists.
"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
temperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
differences lead to
heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
flows from the warmer to the colder parts of the system; similarly,
pressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure. Definition :...
differences will lead to
matterMatter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...
flow from highpressure to lowpressure regions. What is remarkable is the observation that, when both pressure and temperature vary, temperature differences at constant pressure can cause matter flow (as in
convectionConvection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....
) and pressure differences at constant temperature can cause heat flow. Perhaps surprisingly, the heat flow per unit of pressure difference and the
densityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
(matter) flow per unit of temperature difference are equal. This equality was shown to be necessary by
Lars OnsagerLars Onsager was a Norwegianborn American physical chemist and theoretical physicist, winner of the 1968 Nobel Prize in Chemistry.He held the Gibbs Professorship of Theoretical Chemistry at Yale University....
using
statistical mechanicsStatistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
as a consequence of the
time reversibilityTime reversibility is an attribute of some stochastic processes and some deterministic processes.If a stochastic process is time reversible, then it is not possible to determine, given the states at a number of points in time after running the stochastic process, which state came first and which...
of microscopic dynamics (
microscopic reversibilityThe principle of Microscopic reversibility in physics and chemistry is twofold:* First, it states that the microscopic detailed dynamics of particles and fields is timereversible because the microscopic equations of motion are symmetric with respect to inversion in time ;* Second, it relates to...
). 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".
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 refers to
thermoelectricity and transport phenomena in
electrolytesIn chemistry and manufacturing, electrolysis is a method of using a direct electric current to drive an otherwise nonspontaneous chemical reaction...
as wellknown from the 19th century, including "quasithermodynamic" 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 (electrical current caused by a temperature difference) coefficients of a thermoelectric material. Similarly, the socalled "direct piezoelectric" (electrical current produced by mechanical stress) and "reverse piezoelectric" (deformation produced by a voltage difference) coefficients are equal. For many kinetic systems, like the
Boltzmann equationThe Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of one particle in rarefied gas...
or
chemical kineticsChemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition...
, the Onsager relations are closely connected to the principle of detailed balance and follow from them in the linear approximation near equilibrium.
For his discovery of these reciprocal relations,
Lars OnsagerLars Onsager was a Norwegianborn American physical chemist and theoretical physicist, winner of the 1968 Nobel Prize in Chemistry.He held the Gibbs Professorship of Theoretical Chemistry at Yale University....
was awarded the 1968
Nobel Prize in ChemistryThe Nobel Prize in Chemistry is awarded annually by the Royal Swedish Academy of Sciences to scientists in the various fields of chemistry. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895, awarded for outstanding contributions in chemistry, physics, literature,...
. 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 possible a thermodynamic study of irreversible processes." Some authors have even described Onsager's relations as the "Fourth law of thermodynamics".
Thermodynamic potentials, forces and flows
The basic thermodynamic potential is internal
energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
. In a
fluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
system, changes in the energy density
depend on changes in matter density
and
entropyEntropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
density
in the following way:

where
is temperature and
is a combination of pressure and
chemical potentialChemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
. This formula is simply an expression of the
first law of thermodynamicsThe first law of thermodynamics is an expression of the principle of conservation of work.The law states that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed...
, namely energy conservation, where the
term represents
heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
exchange, and the
term represents (mechanical and chemical)
workIn thermodynamics, work performed by a system is the energy transferred to another system that is measured by the external generalized mechanical constraints on the system. As such, thermodynamic work is a generalization of the concept of mechanical work in mechanics. Thermodynamic work encompasses...
done. For nonfluid or more complex systems there will be a different collection of variables describing the work term, but the principle is the same. We can write

The extensive quantities
and
are
conservedIn physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
and their flows satisfy
continuity equationA continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
s:

and
 ,
where
indicates the
partial derivativeIn mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
with respect to
timeTime is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
(that is, the local timerate of change), and
indicates the
divergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of the
flux densitiesFormal Statement:The flux density is simply defined as the amount of flux passing through a unitarea. Mathematical Statement:The flux density would essentially be the number of field lines passing through a defined unitarea...
.
The above expression of the first law in terms of entropy change defines the
conjugate variablesIn thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature/entropy or pressure/volume. In fact all thermodynamic potentials are expressed in terms of conjugate pairs....
of
and
, which are
and
and are intensive quantities analogous to
potential energiesIn physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
; their gradients of are called thermodynamic forces as they cause flows of the corresponding extensive variables as expressed in the following equations.
In the absence of matter flows, we have a version of Fourier's law
 ;
and, in the absence of heat flows, we have a version of Fick's law
 ,
where
now indicates the
gradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
.
The reciprocity relations
In this example, when there are both heat and matter flows, there are "crossterms" in the relationship between flows and forces (the proportionality coefficients are customarily denoted by
):

and

The Onsager reciprocity relations state the equality of the crosscoefficients
and
. Proportionality follows from simple
dimensional analysisIn physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
(
i.e., both coefficients are measured in the same units of temperature times mass density).
Abstract formulation
Let
denote fluctuations from equilibrium values in several thermodynamic quantities, and let
be the entropy. Then, assuming the fluctuations are small (
gaussianGaussian is a computational chemistry software program initially released in 1970 by John Pople and his research group at CarnegieMellon University as Gaussian 70. It has been continuously updated since then...
), the probability
distribution functionIn molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...
can be expressed as
(where we are using Einstein summation convention and
is some symmetric matrix)
Using the quasistationary equilibrium approximation, that is, assuming that the system is only slightly nonequilibrium, we have
Suppose we define
thermodynamic conjugate quantities as
, which can also be expressed as linear functions (for small fluctuations):
Thus, we can write
where
are called
kinetic coefficients
The
principle of symmetry of kinetic coefficients or the
Onsager's principle states that
is a symmetric matrix, that is
Proof
Define mean values
and
of fluctuating quantities
and
respectively such that they take given values
at
Note that
Symmetry of fluctuations under time reversal implies that
or, with
, we have
Differentiating with respect to
and substituting, we get
Putting
in the above equation,
It can be easily shown from the definition that
, and hence, we have the required result.