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{{Redirect|Fourth law of thermodynamics|fourth principle of energetics proposed by H. T. Odum|Maximum power principle}}
{{thermodynamics|cTopic=[[Thermodynamic equations|Equations]]}}
In [[thermodynamics]], the '''Onsager reciprocal relations''' express the equality of certain ratios between [[flux|flow]]s and [[force]]s in [[thermodynamic system]]s out of [[equilibrium (thermo)|equilibrium]], but where a notion of [[local thermodynamic equilibrium|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 [[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. What is remarkable is the observation that, when both pressure and temperature vary, temperature differences at constant pressure can cause matter flow (as in [[convection]]) and pressure differences at constant temperature can cause heat flow. Perhaps surprisingly, the heat flow per unit of pressure difference and the [[density]] (matter) flow per unit of temperature difference are equal. This equality was shown to be necessary by [[Lars Onsager]] using [[statistical mechanics]] as a consequence of the [[time reversibility]] of microscopic dynamics ([[microscopic reversibility]]). 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".<ref name="onsager">L. Onsager, [http://prola.aps.org/abstract/PR/v37/i4/p405_1 Reciprocal Relations in Irreversible Processes. I., Phys. Rev. 37, 405 - 426 (1931)]</ref> 

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<ref name="onsager" /> refers to [[thermoelectricity]] and transport phenomena in [[electrolysis|electrolytes]] as well-known from the 19th century, including "quasi-thermodynamic" theories by [[Thomson_effect#Thomson_effect|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 so-called "direct [[piezoelectric effect|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 equation]] or [[chemical kinetics]], the Onsager relations are closely connected to the principle of [[detailed balance#Onsager reciprocal relations and detailed balance|detailed balance]]<ref name="onsager" /> and follow from them in the linear approximation near equilibrium.

[[Experiment]]al verifications of the Onsager reciprocal relations were collected and analyzed by D.G. Miller<ref>D.G. Miller,  Thermodynamics of irreversible processes. The experimental verification of the Onsager reciprocal relations, Chem. Rev. 60 (1960), 15-37.</ref> for many classes of irreversible processes, namely for [[thermoelectricity]], [[Electrokinetic phenomena|electrokinetics]], transference in [[electrolyte|electrolytic]] [[solution]]s, [[diffusion]], [[heat conduction|conduction of heat]] and [[conduction of electricity|electricity]] in [[Anisotropy|anisotropic]] [[Solid-state physics|solids]], [[thermomagnetism]] and [[galvanomagnetism]]. In this classical review, [[chemical kinetics|chemical reactions]] are considered as "cases with meager" and with inconclusive evidence. Further theoretical analysis and experiments support the reciprocal relations for chemical kinetics with transport.<ref>[[Grigoriy Yablonsky|G.S. Yablonsky]], A.N. Gorban, D. Constales, V.V. Galvita and G.B. Marin, [http://arxiv.org/pdf/1008.1056v2.pdf Reciprocal relations between kinetic curves], EPL, 93 (2011) 20004.</ref> 

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."<ref>[http://nobelprize.org/nobel_prizes/chemistry/laureates/1968/press.html The Nobel Prize in Chemistry 1968. Presentation Speech.]</ref> Some authors have even described Onsager's relations as the "Fourth law of thermodynamics".<ref>For example Richard P. Wendt, Journal of Chemical Education v.51, p.646 (1974) "Sîmplified Transport Theory for Electrolyte Solutions"</ref>

== Example: Fluid system ==

=== The fundamental equation ===
The basic [[thermodynamic potential]] is internal [[energy]]. In a simple [[fluid]] system, neglecting the effects of [[viscosity]] the fundamental thermodynamic equation is written:

:<math>dU=T\,dS-P\,dV+\mu\,dM</math>

where ''U'' is the internal energy, ''T'' is temperature, ''S'' is entropy, ''P'' is the hydrostatic pressure, <math>\mu</math> is the chemical potential, and ''M'' mass. In terms of the internal energy density, ''u'', entropy density ''s'', and mass density <math>\rho</math>, the fundamental equation is written:
:<math>du=T\,ds+\mu\,d\rho</math>

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 equation may be solved for the entropy density:
:<math>ds = (1/T)\,du + (-\mu/T)\,d\rho</math>

The above expression of the first law in terms of entropy change defines the entropic [[conjugate variables (thermodynamics)|conjugate variables]] of <math>u</math> and <math>\rho</math>, which are <math>1/T</math> and <math>-\mu/T</math> and are [[intensive quantity|intensive quantities]] analogous to [[potential energy|potential energies]]; their gradients of  are called thermodynamic forces as they cause flows of the corresponding extensive variables as expressed in the following equations.

=== The continuity equations ===

The [[extensive quantity|extensive]] quantities <math>U</math> and <math>M</math> are [[conservation law|conserved]] and their flows satisfy [[continuity equation]]s. The conservation of mass is written:
: <math>\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J}_\rho = 0</math>
and, assuming that fluid velocity makes a negligible contribution to the energy flow, the conservation of energy is simply the conservation of the internal energy:
: <math>\frac{\partial u}{\partial t} + \nabla \cdot \mathbf{J}_u = 0</math>

where <math>\mathbf{J}_\rho </math> is the mass flux vector and <math>\mathbf{J}_u </math> is the heat flux vector.

The entropy is not conserved and its continuity equation is written:
:<math> \frac{\partial s}{\partial t} + \nabla \cdot \mathbf{J}_s = \frac{\partial s_c}{\partial t}</math>

where <math>\frac{\partial s_c}{\partial t}</math> is the rate of increase in entropy density due to the irreversible processes of equilibration occurring in the fluid.

=== The phenomenological equations ===

In the absence of matter flows, [[Fourier's law]] is usually written:

: <math>\mathbf{J}_{u} = -k\,\nabla T</math>;

where ''k'' is the [[thermal conductivity]]. However, this law is just a linear approximation, and holds only for the case where <math>\nabla T \ll T</math>, with the thermal conductivity possibly being a function of the thermodynamic state variables, but not their gradients or time rate of change. Assuming that this is the case, Fourier's law may just as well be written:

: <math>\mathbf{J}_u = k T^2\nabla (1/T)</math>;

In the absence of heat flows, [[Fick's law]] of diffusion is usually written:

: <math> \mathbf{J}_{\rho} = -D\,\nabla\rho</math>,

where ''D'' is the coefficient of diffusion. 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:

: <math> \mathbf{J}_{\rho} = D'\,\nabla(-\mu/T) \! </math>

where, again, ''D' '' is a function of thermodynamic state parameters, but not their gradients or time rate of change. For the general case in which there both mass and energy fluxes, the phenomenological equations may be written as:

: <math> \mathbf{J}_{u} = L_{uu}\, \nabla (1/T) + L_{u\rho}\, \nabla (-\mu/T)</math>
: <math> \mathbf{J}_{\rho} = L_{\rho u}\, \nabla (1/T) - L_{\rho\rho}\, \nabla (-\mu/T)</math>

or, more concisely,
: <math> \mathbf{J}_\alpha = \sum_\beta L_{\alpha\beta}\,\nabla f_\beta</math>

where the entropic "thermodynamic forces" conjugate to the "displacements" <math>u</math> and <math>\mu</math> are <math>f_u=(1/T)</math> and <math>f_\rho=(-\mu/T)</math> and <math>L_{\alpha \beta}</math> is the Onsager matrix of phenomenological coefficients.

=== The rate of entropy production ===

From the fundamental equation, it follows that:
:<math>\frac{\partial s}{\partial t}=(1/T)\frac{\partial u}{\partial t}+(-\mu/T)\frac{\partial \rho}{\partial t}</math>
and
:<math>\mathbf{J}_s=(1/T)\mathbf{J}_u+(-\mu/T)\mathbf{J}_\rho=\sum_\beta \mathbf{J}_\alpha f_\alpha</math>

Using the continuity equations, the rate of entropy production may now be written:
:<math>\frac{\partial s_c}{\partial t}= \mathbf{J}_u \cdot \nabla (1/T)+\mathbf{J}_\rho \cdot \nabla (-\mu/T)=\sum_\alpha \mathbf{J}_\alpha \cdot \nabla f_\alpha </math>

and, incorporating the phenomenological equations:
:<math>\frac{\partial s_c}{\partial t}= \sum_\alpha\sum_\beta L_{\alpha \beta}(\nabla f_\alpha)(\nabla f_\beta)</math>

It can be seen that, since the entropy production must be greater than zero, the Onsager matrix of phenomenological coefficients <math>L_{\alpha \beta}</math> is a [[positive semi-definite matrix]].

=== The Onsager reciprocal relations ===

Onsager's contribution was to demonstrate that not only is <math>L_{\alpha \beta}</math> positive semi-definite, it is also, except in certain special cases{{Clarify|date=August 2013}}, symmetric. In other words, the cross-coefficients <math>\ L_{u\rho}</math> and  <math>\ L_{\rho u}</math> are equal. The fact that they are at least proportional follows from simple [[dimensional analysis]] (''i.e.'', both coefficients are measured in the same [[unit (measurement)|unit]]s of temperature times mass density).

The rate of entropy production for the above simple example uses only two entropic forces, and a 2x2 Onsager phenomenological matrix. The expression for the linear approximation to the fluxes and the rate of entropy production can very often be expressed in an analogous way for many more general and complicated systems.

== Abstract formulation ==
Let <math>x_1,x_2,\ldots,x_n</math> denote fluctuations from equilibrium values in several thermodynamic quantities, and let <math>S(x_1,x_2,\ldots,x_n)</math> be the entropy. Then, [[Boltzmann's entropy formula]] gives for the probability [[distribution function]] <math>w =A\exp(S/k)</math>, ''A''=const, since the probability of a given set of fluctuations <math>{x_1,x_2,\ldots,x_n}</math> is proportional to the number of microstates with that fluctuation. Assuming the fluctuations are small, the probability [[distribution function]] can be expressed through the second differential of the entropy<ref name="landau">{{cite book |title=Statistical Physics, Part 1|last=Landau |first=L. D.|coauthors=Lifshitz, E.M. |year=1975 |publisher=[[Butterworth-Heinemann]] |location=Oxford, UK |isbn=978-81-8147-790-3}}</ref>

:<math>w=Ae^{-\frac{1}{2}\beta_{ik}x_ix_k}\, ; \;\;\;\;\ \beta_{ik}= -\frac{1}{k}\frac{\partial^2 S}{\partial x_i \partial x_k}\, ,</math>
where we are using [[Einstein summation convention]] and <math>\beta_{ik}</math> is a positive definite symmetric matrix.

Using the quasi-stationary equilibrium approximation, that is, Assuming that the system is only slightly [[non-equilibrium]], we have<ref name="landau"/> <math>\dot{x}_i=-\lambda_{ik}x_k</math>

Suppose we define ''thermodynamic conjugate'' quantities as <math>X_i=-\frac{1}{k}\frac{\partial S}{\partial x_i}</math>, which can also be expressed as linear functions (for small fluctuations): <math>X_i= \beta_{ik}x_k</math>

Thus, we can write <math>\dot{x}_i=-\gamma_{ik}X_k</math> where <math>\gamma_{ik}=\lambda_{il}\beta^{-1}_{lk}</math> are called ''kinetic coefficients''

The ''principle of symmetry of kinetic coefficients'' or the ''Onsager's principle'' states that <math>\gamma</math> is a symmetric matrix, that is <math>\gamma_{ik}=\gamma_{ki}</math><ref name="landau"/>

===Proof===

Define mean values <math>\xi_i(t)</math> and <math>\Xi_i(t)</math> of fluctuating quantities <math>x_i</math> and <math>X_i</math> respectively such that they take given values <math>x_1,x_2,\ldots</math> at <math>t=0</math> Note that <math>\dot{\xi}_i(t)=-\gamma_{ik}\Xi_k</math>

Symmetry of fluctuations under time reversal implies that <math>\langle x_i(t)x_k(0)\rangle=\langle x_i(-t)x_k(0)\rangle = \langle x_i(0)x_k(t)\rangle </math>

or, with <math>\xi_i(t)</math>, we have <math>\langle\xi_i(t)x_k\rangle=\langle x_i\xi_k(t)\rangle</math>

Differentiating with respect to <math>t</math> and substituting, we get <math>\gamma_{il}\langle\Xi_l(t)x_k\rangle=\gamma_{kl}\langle x_i\Xi_l(t)\rangle</math>

Putting <math>t=0</math> in the above equation, <math>\gamma_{il}\langle X_lx_k\rangle=\gamma_{kl}\langle X_lx_i\rangle</math>

It can be easily shown from the definition that <math>\langle X_ix_k\rangle=\delta_{ik}</math>, and hence, we have the required result.

==See also==
* [[Lars Onsager]]
* [[Langevin equation]]

==References==
<references/>

{{DEFAULTSORT:Onsager Reciprocal Relations}}
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