Editing Onsager reciprocal relations (section)

== 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 display="block">\mathrm{d}U = T \, \mathrm{d}S - P \, \mathrm{d}V + \mu \, \mathrm{d}M</math>
where ''U'' is the internal energy, ''T'' is temperature, ''S'' is entropy, ''P'' is the hydrostatic pressure, ''V'' is the volume, <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 at fixed volume is written:
<math display="block">\mathrm{d}u = T \, \mathrm{d}s + \mu \, \mathrm{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 display="block">\mathrm{d}s = \frac 1 T \, \mathrm{d}u + \frac {-\mu} T \, \mathrm{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 are called thermodynamic forces as they cause flows of the corresponding extensive variables as expressed in the following equations.

=== The continuity equations ===

The conservation of mass is expressed locally by the fact that the flow of mass density <math>\rho</math> satisfies the [[continuity equation]]:
<math display="block">\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J}_\rho = 0,</math>
where <math>\mathbf{J}_\rho</math> is the mass flux vector. The formulation of energy conservation is generally not in the form of a continuity equation because it includes contributions both from the macroscopic mechanical energy of the fluid flow and of the microscopic internal energy. However, if we assume that the macroscopic velocity of the fluid is negligible, we obtain energy conservation in the following form:
<math display="block">\frac{\partial u}{\partial t} + \nabla \cdot \mathbf{J}_u = 0,</math>
where <math>u</math> is the internal energy density and <math>\mathbf{J}_u</math> is the internal energy flux.

Since we are interested in a general imperfect fluid, entropy is locally not conserved and its local evolution can be given in the form of entropy density <math>s</math> as
<math display="block"> \frac{\partial s}{\partial t} + \nabla \cdot \mathbf{J}_s = \frac{\partial s_c}{\partial t}</math>
where <math display="inline">{\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 and <math>\mathbf{J}_s</math> is the entropy flux.

=== The phenomenological equations ===

In the absence of matter flows, [[Fourier's law]] is usually written:
<math display="block">\mathbf{J}_{u} = -k\,\nabla T;</math>
where <math>k</math> 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.{{Dubious|date=January 2022}} Assuming that this is the case, Fourier's law may just as well be written:
<math display="block">\mathbf{J}_u = k T^2 \nabla \frac 1 T;</math>

In the absence of heat flows, [[Fick's law]] of diffusion is usually written:
<math display="block"> \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 display="block"> \mathbf{J}_{\rho} = D'\,\nabla \frac {-\mu} T </math>
where, again, <math>D'</math> 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:
<math display="block"> \mathbf{J}_{u} = L_{uu} \, \nabla \frac 1 T + L_{u\rho} \, \nabla \frac {-\mu} T</math>
<math display="block"> \mathbf{J}_{\rho} = L_{\rho u} \, \nabla \frac 1 T + L_{\rho\rho} \, \nabla \frac{-\mu} T</math>
or, more concisely,
<math display="block"> \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>\rho</math> are <math display="inline">\nabla f_u = \nabla \frac 1 T</math> and <math display="inline">\nabla f_\rho = \nabla \frac {-\mu} T</math> and <math>L_{\alpha \beta}</math> is the Onsager matrix of [[transport coefficient]]s.

=== The rate of entropy production ===

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

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

It can be seen that, since the entropy production must be non-negative, 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 symmetric, except in cases where time-reversal symmetry is broken. 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 is suggested by 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 2×2 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.