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:

$${\displaystyle \mathrm {d} U=T\,\mathrm {d} S-P\,\mathrm {d} V+\mu \,\mathrm {d} M}$$

where U is the internal energy, T is temperature, S is entropy, P is the hydrostatic pressure, V is the volume, ${\displaystyle \mu }$ is the chemical potential, and M mass. In terms of the internal energy density, u, entropy density s, and mass density ${\displaystyle \rho }$, the fundamental equation at fixed volume is written:

$${\displaystyle \mathrm {d} u=T\,\mathrm {d} s+\mu \,\mathrm {d} \rho }$$

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:

$${\displaystyle \mathrm {d} s={\frac {1}{T}}\,\mathrm {d} u+{\frac {-\mu }{T}}\,\mathrm {d} \rho }$$

The above expression of the first law in terms of entropy change defines the entropic conjugate variables of ${\displaystyle u}$ and ${\displaystyle \rho }$, which are ${\displaystyle 1/T}$ and ${\displaystyle -\mu /T}$ 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.

The continuity equations

The conservation of mass is expressed locally by the fact that the flow of mass density ${\displaystyle \rho }$ satisfies the continuity equation:

$${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} _{\rho }=0,}$$

where ${\displaystyle \mathbf {J} _{\rho }}$ 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:

$${\displaystyle {\frac {\partial u}{\partial t}}+\nabla \cdot \mathbf {J} _{u}=0,}$$

where ${\displaystyle u}$ is the internal energy density and ${\displaystyle \mathbf {J} _{u}}$ 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 ${\displaystyle s}$ as

$${\displaystyle {\frac {\partial s}{\partial t}}+\nabla \cdot \mathbf {J} _{s}={\frac {\partial s_{c}}{\partial t}}}$$

where ${\textstyle {\partial s_{c}}/{\partial t}}$ is the rate of increase in entropy density due to the irreversible processes of equilibration occurring in the fluid and ${\displaystyle \mathbf {J} _{s}}$ is the entropy flux.

The phenomenological equations

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

$${\displaystyle \mathbf {J} _{u}=-k\,\nabla T;}$$

where ${\displaystyle k}$ is the thermal conductivity. However, this law is just a linear approximation, and holds only for the case where ${\displaystyle \nabla T\ll T}$, 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:

$${\displaystyle \mathbf {J} _{u}=kT^{2}\nabla {\frac {1}{T}};}$$

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

$${\displaystyle \mathbf {J} _{\rho }=-D\,\nabla \rho ,}$$

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:

$${\displaystyle \mathbf {J} _{\rho }=D'\,\nabla {\frac {-\mu }{T}}}$$

where, again, ${\displaystyle D'}$ 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:

$${\displaystyle \mathbf {J} _{u}=L_{uu}\,\nabla {\frac {1}{T}}+L_{u\rho }\,\nabla {\frac {-\mu }{T}}}$$

$${\displaystyle \mathbf {J} _{\rho }=L_{\rho u}\,\nabla {\frac {1}{T}}+L_{\rho \rho }\,\nabla {\frac {-\mu }{T}}}$$

or, more concisely,

$${\displaystyle \mathbf {J} _{\alpha }=\sum _{\beta }L_{\alpha \beta }\,\nabla f_{\beta }}$$

where the entropic "thermodynamic forces" conjugate to the "displacements" ${\displaystyle u}$ and ${\displaystyle \rho }$ are ${\textstyle \nabla f_{u}=\nabla {\frac {1}{T}}}$ and ${\textstyle \nabla f_{\rho }=\nabla {\frac {-\mu }{T}}}$ and ${\displaystyle L_{\alpha \beta }}$ is the Onsager matrix of transport coefficients.

The rate of entropy production

From the fundamental equation, it follows that:

$${\displaystyle {\frac {\partial s}{\partial t}}={\frac {1}{T}}{\frac {\partial u}{\partial t}}+{\frac {-\mu }{T}}{\frac {\partial \rho }{\partial t}}}$$

and

$${\displaystyle \mathbf {J} _{s}={\frac {1}{T}}\mathbf {J} _{u}+{\frac {-\mu }{T}}\mathbf {J} _{\rho }=\sum _{\alpha }\mathbf {J} _{\alpha }f_{\alpha }}$$

Using the continuity equations, the rate of entropy production may now be written:

$${\displaystyle {\frac {\partial s_{c}}{\partial t}}=\mathbf {J} _{u}\cdot \nabla {\frac {1}{T}}+\mathbf {J} _{\rho }\cdot \nabla {\frac {-\mu }{T}}=\sum _{\alpha }\mathbf {J} _{\alpha }\cdot \nabla f_{\alpha }}$$

and, incorporating the phenomenological equations:

$${\displaystyle {\frac {\partial s_{c}}{\partial t}}=\sum _{\alpha }\sum _{\beta }L_{\alpha \beta }(\nabla f_{\alpha })\cdot (\nabla f_{\beta })}$$

It can be seen that, since the entropy production must be non-negative, the Onsager matrix of phenomenological coefficients ${\displaystyle L_{\alpha \beta }}$ is a positive semi-definite matrix.

The Onsager reciprocal relations

Onsager's contribution was to demonstrate that not only is ${\displaystyle L_{\alpha \beta }}$ positive semi-definite, it is also symmetric, except in cases where time-reversal symmetry is broken. In other words, the cross-coefficients ${\displaystyle \ L_{u\rho }}$ and ${\displaystyle \ L_{\rho u}}$ 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 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. 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.

Proof

Define mean values ${\displaystyle \xi _{i}(t)}$ and ${\displaystyle \Xi _{i}(t)}$ of fluctuating quantities ${\displaystyle x_{i}}$ and ${\displaystyle X_{i}}$ respectively such that they take given values ${\displaystyle x_{1},x_{2},\ldots }$ at ${\displaystyle t=0}$. Note that

$${\displaystyle {\dot {\xi }}_{i}(t)=-\gamma _{ik}\Xi _{k}(t).}$$

Symmetry of fluctuations under time reversal implies that

$${\displaystyle \langle x_{i}(t)x_{k}(0)\rangle =\langle x_{i}(-t)x_{k}(0)\rangle =\langle x_{i}(0)x_{k}(t)\rangle .}$$

or, with ${\displaystyle \xi _{i}(t)}$, we have

$${\displaystyle \langle \xi _{i}(t)x_{k}\rangle =\langle x_{i}\xi _{k}(t)\rangle .}$$

Differentiating with respect to ${\displaystyle t}$ and substituting, we get

$${\displaystyle \gamma _{il}\langle \Xi _{l}(t)x_{k}\rangle =\gamma _{kl}\langle x_{i}\Xi _{l}(t)\rangle .}$$

Putting ${\displaystyle t=0}$ in the above equation,

$${\displaystyle \gamma _{il}\langle X_{l}x_{k}\rangle =\gamma _{kl}\langle X_{l}x_{i}\rangle .}$$

It can be easily shown from the definition that ${\displaystyle \langle X_{i}x_{k}\rangle =\delta _{ik}}$, and hence, we have the required result.