**Mathematical Background: Types of Fields**

**Fluxes and Accumulation**

**Conserved and Non-conserved Quantities**

**Fundamental Postulate: Entropy Production Density is Non-Negative**

**Assumption of Local Equilibrium**

**Form of the Entropy Production Density**

Conjugate Forces, Fluxes and Empirical Flux Laws for Unconstrained Components | ||||

Quantity | Flux | Conjugate Force | Empirical Flux Law | |

Heat | Fourier's | |||

Mass | Modified^{1}Fick's form |
|||

Charge | Ohm's |

__Entropy Production for Simple Cases__

If heat is the only quantity that is flowing:

(03-1) |

If diffusion is the only operating process:

(03-2) |

In general, the entropy production is the sum of
all operating fluxes dotted into (minus) the gradient
of the associated potential.^{2}

__Generalized Coupling for the Near-Equilibrium Case__

Let represent the generalized driving forces for a system near equilibrium. A system near equilibrium is one where the driving forces are all small, therefore we can expand the fluxes in terms of these small driving forces:

(03-6) |

or,

It is important to remember the origin of the . They are derived as the linear coefficients of driving forces around the equilibrium state-i.e. the case of condition of small driving forces. Remember that if a function, is expanded around a particular point up to linear terms:

(03-8) |

The values of the linear terms are functions of the point about which they
are expanded (
), so in the
expansion in Eq. 3-7, the linear coefficients
are
also functions of the particular equilibrium state about which the
system is expanded. In other words, we should *expect* the
to be functions of temperature, equilibrium chemical potential, pressure, etc.

The entropy production for the near-equilibrium case is given by:

(03-9) |

Because the term on the right hand side must be positive definite and because each term is real, it is necessary that the matrix is symmetric; this is :

(03-10) |

__Example: Thermal and Ionic Conducting Bar__

Consider heat transport in a bar that can conduct both heat and electricity via ionic conductivity:

Suppose there is no electric current (perfect voltmeter), then

(03-12) |

thus for the case of no electric current density,

(03-13) |

Therefore, the heat flux has two identifiable components, one that comes from the electrostatic potential difference and one the comes from temperature difference. The ``kinetic coefficients'' of the flux are related to combinations of the ``direct effect coefficients'' and and the indirect coefficients and . Presumably, experiments could be performed on such a system to verify whether the Onsager symmetry relation applies, i.e. if .

A set of such physical experiments is considered below.

__Seebeck, Peltier Effects and Thomson's Second Relation__

Consider the following experimental set-up:

In the Seebeck a potential difference is set up in response to the flow of heat between two reservoirs.

The thermoelectric power is a relation between the potential difference and the temperature difference:

(03-14) |

The indicates that the potentiometer is ideal. This quantity can be calculated using equations 3-11 using an approximation for the gradients , etc.

(03-15) |

For the Peltier effect, the experimental set up is illustrated by:

The Peltier coefficient is related to the ratio of the heat flux to the electric current:

(03-16) |

Using equations 3-11, the Peltier coefficient can be calculated in terms of the Onsager coefficients:

(03-17) |

If Onsager's symmetry relation holds ( ), then there must be a relation between the Peltier and Seebeck coefficients:

(03-18) |

This relation is called Thomson's second relation and has been repeatedly experimentally verified and this can be considered experimental verification of Onsager's symmetry relation.

__One Independent Mobile Species__

Consider the case of one chemical species that can diffuse independently of all the others, such as an interstitial carbon atom diffusing in BCC iron, or the case where a gaseous species is diffusing through a quiescent gas mixture.

Suppose that the only driving force is the gradient in chemical potential of the interstitial species, then

(03-19) |

The chemical potential can be related to local concentration through the activity coefficient :

(03-20) |

Therefore, can be related to :

For the ideal case, the activity coefficient is independent of concentration, so

(03-21) |

One would expect this relation to hold for very dilute alloys (Henry's law) or self-interstitial diffusion in a very pure alloy (Raoult's law).

For the case of a non-ideal solution:

(03-22) |

If this is compared to the most simple version of Fick's first law, , is called the intrinsic diffusivity and it is related to the Onsager coefficient as:

(03-23) |

The atomic mobility be defined by the the Einstein relation between the average drift velocity and the driving force, .

(03-24) |

Using the above equation, the flux must be related to the average velocity through the relation,

(03-25) |

Therefore, using the Einstein drift velocity,

(03-26) |

and the relation between intrinsic diffusivity and mobility is

(03-27) |

If the solution is ideal--as in the case of mixture of radioisotopes of an otherwise identical atomic species--then the diffusivity is called the self-diffusivity and since the activity coefficient is constant:

(03-28) |