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Consider sigma d epsilon. A terms consist of something acting as a potential and a differential of an extensive quantity
Consider Equation 2.6 in Kinetics of Materials.
Use the continuity equations and the first law to derive an expression that relates entropy production to local fluxes at a point in a system.
Consider Equation 2.15 in Kinetics of Materials
Entropy production rate is related to the flux of heat.
Units are in terms of energy density per time
Determine how fast the energy density is changing.
Energy density change is termed dissipation
Units of energy density dissipation is J m -3 s-1
There is an unstated assumption that temperature does not vary with time
Nonuniform may refer to variations in space while not constant refers to changes in time
Imagine a system with spatial variations of temperature
Consider heat transfer and maintaining temperature. Find temperature at any point. The system is not in equilibrium but is changing with time. There is heat constantly added to one end and extracted to maintain a temperature gradient.
Terms on the right involve fluxes dotted with a gradient
T sigma dot = - JQ / T grad T - sum J grad Psi
Flux of chemical component, heat, etc. involves a pairing of fluxes and forces. There are gradients of some potential
There is a postulate that entropy production is positive
Entropy Production in Heat Flow
Fourier's law relates heat flux
JQ = - K grad T
K is thermal conductivity
If heat is the only thing that is flowing, the entropy production is minus the heat flux dotted with grad T / T
sigma dot = - JQ grad T / T
sigma dot = K (grad T) ^2 / T
The entropy production is postulated to be greater than zero, which means that the thermal conductivity is greater than zero
K > 0
A positive thermal conductivity means that heat flows from a positive body to a negative body.
Fick's Law
J = M c grad mu
when sigma dot is greater than zero, the mobility is always positive
The diffusion coefficient in traditional form of Fick's law
D is either positive or negative
There is a relationship between flux and chemical potential
Must apply to sum of components or each individual component
Linear Irreversible Thermodynamics
Systems create entropy locally due to fluxes of many things
Predict what fluxes are and define kinetic models
There is a relationship between fluxes and driving forces especially with multiple driving forces (applied simultaneously), most generally flux of heat (q), charge (q), components 
JQ = JQ (FQ, Fq, Fi)
Look at systems not far from equilibrium
Fi: components carry energy
Fq: electronic contribution to heat flow