Variational Interpretation of the Diffusion Equation

The rate of entropy production, (Eq. 2.19), for one-dimensional diffusion becomes 

For an adiabatic system, the rate of total entropy production 5tot is a functional of the concentration field c(x), 

The functional gradient of 5tot indicates the function pointing in the direction of fastest increase. Gradients depend on an inner product because it provides a measure of distance for functions [2]. One choice of an inner product for functions is the L2 inner product, defined by 

The functional gradient of F (or gradient of a vector function) can be defined by Gp, and the inner product with a velocity field v  

That is, of all possible functions v(x), those that are parallel, subject to choice of norm or inner product, to Gjr give the fastest increase in F. For the entropy production with D = constant, 

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