The impact of non-Fickian diffusion on entropy production in a simple model

Recent theoretical work has suggested that the standard model of Fickian diffusion is not appropriate for inhomogeneous systems [B. Ph. van Milligen, {\it et al.}, Eur. J. Phys. {\bf 26}, 913 (2005)]. As an alternative, van Milligen {\it et al.} suggested a Fokker-Planck diffusivity law. The flux from Fick's law is given by $\Gamma \left( x,t \right) = - D\left( x,t \right) \partial n\left( x,t \right) /\partial x$ while the flux from the Fokker-Planck diffusivity law is $\Gamma \left( x,t \right) = - \partial \left[ D\left( x,t \right) n \left( x ,t\right)\right]/\partial x$. In this work, a simple model is used to analyze the effect of the two different diffusivity laws on the production of entropy. Three cases are considered: (1) the spatial dependence of the diffusivity is due solely to a density-dependent diffusivity, $D = D_0 n^\alpha$; (2) an arbitrary spatial dependence in the diffusivity, $D=D\left( x \right)$; and (3) a coupled density and temperature model with both the diffusivity and the conductivity as functions of the density and temperature, $D = D_0 n^{\alpha 1} T^{\alpha 2}$ and $\chi = \chi_0 n^{\alpha 3} T^{\alpha 4}$. Analytic and numerical results for each of these cases will be presented with a focus on the transport and production of entropy.