A Differential Geometry-Based Poisson-Nernst-Planck Model

1 A Differential Geometry-Based Poisson-Nernst-Planck Model 

The GPB and Laplace-Beltrami models discussed in the previous section were obtained from a variational principle applied to equilibrium systems. For chemical and biological systems far from equilibrium, it is necessary to incorporate additional equations (e.g., the Nernst-Planck equation) to describe the dynamics of charged particles. Various DG-based Nernst-Planck equations have derived from mass conservation laws in earlier work by Wei and co-workers [74, 75]. We outline the basic derivation here. For simplicity in derivation, we assume that the flow stream velocity vanishes ( v = 0) and we omit the chemical reactions in our present discussion. 

The chemical potential contribution to the free energy consists a homogeneous reference term and the entropy of mbcing [188]  

We first derive the generalized Poisson equation by the variation of the total free energy functional with respect to the electrostatic potential J). The resulting generalized Poisson equation is 

Using these relations, the relative generalized chemical potential gen jjg rewritten as  

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