Transport of Charged Particles: Entropy Production and Maximum   Dissipation Principle
  In order to describe the dynamics of crowded ions (charged particles), we use an energetic variation approach to derive a modified Poisson-Nernst-Planck (PNP) system which includes an extra dissipation due to the effective velocity differences between ion species. Such a system is more complicated than the original PNP system but with the same equilibrium states. Using Schauder's fixed-point theorem, we develop a local existence theorem of classical solutions for the modified PNP system. Different dynamics (but same equilibrium states) between the original and modified PNP systems can be represented by numerical simulations using finite element method techniques. 
Chia-Yu Hsieh, YunKyong Hyon, Hijin Lee, Tai-Chia Lin and Chun Liu
math-ph math.AP math.MP