Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusion
should be nonlinear if there exist non-diagonal terms. The vast variety of nonlinear
multicomponent diffusion equations should be ordered and special tools are needed to
provide the systematic construction of the nonlinear diffusion equations for
multicomponent mixtures with significant interaction between components. We develop an
approach to nonlinear multicomponent diffusion based on the idea of the reaction mechanism
borrowed from chemical kinetics.
Chemical kinetics gave rise to very seminal tools for the modeling of processes. This is
the stoichiometric algebra supplemented by the simple kinetic law. The results of this
invention are now applied in many areas of science, from particle physics to sociology. In
our work we extend the area of applications onto nonlinear multicomponent diffusion.
We demonstrate, how the mechanism based approach to multicomponent diffusion can be
included into the general thermodynamic framework, and prove the corresponding dissipation
inequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementary
process cannot have an arbitrary form. For the general kinetic law (the generalized Mass
Action Law), additional conditions are proved. The cell–jump formalism gives an
intuitively clear representation of the elementary transport processes and, at the same
time, produces kinetic finite elements, a tool for numerical simulation.