Spatially non-uniform steady states in concentration-dependent diffusion

Diffusive processes in dense multi-component mixtures should be described with special concentration-dependent terms. Traditional systems of diffusion equations with constant matrices either ignore interaction between the mixture components (when the diffusion matrix is diagonal), or do not guarantee the positivity of concentrations (even if this matrix is kept symmetric and positively defined). A very interesting approach to this problem consists in using the cell-jump formalism. It produces systems of differential equations with concentration-dependent diffusion matrices that yield positive and bounded solutions.

We apply this method to the description of diffusion in the two-component liquid mixture. The obtained model of the binary mixture diffusion possesses an interesting feature: it has an infinite set of spatially non-uniform steady state solutions. We study the stability of these solutions both analytically and numerically. We outline the criterion of stability and check it for several examples.