A generalized linear equation for nonlinear diffusion in external fields and nonideal systems
Abstract
Based on macroscopic thermodynamic analysis, we have established a generalized linear theory for nonlinear diffusion in external fields and nonideal systems, which was classically described by the FokkerPlanck equation (or the Smoluchowski equation) and the nonlinear Fickian equation, respectively. The new theory includes three basic equations expressed in 'apparent variables' as defined in this paper: (i) a generalized linear flux equation for nonlinear diffusion; (ii) an apparent mass conservation equation and (iii) a generalized linear nonsteady state equation for nonlinear diffusion. Our analysis shows that (i) all of the existing linear and nonlinear equations are the special cases of the new nonsteady state general diffusion equation. It was also demonstrated that the general equation of the nonsteady state is equivalent to the FokkerPlanck equation; (ii) coupling diffusion with multiple driving forces can be unified to a single force: the apparent concentration gradient; (iii) the exact relationship between diffusion coefficient and concentration in the nonlinear Fickian equation under nonideal conditions could be established and (iv) the potential energy is conservative in a diffusion process. An application of the generalized linear equation showed that the solution is simple. For the first time, an analytic solution of the Smoluchowski equation with a timedependent potential in the algebraic form was obtained.
 Publication:

New Journal of Physics
 Pub Date:
 October 2007
 DOI:
 10.1088/13672630/9/10/357
 Bibcode:
 2007NJPh....9..357L