A generalized linear equation for non-linear diffusion in external fields and non-ideal systems

Details A generalized linear equation for non-linear diffusion in external fields and non-ideal systems A generalized linear equation for non-linear diffusion in external fields and non-ideal systems

Oct 2007

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2007njph....9..357l&link_type=abstract

New Journal of Physics, Volume 9, Issue 10, pp. 357 (2007).

Physics

1

Scientific paper

Based on macroscopic thermodynamic analysis, we have established a generalized linear theory for non-linear diffusion in external fields and non-ideal systems, which was classically described by the Fokker Planck equation (or the Smoluchowski equation) and the non-linear 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 non-linear diffusion; (ii) an apparent mass conservation equation and (iii) a generalized linear non-steady state equation for non-linear diffusion. Our analysis shows that (i) all of the existing linear and non-linear equations are the special cases of the new non-steady state general diffusion equation. It was also demonstrated that the general equation of the non-steady state is equivalent to the Fokker Planck 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 non-linear Fickian equation under non-ideal 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 time-dependent potential in the algebraic form was obtained.

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