INSTITUTE OF PHYSICS PUBLISHING EUROPEAN JOURNAL OF PHYSICS Eur. J. Phys. 26 (2005) 913–925 doi:10.1088/0143-0807/26/5/023 On the applicability of Fick’s law to diffusion in inhomogeneous systems B Ph van Milligen 1 , P D Bons 2 , B A Carreras 3 and R S ´ anchez 4 1 Asociaci´ on EURATOM-CIEMAT para Fusi´ on, Avda Complutense 22, 28040 Madrid, Spain 2 Institut f ¨ ur Geowissenschaften, Eberhard Karls Universit¨ at T¨ ubingen, Sigwartstrasse 10, D-72076 T¨ ubingen, Germany 3 Fusion Energy Division, Oak Ridge National Laboratory, PO Box 2001, Oak Ridge, TN 37831-2001, USA 4 Departamento de F´ ısica, Universidad Carlos III, Avda de la Universidad 30, 28911 Legan´ es, Spain Received 15 April 2005 Published 29 July 2005 Online at stacks.iop.org/EJP/26/913 Abstract Two alternative expressions exist for the diffusive flux in inhomogeneous systems: Fick’s law and the Fokker–Planck law. Here we re-examine the origin of these expressions and perform numerical and physical experiments to shed light on this duality. We conclude that in general the Fokker–Planck expression should be conceded preference, in spite of the fact that Fick’s law seems to be more popular. (Some figures in this article are in colour only in the electronic version) 1. Introduction Diffusion is one of the most fundamental phenomena in physics. The modelling of (particle) diffusion is normally based on two assumptions. The first is the conservation of the number of particles: ∂n(x,t) ∂t =− ∂Ŵ(x,t) ∂x , (1) where n(x,t) is the particle density and Ŵ(x,t) is the particle flux through the point x. Here and in the following, for simplicity we write all equations for one-dimensional systems, the generalization of the above and following equations to three dimensions being evident. The second assumption is a relation between the flux, Ŵ, and the density. If one has a full theory for the particle motion (kinetic or microscopic), one may derive Ŵ directly from the particle dynamics. However, if a theory is lacking or incomplete, it is customary to apply a phenomenological relation. The most common assumption is Fick’s law [1]: Ŵ(x,t) =−D ∂n(x,t) ∂x , (2) 0143-0807/05/050913+13$30.00 c  2005 IOP Publishing Ltd Printed in the UK 913