Editing Fick's laws of diffusion (section)

=== Generalizations ===
* In ''non-homogeneous media'', the diffusion coefficient varies in space, {{math|1=''D'' = ''D''(''x'')}}. This dependence does not affect Fick's first law but the second law changes: <math display="block">\frac{\partial \varphi(x,t)}{\partial t}=\nabla\cdot \bigl(D(x) \nabla \varphi(x,t)\bigr)=D(x) \Delta \varphi(x,t)+\sum_{i=1}^3 \frac{\partial D(x)}{\partial x_i} \frac{\partial \varphi(x,t)}{\partial x_i}. </math>
* In ''[[anisotropic]] media'', the diffusion coefficient depends on the direction. It is a symmetric [[tensor]] {{math|1=''D<sub>ji</sub>'' = ''D<sub>ij</sub>''}}. Fick's first law changes to <math display="block">J=-D \nabla \varphi ,</math> it is the product of a tensor and a vector: <math display="block"> J_i=-\sum_{j=1}^3 D_{ij} \frac{\partial \varphi}{\partial x_j}.</math> For the diffusion equation this formula gives <math display="block">\frac{\partial \varphi(x,t)}{\partial t}=\nabla\cdot \bigl(D \nabla \varphi(x,t)\bigr)=\sum_{i=1}^3\sum_{j=1}^3D_{ij} \frac{\partial^2 \varphi(x,t)}{\partial x_i \partial x_j}. </math> The symmetric matrix of diffusion coefficients {{math|''D<sub>ij</sub>''}} should be [[Positive-definite matrix|positive definite]]. It is needed to make the right-hand side operator [[Elliptic operator|elliptic]].
* For ''inhomogeneous anisotropic media'' these two forms of the diffusion equation should be combined in <math display="block">\frac{\partial \varphi(x,t)}{\partial t}=\nabla\cdot \bigl(D(x) \nabla \varphi(x,t)\bigr)=\sum_{i,j=1}^3\left(D_{ij}(x) \frac{\partial^2 \varphi(x,t)}{\partial x_i \partial x_j}+ \frac{\partial D_{ij}(x)}{\partial x_i }  \frac{\partial \varphi(x,t)}{\partial x_i}\right). </math>
* The approach based on [[Diffusion#Einstein's mobility and Teorell formula|Einstein's mobility and Teorell formula]] gives the following generalization of Fick's equation for the ''multicomponent diffusion'' of the perfect components: <math display="block">\frac{\partial \varphi_i}{\partial t} = \sum_j \nabla\cdot\left(D_{ij} \frac{\varphi_i}{\varphi_j} \nabla \, \varphi_j\right) ,</math> where {{mvar|φ<sub>i</sub>}} are concentrations of the components and {{mvar|D<sub>ij</sub>}} is the matrix of coefficients. Here, indices {{mvar|i}} and {{mvar|j}} are related to the various components and not to the space coordinates.

The [[Diffusion#The theory of diffusion in gases based on Boltzmann's equation|Chapman–Enskog formulae for diffusion in gases]] include exactly the same terms. These physical models of diffusion are different from the test models {{math|1=∂<sub>''t''</sub>''φ<sub>i</sub>'' = Σ<sub>''j''</sub> ''D<sub>ij</sub>'' Δ''φ<sub>j</sub>''}} which are valid for very small deviations from the uniform equilibrium. Earlier, such terms were introduced in the [[Maxwell–Stefan diffusion]] equation.

For anisotropic multicomponent diffusion coefficients one needs a rank-four tensor, for example {{math|''D''<sub>''ij'',''αβ''</sub>}}, where {{math|''i'', ''j''}} refer to the components and {{math|1=''α'', ''β'' = 1, 2, 3}} correspond to the space coordinates.