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I've recently been reviewing some concepts, including diffusion. Fick's 1st law:

$$J = -D\frac{\partial C(x,t)}{\partial x}$$

as I understand it, applies to the steady state.

For 1-D diffusion and in the steady state, why is the expression not

$$J = -D\frac{\mathrm dc(x)}{\mathrm dx}$$

s there is only one spatial variable and no time dependence (steady state)?

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  • $\begingroup$ Fick's law also applies to non-steady states. The only requirement is that the $t$ and $x$ dependence is slowly varying compared to collisional time and length scales. $\endgroup$
    – Thomas
    Commented Feb 3, 2016 at 2:55
  • $\begingroup$ I'm not sure I understand your comment. I thought Fick's second law applies to the unsteady state? $\endgroup$
    – user105959
    Commented Feb 4, 2016 at 15:24

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The distinction between Fick's first and second law is mostly historical (it is not a distinction you will find in any reasonably modern treatment of non-equilibrium physics). Fick's law is $$ \vec{\jmath}= -D\vec{\nabla}c(\vec{x},t). $$ Then Fick's first law is the trivial special case $c(\vec{x})$, and Fick's second law is Fick's law combined with current conservation $$ \partial_0 c+\vec{\nabla}\cdot\vec{\jmath}=0. $$ As to why we think Fick's law is correct (for general, but smooth, non-equilibrium states), see, for example, here: Rigorous derivation of Fick's first law .

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  • $\begingroup$ OK. I have never seen this form of Fick's Law before. I should mention that my background is in materials science and electrochemistry, where it is still common to treat Fick's 1st and 2nd laws separately (hence my comment). Thank you for the clarification $\endgroup$
    – user105959
    Commented Feb 4, 2016 at 21:04

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