I am trying to write the formulation for fluctuation in diffusion flux (Fick's law):

$$ \vec{j}= - \rho D\vec{\nabla} c $$

Then I describe fluctuation in concentration and density as the following:

\begin{align} c ( \vec{x},t ) &= \langle c ( \vec{x},t ) \rangle + \delta c ( \vec{x},t )\\[.15cm] \rho ( \vec{x},t ) &= \langle \rho ( \vec{x},t ) \rangle + \delta \rho ( \vec{x},t )\\ &= \langle \rho ( \vec{x},t ) \rangle \left[ 1+ \beta \delta c ( \vec{x},t ) \right] \end{align}

and substitute them in diffusion equation: \begin{align} \vec{j}+ \delta \vec{j} =&- \langle \rho \rangle \left[ 1+ \beta \delta c \right] D\vec{\nabla } \left( \langle c \rangle + \delta c \right)\\ =&- \langle \rho \rangle D\vec{\nabla } \langle c \rangle\\ &- \langle \rho \rangle D\vec{\nabla } \langle \delta c \rangle\\ &- \langle \rho \rangle D \beta \delta c\vec{\nabla } \langle c \rangle\\ &- \langle \rho \rangle D \beta \delta c\vec{\nabla } \langle \delta c \rangle \end{align}

while ignoring the last term (is it a correct assumption?) I use the gradient operation on it : \begin{align} \vec{{\nabla }}\cdot\left(\vec{j}+\delta\vec{j}\right) =&-\left\langle \rho\right\rangle D{\nabla }^2\left\langle c\right\rangle\\ & -\left\langle \rho\right\rangle D{\nabla }^2\left\langle \delta c \right\rangle \\ &-\left\langle \rho\right\rangle D\beta\vec{{\nabla }}(\delta c)\cdot\vec{\nabla }\left\langle c\right\rangle\\ & -\left\langle \rho\right\rangle D\beta\delta c{\nabla }^2\left\langle c\right\rangle \end{align}

If we substitute $ \vec{\nabla }\cdot \vec{j} =- \rho D\nabla^{2}c $ , then we have:

\begin{align} \vec{{\nabla }}\cdot\delta \vec{j} =& -\left\langle \rho \right\rangle D\nabla^2\left\langle \delta c\right\rangle\\ &-\left\langle \rho\right\rangle D\beta\vec{{\nabla }}(\delta c) \cdot\vec\nabla\left\langle c\right\rangle\\ &-\left\langle \rho\right\rangle D\beta\delta c\nabla^2\left\langle c\right\rangle \end{align}

Does the end result seem correct?

  • $\begingroup$ This seems correct to me, but please be careful about formatting as your notations are not consistent across the post. $\endgroup$ – QuantumApple May 12 '20 at 11:46
  • $\begingroup$ I'm not very familiar with Latex, the misused notations are the result of that probably. Thanks for looking at it, do you know of some reference book, paper... that I can find same equation derivation? $\endgroup$ – mojijoon May 12 '20 at 13:58
  • $\begingroup$ I don't remember studying something like this personally, but this has probably been done before. Nonetheless the calculation seems consistent with the initial assumptions if you neglect the second order term as you did. $\endgroup$ – QuantumApple May 12 '20 at 14:34

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