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I know the steady state for $\frac{\partial f}{\partial t}+\nabla J=0$ (with ICs: $f(x,0)=0$, BCs: $J(0,t)=J(L,t)=0$) is when $\nabla J =0$, i.e. atomic flux divergence should be zero. This equivalent to say the total flux should be constant, i.e.

$$J=constant$$

For my problem, space is 1D and bounded from two ends and $J$ consists of two fluxes: $J=\alpha \nabla f+\beta \nabla g$. Equivalently, it can be seen as two opposite forces fighting against each other and balance in steady state. This is, the fluxes generated by the gradients of $f$ and $g$ have the same magnitude but opposite sign, i.e.: $\alpha \nabla f=-\beta \nabla g$. This leads to:

$$J=0$$

This has been giving me headache for a week that what is the necessary condition for the steady state? If $J=constant$ is the essential condition why in my problem $J\neq 0$ never happens (while it could). Basically, I always get $J=0$ (for any $\alpha, \beta, f$ and $g$).

Ali Abbasinasab
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  • I should mention I've already studied this https://math.stackexchange.com/questions/149283/steady-state-solution-of-diffusion-decay-pde. My question is different. – Ali Abbasinasab Jul 21 '17 at 09:10
  • There must be something to do with the boundary conditions or the fact that my space is bounded by two ends (e.g. x=0 and x=L), but I can't fully get it. – Ali Abbasinasab Jul 21 '17 at 09:16
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    The condition $\nabla\cdot{\bf J}=0$ implies ${\bf J}=\nabla\times{\bf A}$ and then you can get a constant or zero depending on the field ${\bf A}$. – Jon Jul 21 '17 at 09:35
  • @Jon Current BCs: $J(0,t)=J(L,t)=0$ equivalently $\alpha \nabla V=-\beta \nabla \sigma$ but I'm not sure if it is correctly modeling the phenomenon. Any advice, based on the description in my previous comments, is greatly appreciated. – Ali Abbasinasab Jul 22 '17 at 05:14
  • Are you aware that what you wrote implies $\alpha\Delta_2f+\beta\Delta_2g=0$, assuming $\alpha,\ \beta$ constants, in the stationary case? – Jon Jul 22 '17 at 16:56
  • @Jon Unfortunately not. I do not fully get it why boundary conditions (only at x=0 and x=L) force the flux to be zero "everywhere" in steady state. Can you kindly explain why? – Ali Abbasinasab Jul 22 '17 at 22:42
  • In one dimension you can only have $J=constant$ and, with the given boundary conditions, such a constant can only be 0. But yours is a physical problem and a width of the wire should be considered even if is negligible with respect to the length. – Jon Jul 23 '17 at 19:38

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