# Diffusion in a chemical potential gradient

I am having troubles to understand to connection between equilibrium and non equilibrium thermodynamics.

I am studying a mixture of molecules $A,B,C$ and solvent $S$. The free energy $F$ is given by the functional

$F[\{\phi_i\}_i]=\int \left[ f(\{\phi_i(x)\}_{i})+\sum_i \gamma_i(\nabla\phi_i)^2 \right] dx$

with $\phi_i$ the concentrations, $f$ the free energy density, and $\gamma_i$ the surface tension coefficients. The reversible chemical reaction $A+B\Leftrightarrow_{k_2}^{k_1} C$ takes place.

My goal is to write the equations of motion for the concentrations. In a paper I find for the first concentration $\phi_A$:

$\frac{\partial \phi_A}{\partial t}=-\nabla. \mathbf J_A - k_1 \phi_A\phi_B + k_2 \phi_C$

with $\mathbf J_i=-\sum_j M_{ij}\frac{\delta F}{\delta \phi_j}$

I don't get the first term $-\nabla. \mathbf J_A$. Apparently $J_A$ is vector (bold font) but why? From the definition above I don't see.

I roughly understand this is the contribution to the chemical potential gradient but I wish to understand how to derive it. I don't know where to look and I don't find so far in the non-equilibrium books I checked. I'd be happy if you could explain or direct me toward a book.

• By definition $\mu = \delta F/\delta\phi$, and from its gradient you get the flux (using Onsager's reciprocal relations). Now for diffusive processes, you're going to have to take the divergence of this, like in regular heat/diffusion equation due tot he continuity equation (so I think you're missing one nabla in your equations). For more, Google "time dependent Ginzburg-Landau", and "model B" (if a nondiffusive process if that's what you want instead, search for "model A" which is to grand canonical ensemble what model B is to canonical). – alarge Jan 11 '15 at 19:40
• No nabla is missing, at least this is how it is in the paper. Beside $\mathbf J_A$ is written as a vector, but I don't why from the relations I have written. – David Jan 11 '15 at 19:48
• This, to me, appears to be a typo in the paper; in standard texts, including Wikipedia (en.wikipedia.org/wiki/Onsager_reciprocal_relations), there is a nabla there (because to get the force, you take the gradient of the potential). Indeed, Eq. (4) of the paper has $\nabla^2 = \nabla\cdot\nabla$, so one nabla must've appeared from somewhere. Note that you have a further typo in your equation with the indices (the subscript $i$ on $\phi$ should be $j$). – alarge Jan 11 '15 at 20:07
• Thank you I corrected the last typo. I am also confused with the multicomponent thing. Could you please write this term explicitly? – David Jan 11 '15 at 20:10
• Could you perhaps be more specific than "the multicomponent thing"? If you mean the fact that $M_{ij}$ in general is a matrix, you needn't worry too much and assume it to be diagonal. As to how to derive the equations: As far as I know, they are, like so many things in non-equilibrium, phenomenological, characterizing approximative linear response; There is no rigorous derivation. – alarge Jan 11 '15 at 20:23