Chemical Potential of Interacting Particles I'm interested in studying the diffusion of (classical) particles that have some interaction with each other. More specifically, let the potential energy of any two of particles separated by a distance $r$ be given by a function $\Phi(r)$. Let the distribution of these particles be described by a continuous concentration field $c(\mathbf{x})$.
How then can I find the corresponding chemical potential of this system so I can plug it in to the diffusion equation?
$$\frac{\partial c}{\partial t} = \nabla \cdot \bigg(\frac{Dc}{RT}\nabla \mu\bigg)$$
 A: 1) You don't provide quite enough information about what you mean by $c$ and $\mu$. Let me assume that there are two species with densities $n_1$ and $n_2$. I will denote by $c$ the concentration of $n_1$, that is $c=n_1/n_2$ (assuming that $n_1\ll n_2$; we can also consider $c=n_1/n$ with $n=n_1+n_2$). I will use $\mu$ to denote the chemical potential associated with $n_1$, so that $dn_1=Pd\mu$. 
2) Let us denote the susceptibility by $\chi$,
$$ 
\chi=\chi_{11}=\frac{\partial n_1}{\partial\mu}.
$$
Then 
$$
\frac{Dc}{T}\nabla\mu = \frac{Dc}{\chi T}\nabla n_1. 
$$
If $n$ is approximately constant then $\nabla n_1=n\nabla c$. If not, additional terms appear in the diffusion equation (this effect is real). Now
we can define
$$
\bar{D}=\frac{Dn_1}{\chi T}
$$
and get the usual diffusion equation
$$
\partial_0 c - \nabla\left[ \bar{D}\nabla c\right]=0.
$$
3) This whole derivation is frequently side-stepped by defining $\bar{D}$ to be the diffusion coefficient. Also note that in a dilute gas $\chi\sim n_1$, so that, despite appearance, $\bar{D}$ is approximately independent of concentration. There is, however, nothing fundamentally wrong with $\bar{D}=\bar{D}(c)$.
4) If $\Phi(r)$ is the potential between particles of type 1 and 2, then it is possible (in principle) to compute $D$ using kinetic theory. This is not entirely trivial, but explained in standard textbooks on kinetic theory. The results depends on whether you are in the quantum or classical regime. In either case we have to compute the cross section for $1+2\to 1+2$ scattering, and insert the result into the Boltzmann equation. The linearized solution of the Boltzmann equation in the limit of small gradients of $\mu$ determines $D$. 
