Reconciling various expressions of flux and flow in kinetic theory There are a few expressions I have been shown to describe the flux $\vec{\Phi}_A$ of some quantity $A$ described in kinetic theory. I am having trouble understanding how they are related or if they are even correct. 
Assume there is some underlying probability distribution given by the measure $\text{d}F$ and an associated flux measure $\text{d}\vec{\Phi}\propto n\vec{v}\text{d}F$ (properly normalized of course). The following are descriptions I have heard.
Continuity equation: $\vec{\nabla}\cdot\vec{\Phi}_A + \frac{\partial (nA)}{\partial t} = 0$
Something else: $\vec{\Phi}_A=\int A\text{d}\vec{\Phi}$
Something I am unsure of: $\vec{\Phi}_A=-\frac{n\lambda_{mfp}\langle v\rangle}{3}\vec{\nabla} A$
Please let me know if you understand the relationships between the three and/or if you know the right framework or mathematical structure to think about these ideas in. Thanks!
 A: It might help to have an example of each of the three types of statement. So to be concrete, let $F(\vec{v})$ be the probability that a given particle has speed $\vec{v}\in[\vec{v},\vec{v}+d\vec{v}]$ and thus the differential particle flux is given by your expression:
$$ d\vec{\phi}(\vec{v}) = n\vec{v}F(\vec{v})|d\vec{v}| $$
This is some function on position space (as is $F$). Integrating over all positions gives us the total particle flux:
$$\vec{\phi} = \int d\vec{\phi} = \int_{R^3}n\vec{v}F$$
Here the particle number density is $n$ and its transport is entirely captured by $\phi$. We might want to consider other quantities such as $g$ the momentum density. Then it has its own flux $\phi_g$. Actually, momentum is a vector $\vec{g}$ and its flux must be a rank 2 tensor $\hat{\Pi}$ but we can consider a 1D case for simplicity.
For an ideal gas we have that $\vec{g}=nm\vec{v}$ and (in 1D) that $d\phi_g = mvd\phi$. This is because momentum is only transported ballistically by particles carrying their own momentum from place to place. Thus we get expressions like:
$$\phi_g = \int (mv)d\phi $$
Hopefully this clarifies the second of your three relations by giving a specific example. I think we have assumed their is only ballistic transport (on the kinetic level I'm not sure what else there could be. Perhaps transport due to long ranged fields?)
The first equation you give is much more general, and is an expression of continuity for the quantity $A$. It is not an equation for $\phi_A$ but a seperate statement that $A$ is conserved locally. We can use it in conjunction with the third statement to get a diffusion equation for $A$. The third statement is I think just the evaluation of $\phi_A$ within some specific model of the gas - it's worth noting that the one you have presented is the result of a very standard textbook calculation that ignores many subtleties and as such the exact factor of $1/3$ is highly dubious.
