# 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 general rule, unit-wise, is that for continuity-like equations the flux is just the density in question (e.g., mass, energy, momentum, etc.) times the velocity. Another way of thinking about it is that the flux is a rate of transfer of some density. – honeste_vivere Dec 5 '18 at 17:19
• @honeste_vivere yes, this captured by the second relation presented – Aakash Lakshmanan Dec 5 '18 at 17:22
• en.wikipedia.org/wiki/Flux – honeste_vivere Dec 5 '18 at 17:30

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.

• Okay so the second is basically a definition in a formal sense, the first is the assertion of conservation, and the third is an approximation given various assumptions. Would you know how to prove the third from the second I.e. do you know the assumptions and deduction? – Aakash Lakshmanan Dec 5 '18 at 18:11
• @AakashLakshmanan the derivation of (3) is hopefully contained wherever you found it? If I recall correctly it is what one derives if you assume every particle travels exactly a distance $\lambda$ rather than just a mean distance $\lambda$. (2) is not a definition I think but rather a statement of what it means to be ballistically transported. It could be shown in the same way that you might have been shown how to express $d\vec{\phi}$ in terms of $n,F$ etc. – jacob1729 Dec 5 '18 at 18:16
• Sure, I just mean definition in that this is the mathematical abstraction that matches our intuitive notion of what flux really is. Also, the derivation of 3 In the book i found was very lousy but it seems almost essential to reach the transport equations so was hoping for a nicer deduction. – Aakash Lakshmanan Dec 5 '18 at 18:18
• @AakashLakshmanan I'm afraid I don't have a reference to hand. I think the standard text for this sort of stuff is Chapman and Cowling though. – jacob1729 Dec 5 '18 at 18:30