In short, I think there are two types of comoving frame when talking about distribution function, since it is defined in phase space. Which one should be the real one? I suppose this question is admittedly hard to understand, so I really appreciate it if you can share your opinions, or just finish reading. Thanks in advance for your time!
I am working on a multi-fluid system, in which background fluid $A$ scatters particle $B$ (in reality $B$ is cosmic-ray, but that's not quite important). Let's describe $A$ by a velocity field $\vec{U}(\vec{x},t)$, and describe $B$ by a distribution function $f(\vec{x},\vec{v},t)$. For simplicity we assume Newtonian mechanics. Now the final goal is to write down a Boltzmann equation that describes the transport process:
$$\frac{\partial f}{\partial t}+\nabla\cdot(f\vec{v})+\nabla_v \cdot(f\vec{a})=\left(\frac{\delta f}{\delta t}\right)_{\rm scatter}$$
Assume we know how to write it down in a frame where $A$ is static, in other words we know how $A$ scatters $B$ when the fluid element $A$ is static relative to the particle $B$. Now we hope to write the equation in lab frame, and the typical operation in the textbook is to introduce a "comoving frame". But after some consideration, I began to think there are two types of comoving frame, because we are talking about a distribution in phase space. And I can't distinguish them in terms of physical essence.
In the following I will show why I think there are two types of comoving frames. I will use the transformation rules of partial time derivative to help me demonstrate my point. Before that let's agree on the notation, $\vec{v}$ is the particle velocity in lab frame, and $\vec{v}'$ is the particle velocity relative to the local fluid element, naturally $\vec{v}=\vec{v}'+\vec{U}$
comoving frame 1
In my textbook, it introduces a comoving frame, where the transformation of time derivative is $$\left.\frac{\partial f}{\partial t}\right|_{\rm lab}\rightarrow\left.\frac{\partial f}{\partial t}\right|_{\rm comoving}-\frac{\partial \vec{U}}{\partial t}\cdot\nabla_v f$$ From this transformation, I think here "comoving" means that, we fix the coordinate rods at a certain spacial point, but at each moment adjust its velocity to equal the local fluid. The frame itself does not really move, despite it has a reference velocity. So this "comoving frame" simply changes our measurements of velocity, but is spacially static. I am rather unfamiliar with this definition of "comoving", and I think it is really different from what we usually mean.
comoving frame 2
This is just what I believe should be the definition of comoving: we move along with the local fluid element. In this case the transformation rule for partial time derivative should be $$\left.\frac{\partial f}{\partial t}\right|_{\rm comoving}\rightarrow\left.\frac{\partial f}{\partial t}\right|_{\rm lab}+\vec{U}\cdot\nabla f+\frac{D\vec{U}}{Dt}\cdot\nabla_v f$$
It is simply the Lagrangian description, slightly different because the change in fluid velocity also changes our measurements.
Now I am a little confused, both types look like reasonable pictures to me, in terms of writing down scattering terms in a frame where fluid $A$ is static. But which one is correct? Of course maybe they are both right, and both work, only if they have the same results when we ultimately transform everything back to the lab frame. I gave it a try but failed to get good results, probably because of my poor math skills. Anyway, even if they are both correct, I still feel I am missing something, about the essential difference between these two types, like what is the consequence if you use one of them instead of the other? What do we mean when we usually say "reference frame"? Does the frame have to move with the reference velocity of the frame? Why in type 1 the frame has a velocity of reference but can be fixed?
This topic is in the book Transport Processes in Space Physics and Astrophysics by Gary P. Zank, section 5.1. By the way this method in the book is called mixed coordinate or mixed frame, I wonder if there is anything to recommend for reading?
