What frame of refernce to select in statistical mechanics?
Suppose we have a solid particle suspended inside a fluid such as an ideal gas, as shown in the following picture:
Our system is the solid particle and the environment is the gas (which acts as a heat bath). Our frame of reference is attached on the edge of the container (shown with black color, ignore the other origin for the moment).
The energy of the solid particle, composed of $N$ atoms, in this frame of reference is (for sake of simplicity we assume we have a monoatomic solid and neglect the potential energy terms):
$$ E = \sum_{i=1}^N \frac{1}{2}mV_{i}^2 = \underbrace{\frac{1}{2}MV_{\mathrm{cm}}^2}{\text{KE of the center of mass}} + \underbrace{\sum{i=1}^N \frac{1}{2}mu_{i}^2}_{\text{KE with respect to center of mass}} $$
Selecting a frame of reference
What is the correct frame of reference to apply statistical mechanics? The center of mass of the system or the one attached on the edge of the container?
If it is the second, then that means that even macroscopic objects such as a rock emerged on a fluid (e.g. sea), they have on average $\frac{3}{2}k_\mathrm{B}T$ energy associated with their center of mass, which means that they move a little bit (because of the very high mass). Is that correct?
I am giving below a gif from Wikipedia which can help visualizing the process. 453
In this gif the yellow "ball" is a dust particle.
Frame of reference and net velocity
Suppose now that we describe our system based on the origin with blue color, which happens to move relatively to the container (and also to the gas). This means that now $\langle V_\mathrm{cm} \rangle \neq 0$. This is not a proper frame of reference to apply statistical mechanics since if the relative velocity is increased, the temperature of the solid particle will increase which doesn't make sense. Is that also correct?
In summary, when we want to describe our system in statistical mechanics, what frame of reference should we use? 453