Only five collisional invariants, why? and more

[lalbatros]

Hello,

I did not care much for that point, believing it was an easy matter.
Now I realize that I simply don't know and can't prove why there are only five collisional invariants.
This point is crucial to derive the Maxwell distribution from the Boltzmann equation.
For sure this must have been well known 160 years ago, yet I don't have any clue.

Could you relief me from this bad feeling?

Thanks,

Michel

PS (more):
In addition I wonder if there is any relation between being a "collisional invariant" and being an additive constant of motion.
Being an additive constant of motion is the key point to derive the canonical ensemble.

 

[eljose79]

Dear Michel,

Thank you for reaching out and expressing your concerns. It is completely understandable to feel frustrated when faced with a concept that may seem simple but is actually quite complex. Let me try to alleviate your worries and provide some clarification on the topic of collisional invariants and their relation to the Maxwell distribution and the Boltzmann equation.

Firstly, let's define what a collisional invariant is. In simple terms, a collisional invariant is a physical quantity that remains constant during collisions between particles. In other words, it is a property that is conserved during interactions between particles. This concept is crucial in understanding the behavior of particles in a gas and is a fundamental principle in the derivation of the Maxwell distribution and the Boltzmann equation.

Now, to answer your question about why there are only five collisional invariants, it is important to note that this number is not a limitation, but rather a result of the conditions and assumptions made in the derivation process. The five collisional invariants are a result of considering a gas in a thermodynamic equilibrium state, where the particles are assumed to be in random motion and have no preferential direction of movement. These assumptions are necessary in order to simplify the calculations and make the derivation process manageable. However, in more complex systems, there may be more than five collisional invariants that need to be considered.

Furthermore, the relation between being a collisional invariant and being an additive constant of motion is closely connected. In fact, a collisional invariant can also be referred to as a constant of motion. This is because the quantity remains constant during collisions, which are essentially changes in the state of motion of the particles. This concept is fundamental in the derivation of the canonical ensemble, as you have correctly pointed out.

In conclusion, please do not feel discouraged or overwhelmed by the concept of collisional invariants. It is a complex topic, but with patience and further study, you will gain a better understanding and be able to apply it in your research. I hope this explanation has helped to alleviate your concerns and I wish you all the best in your scientific endeavors.