Why Does This Seemingly Nonsensical Argument Hold?

[aaaa202]

I've sometimes seen this argument being used:

The amount of vectors with a given velocity is propotional to the area of the sphere given by:
4πv², because there are more vectors corresponding to bigger speeds.

But mathematically this is nonsense to me, pretty much like comparing infinities. There are an infinite amount of vectors corresponding to any speed apart from zero speaking strictly mathematical.

So why is that on a deeper level makes this argument of "nonsense" hold?

 

[Philip Wood]

It's the shell volume 4$\pi v^2 dv$ which is larger. If we imagine different vectors v, distributed as a uniform fine lattice of points in velocity-space, then the number of points in the shell will be proportional to the shell volume.

I realize this 'answer' raises other issues, but I hope it is of some help.

 

[aaaa202]

Yes exactly, and it is probably these other "questions" that I think about. Is it something quantum mechanical?

 

[Philip Wood]

Yes. Boltzmann (working before quantum theory) did effectively use a lattice of points, but it was arbitrary. How brilliant! Now we can justify the lattice quantum mechanically. In a crude treatment the molecules are matter waves of wavelength related to particle velocity by de Broglie's relation,
$$mv=\frac{h}{\lambda}.$$ The wavelengths, $\lambda$, are fixed by boundary conditions for standing waves in a box. The lattice of points in velocity space emerges very simply from this.

 

[PJC01]

I understand the frustration of encountering seemingly nonsensical arguments. However, it is important to approach these situations with an open mind and a critical eye. In this case, the argument may seem nonsensical because it is not being presented in a clear and logical manner.

Firstly, it is important to note that the statement "the amount of vectors with a given velocity is proportional to the area of the sphere given by 4πv2" is not entirely accurate. It should be clarified that this statement is referring to the number of vectors with a given velocity that are contained within a certain area of the sphere. This is a subtle but important distinction.

Secondly, it is true that there are an infinite number of vectors corresponding to any speed apart from zero. However, this does not mean that all of these vectors have the same magnitude or direction. In fact, there are an infinite number of possible combinations of magnitude and direction for a given speed, which can be represented by the points on a sphere.

Now, let's consider the statement that there are more vectors corresponding to bigger speeds. This is true, as the magnitude of the vector increases, the area of the sphere also increases. This can be seen by the equation 4πv2, where the v2 term represents the magnitude of the vector.

So, while the argument may seem nonsensical at first glance, it is actually based on mathematical principles and can be explained and understood on a deeper level. It is important to carefully examine and clarify the statements being made and to consider the underlying principles and mathematical relationships involved.