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.