# PHYS6562 M3 Daily Question

## Weirdness in high dimensions

Most of the volume of high dimensional spheres is near the surface. As we can approximate the volume as $$V \approx r^{3N}$$ in 3N dimensions, near the center, $$r\approx 0$$ and $$V \approx 0$$. Near the surface, where $$r$$ becomes larger and contributes to most of the volume.

For statistical mechanics, instead of taking the whole E sphere into account, which might involve high-dimensional integral, it’s easier to focus on the shell of E sphere. Then the momenta can be considered to be $$p = \sqrt{2mE}$$ instead of doing

$\int_0^{\sqrt{2mE}} (....)dp$