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

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

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