The Gibbs rotational ensemble is useful for calculations regarding rotating systems. It is commonly used for describing particle distribution in centrifuges. For example, take a rotating cylinder (height Z , radius R ) with fixed particle number N , fixed volume V , fixed average energy ⟨ ⟨ E ⟩ ⟩ , and average angular momentum ⟨ ⟨ J → → ⟩ ⟩ . The expectation value of number density of particles ⟨ ⟨ n ( r ) ⟩ ⟩ at radius r can be written as