Derivation of mean field equations : Vlasov and nonlinear Schrödinger
Partial differential equations

The derivation of the Boltzmann equation from Hamiltonian mechanics focuses on two-body interactions between molecules.

A mathematically different situation, but important for electromagnetism and plasma problems, is that where each particle is subject to the "average of the actions of the others". This corresponds, in the case of classical mechanics, to the Hamiltonian system, for the usual variables X _N = (x ₁ , x ₂ ... x _N ), V _N = (v ₁ , v ₂ ... v _N ),