where the third term includes the particle acceleration due to the Lorentz force and the Fokker–Planck term at the right-hand side represents the effects of particle collisions. The quantities ⟨ ⟨ Δ Δ v i ⟩ ⟩ and ⟨ ⟨ Δ Δ v i Δ Δ v j ⟩ ⟩ are the average change in velocity a particle of type s experiences due to collisions with all other particle species in unit time. Expressions for these quantities are given elsewhere. If collisions are ignored, the Boltzmann equation reduces to the Vlasov equation.