Instead of collision-based kinetic description for interaction of charged particles in plasma, Vlasov utilizes a self-consistent collective field created by the charged plasma particles. Such a description uses distribution functions f e ( r , p , t ) and f i ( r , p , t ) for electrons and (positive) plasma ions. The distribution function f α α ( r , p , t ) for species α describes the number of particles of the species α having approximately the momentum p near the position r at time t. Instead of the Boltzmann equation, the following system of equations was proposed for description of charged components of plasma (electrons and positive ions): ∂ ∂ f e ∂ ∂ t + v e ⋅ ⋅ ∇ ∇ f e − − e ( E + v e c × × B ) ⋅ ⋅ ∂ ∂ f e ∂ ∂ p = 0 ∂ ∂ f i ∂ ∂ t + v i ⋅ ⋅ ∇ ∇ f i + Z i e ( E + v i c × × B ) ⋅ ⋅ ∂ ∂ f i ∂ ∂ p = 0 ∇ ∇ × × B = 4 π π j c + 1 c ∂ ∂ E ∂ ∂ t ∇ ∇ × × E = − − 1 c ∂ ∂ B ∂ ∂ t ∇ ∇ ⋅ ⋅ E = 4 π π ρ ρ ∇ ∇ ⋅ ⋅ B = 0 ρ ρ = e ∫ ∫ ( Z i f i − − f e ) d 3 p , j = e ∫ ∫ ( Z i f i v i − − f e v e ) d 3 p , v α α = p m α α ( 1 + p 2 ( m α α c ) 2 ) 1 / 2