First, the expected value of the microdynamical equation s m ( r + c m , k + 1 ) = s m I ( r , k ) is obtained f m ( r + c m , k + 1 ) = ⟨ s m I ( r , k ) ⟩ where ⟨ ⟨ ⋅ ⋅ ⟩ ⟩ denotes the expected value, and f m ( r , k ) := ⟨ s m ( r , k ) ⟩ is the expected value of the m -th channel occupation number of the lattice site at r at time step k . However, the term on the right, ⟨ s m I ( r , k ) ⟩ is highly nonlinear on the occupation numbers of both the lattice site r and the lattice sites within the interaction neighborhood N , due to the form of the transition probability P ( s → → s I | s N ) and the statistics of particle placement within velocity channels (for example, arising from an exclusion principle imposed on channel occupations). This non-linearity would result in high-order correlations and moments among all channel occupations involved. Instead, a mean-field approximation is usually assumed, wherein all correlations and high order moments are neglected, such that direct particle-particle interactions are substituted by interactions with the respective expected values. In other words, if X 1 , X 2 , … … , X n are random variables, and F : R n ↦ ↦ R is a function, then ⟨ F ( X 1 , X 2 , … … , X n ) ⟩ ≈ ≈ F ( ⟨ X 1 ⟩ , ⟨ X 2 ⟩ , … … , ⟨ X n ⟩ ) under this approximation. Thus, we can simplify the equation to f m ( r + c m , k + 1 ) = C ( f ( r , k ) , f N ( r , k ) ) where C ( f ( r , k ) , f N ( r , k ) ) is a nonlinear function of the expected lattice site configuration f ( r , k ) and the expected neighborhood configuration f N ( r , k ) dependent on the transition probabilities and in-node particle statistics.