From this nonlinear FDE, one may identify several homogeneous steady states, or constants f ¯ ¯ m independent of r and k which are solutions to the FDE. To study the stability conditions of these steady states and the pattern formation potential of the model, a linear stability analysis can be performed. To do so, the nonlinear FDE is linearized as f m ( r + c m , k + 1 ) = ∑ ∑ j = 1 K ∂ ∂ C ∂ ∂ f j ( r , k ) | s s f j ( r , k ) + ∑ ∑ j = 1 K ∑ ∑ p = 1 K ∂ ∂ C ∂ ∂ f j ( r + c p , k ) | s s f j ( r + c p , k ) where s s denotes the homogeneous steady state f m ( r , k ) = f ¯ ¯ m , m ∈ ∈ { 1 , … … , K } , and a von Neumann neighborhood was assumed. In order to cast it into a more familiar finite difference equation with temporal increments only, a discrete Fourier transform can be applied on both sides of the equation. After applying the shift theorem and isolating the term with a temporal increment on the left, one obtains the lattice-Boltzmann equation f ^ ^ m ( q , k + 1 ) = e − − 2 π π i L q ⋅ ⋅ c m { ∑ ∑ j = 1 K [ ∂ ∂ C ∂ ∂ f j ( r , k ) | s s + ∑ ∑ p = 1 K ∂ ∂ C ∂ ∂ f j ( r + c p , k ) | s s e 2 π π i L q ⋅ ⋅ c p ] f ^ ^ j ( q , k ) } where i = − − 1 is the imaginary unit, L is the size of the lattice along one dimension, q ∈ ∈ { 1 , 2 , … … , L } d is the Fourier wave number, and ⋅ ⋅ ^ ^ = F { ⋅ ⋅ } denotes the discrete Fourier transform. In matrix notation, this equation is simplified to f ^ ^ ( q , k + 1 ) = Γ Γ f ^ ^ ( q , k ) , where the matrix Γ Γ is called the Boltzmann propagator and is defined as Γ Γ m , j = e − − 2 π π i L q ⋅ ⋅ c m [ ∂ ∂ C ∂ ∂ f j ( r , k ) | s s + ∑ ∑ p = 1 K ∂ ∂ C ∂ ∂ f j ( r + c p , k ) | s s e 2 π π i L q ⋅ ⋅ c p ] . The eigenvalues λ λ ( q ) of the Boltzmann propagator dictate the stability properties of the steady state: