The use of the above lattice Boltzmann equation removes the statistical noise
from the lattice simulations. The collision operator
still depends on
the 
Boolean input and output states, where m=b for a model
with b links and no `rest-particles' and m=b+1 for a model with
rest-particles. For a Boltzmann simulation based on the two-dimensional
FHP-III model
is a 
matrix, for a face-centred-hypercubic
(FCHC) model, used in three-dimensional simulations,
is a 
matrix. Clearly the computational requirements for handling such a matrix can be limiting. The size of the matrix
can be greatly reduced [40] by expanding the distribution function
where 
is the equilibrium value of the distribution function
and 
is the non-equilibrium part. Expanding the
collision operator about the equilibrium distribution gives
where
Thus, since 
, the lattice Boltzmann equation
can be written [40]
Here 
is the linearised collision matrix and is an 
matrix, a considerable reduction from the 
matrix
. The value of the individual elements of still
depends on the form of the lattice gas collision rules.