The form of the collision operator can be further simplified by considering
the form of 
to depend not on a set of collisions but on
the isotropy of the model and the conservation constraints [41].
Consider first the elements 
which
describe the change in the distribution function
which is induced by a unit change in 
during the collision.
Due to the isotropy of the
model must depend only on the angle between
and 
which,
for a hexagonal grid, must be one of
four angle: 
and 
. Thus there
are only four independent elements in . If `rest-particles' are
also allowed two new independent variables must also be included in
, one to account for the influence of `rest
particles' on themselves ( 
) and one to account for the influence
of `rest particles' on moving `particles' ( 
and 
) [41]. Thus for a hexagonal lattice with `rest
particles'
where 
are the matrix elements linking directions which intersect
at an angle 
and b and c are elements linking `rest particles' to
moving `particles' and to themselves respectively.
The number of independent variables can be further reduced by considering
the conservation of mass (the sum of each column of the
matrix equation (4.8) = 0) and the conservation of momentum [41]
The collision rules described by this greatly reduced collision matrix
are referred to as the enhanced collision rules.
The eigenvectors for the matrix 
can
be calculated. For a two-dimensional hexagonal grid with b=6 there are
three distinct non-zero eigenvalues
The first eigenvalue can then be associated with the viscosity [41, 8] to give
where 
.
Such a model can be shown [41, 42] to satisfy the
Navier-Stokes equation with the additional 
factor and with the
viscosity given by equation (4.11).
