Rather than considering the densities of the two fluids, 
and 
,
the total density, 
, and the density difference or
order parameter,
, are considered.
Two distribution functions
and 
are then used to describe the population of
and 
respectively
on each of the i links. The evolution
of both distribution functions are governed by the single relaxation time
Boltzmann equations:
and
The equilibrium distributions 
and
take the general
form
and
The coefficients 
and
are chosen to satisfy the following equations.
The equilibrium distribution must satisfy the conservation of mass and
momentum equations:
while the higher moments of and are
defined
so that the resulting continuum equations have the correct form for a
binary fluid [9, 64].
Here 
is the mobility and 
is the chemical-potential difference between the two components.
The pressure tensor 
and the chemical potential
determine the thermodynamical properties of the model and
are determined by the free energy [9, 38, 14, 65].
Orlandini et al.
[9] choose the free energy describing the
simplest possible binary fluid; two ideal gases with a repulsive
interaction energy. This corresponds to a free energy [9, 38]
where T is the temperature,
is the interfacial energy
and 
measures the strength of the interaction. For 
the mixture separates into two phases.
With this definition of the free energy the chemical potential and the
pressure tensor are given by [9, 38, 14, 65]
and
were
