Consider a two-dimensional regular lattice where
r is the position vector of any site.
The lattice has b
distinct links 
.
Any particle travelling on link , i=1, ...,b
moves from
one site to a neighbouring site in unit time and so has velocity
.
We label the occupation numbers of the links at a site
r
at time t by 
where
The density and velocity, 
and 
are defined
We also define the collision function
The collision function describes the change in
during a collision at time t at site
.
The mass and momentum must be conserved by the collision function at each site. This can be expressed
The particles on the lattice are updated according to a time evolution operator which can be described as the convolution
where 
describes the collision operator and 
the streaming operator.
Each collision can be considered to be the transition from an in-state 
at time 
just before the collision to
an out-state
at 
just after the collision. We can
assign a probability 
to be the
probability that an in-state s collides to give an out-state 
.
The collision rules are said to satisfy detailed
balance if
and semi-detailed balance if
The FHP models described here satisfy semi-detailed balance but not detailed balance.
