First we expand the equilibrium distribution up to second-order in the velocity :
where
The coefficients and need to be found, subject to isotropy and Galilean invariance, in order to obtain the required equilibrium distribution (to accuracy ). The isotropy conditions required on the second- and fourth-order tensors [47, 48] can be explicitly introduced by defining and :
and
The sum of the equilibrium distribution multiplied by an odd number of 's is zero. Considering in equation (4.16) and in equation (4.17) gives
and
Combining these we see that
We now consider the Galilean invariance requirement. If is the jth moment,
The zeroth moment is
Considering terms and using the isotropy condition, equation (4.16), we get
The first moment is
Considering terms O(u) and using the isotropy condition, equation (4.16), gives
The second moment is
Collecting terms of second-order in gives
Considering separately the terms and we get
and
The third moment is
Collecting terms gives
This gives the final condition
Solving equations (4.20), (4.24), (4.26), (4.29), (4.30) and (4.33) gives and C = 8. The value of is found from equation (4.18) and is obtained by considering the sum of the distribution functions
Thus we have the following equilibrium distribution for a two-dimensional lattice Boltzmann model which, by construction, must be isotropic and Galilean invariant.
where