How does one choose the moments used in MRT relaxation within the Lattice Boltzmann method?

Question

I am trying to understand MRT relaxation in the lattice-Boltzmann method. The central equation in the lattice-Boltzmann method to simulate fluid flows is:

$$f_i(x + c_i\Delta t, t + \Delta t) = f_i(x,t) + \Omega_i(x,t)$$

where, $f_i$ are the components of the distribution function and $\Omega_i$ the collision operator. Common collision operators are the BGK (or Single Relaxation Time, SRT) operator, which relaxes the distribution functions directly at the same rate:

$$\Omega_i = -\Delta t \ \frac{f_i-f_i^{eq}}{\tau}$$

and the more complex MRT operator which relaxes moments of the distribution functions (at potentially different rates):

$$\Omega_i = M^{-1}S \ [Mf_i - Mf_i^{eq}]$$

with $S$, the diagonal matrix of relaxation rates. My question is how the matrix $M$ is chosen, i.e. how do we define the moments $m = Mf$, which we will relax. I read that, in general the rows of $M$ need to be orthogonal to guarantee that each moment can be relaxed independently. Can someone explain why this is the case?

If it helps, I am using the book "The Lattice Boltzmann Method: Principles and Practice" by Krüger et al., where they make this claim.

Answer 1

Those are choosen to represent physical statistical moments associated with the hydrodynamic equations, as moments of 0 order, density $\rho$, 1 order, velocities $(u_x,...)$, 2 order, stress tensor $(u_xu_y,...)$ and so on, them from the collision matrix you can set each of those moments a different relaxation frequency.