Is it true that in Lattice Boltzmann method particles have only one degree of freedom even in 3D case? Can someone explain that fact or provide a link?
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I think that your source is talking about the fact that the lattice Boltzmann methods is numerically solving a scalar equation. And this scalar equation (the Boltzmann equation) implies the mass, momentum and energy conservation of a perfect fluid. (See "The Boltzmann Equation and its applications", Cercignani, 1988)
But particles have many degree of freedom according to the number of dimension you are working on. However, in the lattice Boltzmann method, the particles themselves are not track, only their statistical distribution.