# Units of the PDF in the Lattice Boltzmann Method

In the Lattice Boltzmann method we require (based on mass conservation) that the sum of the distribution functions for a node is equal to the density, i.e.

$$\sum_i f_i = \rho$$

But what units do $f_i$ have that make their sum equal to the density, which has units of $kg/m^3$?

I guess it means that $f_i$ must have same units as the density, but how can I interpret $f_i$ in that case?

• By the definition you write down, the unit must be the same as the density – unsym Dec 18 '13 at 20:39

Suppose you want to use real measurements in this equation, then the density is actually the mass $m_0$ contained in a mesh unit volume (or voxel) divided by the volume of the voxel, $a^3$ such that $\rho=m_0/a^3$. As far as I know in the lattice Boltzmann method there is a finite number of velocities $\vec c_i$ and $f_i$ is the mass $m_i$ of matter moving with velocity $\vec c_i$ divided by $a^3$, $f_i=m_i/a^3$. As $\sum_im_i=m_0$, this explains why $\sum_f_i=\rho$. I hope this answers the interpretation part of your question.
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