I have a question: we have that the density rho is equal to the sum of the populations of the probability mass function, so it must sum up to one by definition. However, rho takes values close to one in a simulation. By example when defining a density gradient, between a inlet and outlet. Because the compressible nature of the model rho has to change…so…the probability mass function in lattice boltzmann is really a probability mass function???This is very confusing…

Probability density functions must indeed to normalised to unity, But f in the Boltzmann equation is the single particle distribution function defined such that the number of particles N in a volume dv centred at position x with velocity between c and dc is given by fdvdc. The density rho that is calculated from f is thus the number density. It’s been a long time since I studied statistics and probability but I suppose the normalised phase-space distribution function (a probability density function) is F=f/n, where n is the configuration space distrubution (or number density, or particle concentration, depending on the application or your preferred terminology). Then integral(f)=1/n*integral(F)=1. n is basically rho (and can depend on position and time).

Thank for your response!. I’m still a little confused because in that case i think that single particle distribution would need to be normalized after a collide-stream step , but this never happens in the lattice boltzmann algorithm.

First of all, my notation in the first post is inconsistent. Or rather, I wrote things the wrong way around at the end end of the message. It should read as follows:

F is the normalised pdf, so integral(F)=1/n*integral(f)=1.

Sorry! Perhaps that clears things up? If not then in answer to your question:

It doesn’t need to be normalised because the LBE, like the Boltzmann equation (BE) in kinetic theory, is a model for f, not F. The BE solves for f and the density, which can vary considerably in this case, is again the zeroth order moment of f. It is certainly true that these sort of questions aren’t addressed in the LB literature, but they are covered extensively in classical kinetic theory, from Boltzmann’s original work to modern text books. If you are interested in such matters then I would recommend the book of Chapman and Cowling, and also ones by Cercignani and the papers by people like Grad.