i am a new rookie and i have some questions to consult.
as you konw,the derivation of the macroscopic navier-stokes equation from the run under the chapman-enskog expansion ,in chapman-enskog expansion, the following multi-scale expansion will be introduced
f_i=f_i(0)+ef_i(1)+ee*f_i(2)
t_1=et , t_2=ee*t
x_1=e*x
where e is the expansion parameter which is the ratio between the molecular mean free path and the characteristic length.
i wanna konw
what are the physical meaning of f_i(0), f_i(1) ,f_i(2) and t_1,t_2 ,x_1, respectively.
the populations f_i(0), f_i(1) and so on have the following interpretation:
f_i(0) is the distribution function of the velocities in equilibrium. Equilibrium means that the entire system has the same macroscopic velocity and there is no shear at all. Of course the (microscopic) velocities of the particles are not the same, they obey the truncated Maxwell distribution.
Now, f_i(1), f_i(2) etc. are the corrections to the equilibrium state. In general, a system is not in equilibrium. The LBM assumes that the system is always near equilibrium. This way you can approximate the current state by equilibrium plus deviations of order e, e^2 etc.
There is a lot of literature on this topic. Maybe you have a look at one of the books, e. g. by Succi or Sukop & Thorne. It is also wise to start with a review paper, e. g. Chen, Doolen: Lattice Boltzmann Method for Fluid Flows (Annu. Rev. Fluid Mech. 1998, 30:329-64), and then read some references therein.
well, there is not really a space expansion, since you only have the e-term.
The time expansion on the other hand has a simple justification. Advection and diffusion phenomena happen on different time scales. Advection is always the fastest process, but diffusion is much slower. This is captured by the separation d / dt = e d / dt_1 + e^2 d / dt_2, where t_1 describes the advection and t_2 the diffusion.