Chapman-Enskog Expansion

Boltzman equation

$$\frac{\partial f}{\partial t} + \frac{\vec{p}}{m}\cdot \nabla f + \vec{F}\cdot \nabla_p f =(\frac{\delta f}{\delta t})_{coll}$$

Assume $f$ can be expansed as 

$$f= f_0 + \epsilon f_1 + \epsilon^2 f_2 + \ldots .$$

If we assume hydrodynamic parameters such as number density, bulk velocity and temperature are expressed through $f_0$ only, 

e.g. $$n = \int f_0 d^3v ,$$

otherwise, the defintion for density tells us 

$$n =  \int f d^3v ,$$

so we get the integration 

\[ \int f_{k} d^3v = 0, \,for\, k>=1 .\ ]

Applying the same argument to bulk density and temperature, we get 

$$\int f_k d^3v = \int \vec{v} f_k d^3v = \int (\vec{v} - \vec{u})^2 f_k d^3v =0, \, for \, k \ge 1 .$$   

Reference:

https://iopscience.iop.org/book/978-0-7503-1200-4/chapter/bk978-0-7503-1200-4ch1