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