# Boltzmann Transport Equation nonlinear terms

Let's take into consideration the classical Boltzmann Equation for gas kinetics (i'm writing down the one obtained in Huang K. - Statistical mechanics, equation (3.36)):

$$\left(\frac{\partial}{\partial t} +\frac{\mathbf p_1}{m}\nabla_r+\mathbf F\nabla_{p_1} \right)f_1=\int d^3p_2 d^3p'_1 d^3p'_2 \delta^4(P_f-P_i)|T_{fi}|^2(f'_2f'_1-f_1f_2)$$

which is an integro-differential nonlinear (due to the $f'_2f'_1$ and $f_1f_2$ terms) equation in the distribution function $f_1$.

In a textbook I'm reading it is said that another feature of such equation is that

the nonlinear terms $f'_2f'_1$ and $f_1f_2$ are calculated for different values of the momentum.

What could be the meaning of this statement? Could "the momentum" be the $\mathbf p_1$ in $f_1(\mathbf x_1, \mathbf p_1, t$)?

• I think it means that $f_1$ is a shortcut notation for $f(\vec{x}, \vec{p}_1,t)$; then $f'_1 = f(\vec{x},\vec{p}'_1,t)$ etc... – Matteo Feb 4 '18 at 17:34

The meaning is that in the scattering term on the RHS of the Boltzmann transport equation you have the products of the distribution functions $f_1$ and $f_2$ for different momenta $p_1$ and $p_2$.
• Thank you. But in the case of $f'_2f'_1$ we take into consideration every value of $p'_1$ and $p'_2$, due to integration... – Lo Scrondo Feb 4 '18 at 21:54