When deriving the Boltzmann equation from the BBGKY hierarchy one of the starting points is the equation: $$ \newcommand{\p}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\f}[2]{\frac{ #1}{ #2}} \newcommand{\l}[0]{\left(} \newcommand{\r}[0]{\right)} \newcommand{\mean}[1]{\langle #1 \rangle}\newcommand{\e}[0]{\varepsilon} \newcommand{\ket}[1]{\left|#1\right>} \newcommand{\bra}[1]{\left<#1\right|} \newcommand{\braoket}[3]{\left<#1\right|#2\left|#3\right>} \l \p{}{t}+\f{\vec{p}_1}{m}\cdot \p{}{\vec{q}_1}+\f{\vec{p}_2}{m}\cdot \p{}{\vec{q}_2}-\p{U(|\vec{q}_1-\vec{q}_2|)}{\vec{q}_1}\cdot \l \p{}{\vec{p}_1}-\p{}{\vec{p}_2}\r\r f_2$$ $$=\int dV_3\l \p{U(|\vec q_1-\vec q_3|)}{\vec q_1}\cdot \p{}{\vec p_1}+\p{U(|\vec{q}_2-\vec{q}_3|)}{\vec{q}_2}\cdot \p{}{\vec p_2}\r f_3$$ The term on the RHS is called the collisional integral and scales as $nd^3/\tau_c$ compared to the last term on the LHS which scales as $1/\tau_c$. Due to this scaling in the derivation of the Boltzmann relation the RHS is often set to zero. I understand completely the mathematics behind this but not the physics. So please can someone explain the physics of why the RHS (concerning collisions with other particles in the system) can be neglected compared to the last term on the LHS (concerning collisions between the two particles under consideration).
EDIT
As suggested in the comments - here is a definition of the terms in this equation:
- $\vec q_i$ is the position of the $i$th particle.
- $\vec p_i$ is the momentum of the $i$th particle.
- $U(|\vec q_i-\vec q_j|)$ is the interaction potential between the $i$th and the $j$th particles.
- $f_n(\vec q_1,...,\vec q_n,\vec p_1,...,\vec p_n)$ (argument dropped in above) is the n-particle distribution function (i.e. the probability that you would find any arbitrary $n$ particles with position $\vec q_1,...,\vec q_n$ and momenta $\vec p_1,...,\vec p_n$).
- $dV_3$ is a phase space volume, $dV_3=d\vec q_3 d\vec p_3$.