I'm studying an opinion formation model as follows. Opinions are given by $w \in \mathcal{I} = [-1,1]$. If an individual with opinion $w$ meets an individual with opinion $v$, the interaction is described by \begin{align} w' = w - \gamma P(|w-v|)(w-v) + \eta_1D(w), \\ v' = v - \gamma P(|v-w|)(v-w) + \eta_2D(w), \end{align} where $(w',v')$ is the pair of opinions after the interaction, $P$ and $D$ are functions relating to compromise and self-thinking respectively, $\gamma \in (0,1/2)$ is a given constant, and $\eta_1$ and $\eta_2$ are random variables with zero mean and the same variance $\sigma^2$. We set the restriction that $0 \leq P,D \leq 1$.
In an attempt to study the time evolution of the opinion distribution function $f(w,t)$, I am required to use "standard methods of kinetic theory" to derive this Boltzmann-like equation: \begin{equation} \frac{\partial}{\partial t}f(w,t) = \frac{1}{\tau}Q(f,f)(w), \end{equation} where the collision operator $Q$ can be written in weak form: \begin{equation} \int_\mathcal{I}Q(f,f)(w)\phi(w)dw = \frac{1}{2}\left\langle\int_{\mathcal{I}^2}(\phi(w')+\phi(v')-\phi(w)-\phi(v))f(w)f(v)dvdw\right\rangle, \end{equation} with $\langle\cdot\rangle$ denoting the operation of mean with respect to the random variables $\eta_i$.
Unfortunately, I am at a loss as to how to do this. I have done lots of reading around the derivation of the Boltzmann equation for dilute gases, but given this 1-dimensional context of opinion, I don't have any idea of how to begin this derivation. If anyone could give me some hints as to how to go about this, or direct me to some relevant reading, that would be much appreciated as I currently can't find anything whatsoever to help me.