$\displaystyle\frac{df}{dt}=\displaystyle\frac{1}{\tau}(f_{in}-f_{out})$

\displaystyle\frac{df}{dt}=\displaystyle\frac{1}{\tau}(f_{in}-f_{out})

$f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v}_2,t)|\vec{v}_2-\vec{v}_1|\sigma(\vec{v}_1,\vec{v}_2\rightarrow\vec{v}_12,\vec{v}_22)d\vec{v}_12d\vec{v}_22$

f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v}_2,t)|\vec{v}_2-\vec{v}_1|\sigma(\vec{v}_1,\vec{v}_2\rightarrow\vec{v}_12,\vec{v}_22)d\vec{v}_12d\vec{v}_22

$\displaystyle\frac{1}{\tau}f_{in}(\vec{v})=\displaystyle\int d\vec{v}_1d\vec{v}_2d\vec{v}_12f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v}_2,t)|\vec{v}_2-\vec{v}_1|\sigma(\vec{v}_1,\vec{v}_2\rightarrow\vec{v}_12,\vec{v})$

\displaystyle\frac{1}{\tau}f_{in}(\vec{v})=\displaystyle\int d\vec{v}_1d\vec{v}_2d\vec{v}_12f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v}_2,t)|\vec{v}_2-\vec{v}_1|\sigma(\vec{v}_1,\vec{v}_2\rightarrow\vec{v}_12,\vec{v})

$\displaystyle\frac{1}{\tau}f_{out}(\vec{v})=\displaystyle\int d\vec{v}_1d\vec{v}_12d\vec{v}_22f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v},t)|\vec{v}-\vec{v}_1|\sigma(\vec{v},\vec{v}_1\rightarrow\vec{v}_12,\vec{v}_22)$

\displaystyle\frac{1}{\tau}f_{out}(\vec{v})=\displaystyle\int d\vec{v}_1d\vec{v}_12d\vec{v}_22f(\vec{x},\vec{v}_1,t)f(\vec{x},\vec{v},t)|\vec{v}-\vec{v}_1|\sigma(\vec{v},\vec{v}_1\rightarrow\vec{v}_12,\vec{v})

$\displaystyle\frac{1}{\tau}(f_{in}-f_{out})=\displaystyle\int d\vec{v}_1d\vec{v}2d\vec{v}_12(f(\vec{x},\vec{v}2,t)f(\vec{x},\vec{v}_12,t)-f(\vec{x},\vec{v},t)f(\vec{x},\vec{v}_1,t))|\vec{v}-\vec{v}_1|\sigma(\vec{v},\vec{v}_1\rightarrow\vec{v}2,\vec{v}_12)$

\displaystyle\frac{1}{\tau}(f_{in}-f_{out})=\displaystyle\int d\vec{v}_1d\vec{v}2d\vec{v}_12(f(\vec{x},\vec{v}2,t)f(\vec{x},\vec{v}_12,t)-f(\vec{x},\vec{v},t)f(\vec{x},\vec{v}_1,t))|\vec{v}-\vec{v}_1|\sigma(\vec{v},\vec{v}_1\rightarrow\vec{v}2,\vec{v}_12)