Velocity-space diffusion (${\partial }_{t}f={\partial }_v(D(v){\partial }_{v}f)$)

Random changes of particle velocity described by a diffusion equation in velocity space. For constant $D$, $⟨(\Delta v{)}^2⟩=2D\Delta t$. With zero boundary flux it conserves number. For a number-normalized one-dimensional distribution, its kinetic energy production is $d\mathcal{K}/dt=-m\int vD(v)f^{\prime }(v)dv$, which reduces to $mnD$ for constant $D$. Variable $D$ must remain inside the derivative.