## FANDOM

190 Pages

The author goes to great lengths to explain the linear advection and diffusion terms in the Boltzmann equation (equation 2.11), but is content with a vague "hand waving" explanation for the last term on the right hand side

$\frac{1}{3} \left( \nabla \cdot \mathbf{U} \right) p \frac{\partial f}{\partial p}$

This term guarantees that the number of particles is conserved

$N = \int d^3 x \int d^3 p \cdot f$

$\frac{\partial N}{\partial t} = \int d^3 x \int d^3 p \cdot \frac{\partial f}{\partial t} =$

$= \int d^3 x \int d^3 p \cdot \left( -\mathbf{U} \cdot \nabla f + \nabla \left( \kappa \nabla f \right) + \frac{1}{3} \nabla \cdot U p \frac{\partial f}{\partial p} \right) =$

$= 4 \pi \int d^3 x \int p^2 dp \cdot \left( -\mathbf{U} \cdot \nabla f + \frac{1}{3} \nabla \cdot U p \frac{\partial f}{\partial p} \right) =$

$= 4 \pi \int d^3 x \int dp \cdot \left( -p^2 \mathbf{U} \cdot \nabla f - \frac{1}{3} \nabla \cdot U \left(\frac{\partial}{\partial p}p^3\right) f \right) =$

$= 4 \pi \int d^3 x \int dp \cdot \left( -p^2 \mathbf{U} \cdot \nabla f - \nabla \cdot U p^2 f \right) =$

$= - 4 \pi \int d^3 x \int p^2 dp \cdot \left( \mathbf{U} \cdot \nabla f + \nabla \cdot U f \right) =$

$= - 4 \pi \int d^3 x \int p^2 dp \cdot \left( \nabla \cdot \left( \mathbf{U} f\right) \right) = 0$

but it is not unique (there may be different terms that restore conservation of particle number). This particular form has to do with adiabatic cooling, which relates the expansion to change in momentum

$\frac{dp}{dt} = -\frac{1}{3} p \nabla \cdot \mathbf{U}$

Hence the change in the density distribution function is

$\left( \frac{\partial f}{\partial t} \right)_{ad} = \frac{\partial f}{\partial p} \frac{dp}{dt} = -\frac{1}{3}\left(\nabla \cdot \mathbf{U} \right)p\frac{\partial f}{\partial p}$

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