Boltzmann's equation

We're interested in finding an equation that describes the distribution of stars in a galaxy. So we want to know is how many stars there are in a small volume $d\hbox{$\bf x$}$ around a position $\hbox{$\bf x$}$, that have velocity in a small interval $d\dot\hbox{$\bf x$}$ around a given velocity $\dot\hbox{$\bf x$}$. This density of stars, $f(\hbox{$\bf x$},\dot\hbox{$\bf x$},t)$, is called the distribution function. If we don't care about the velocity of stars, then we can integrate $f$ over all velocities, to obtain the density of stars,

$$n(\hbox{$\bf x$},t)=\int f(\hbox{$\bf x$},\dot\hbox{$\bf x$},t)\,d\dot\hbox{$\bf x$}\,.$$ (8.15)

Now we only have to realise that stars acts as our cars on the motor way, and hence $f$ should satisfy the continuity equation (8.14). The way to do it is to substitute for the $l$ coordinates in that equation the positions and velocities of the stars,

$$\hbox{$\bf l$} \equiv (\hbox{$\bf x$},\hbox{$\bf v$})$$

$$\dot\hbox{$\bf l$} \equiv (\hbox{$\bf v$},\dot\hbox{$\bf v$}) = (\hbox{$\bf v$},-\nabla\Phi)\,,$$ (8.16)

where I've written $-\nabla\Phi$ for the gravitational acceleration $\dot\hbox{$\bf v$}$ of the star.

So this $\hbox{$\bf l$}$ vector has 6 coordinates, $l_\alpha=\hbox{$\bf x$}$ for $\alpha=1\rightarrow 3$, and $l_\alpha=\hbox{$\bf v$}$ for $\alpha=4\rightarrow 6$. Now these coordinates are very special, since

$$\sum_{\alpha=1}^6 {\partial\dot l_\alpha\over \partial l_\alpha} = \sum_{i=1}^3 \left({\partial v_i\over\partial x_i}+{\partial \dot v_i\over\partial v_i}\right)$$

$$= \sum_{i=1}^3 -{\partial\over\partial v_i}\left({\partial\Phi\over \partial x_i}\right)$$

$$= 0\,.$$ (8.17)

Here, $\partial v_i/\partial x_i=0$ since the positions and velocities of the stars are independent variables, and the second term is zero because the gravitational acceleration $\partial\Phi/\partial x_i$ does not depend on velocity.

We can use this equation to simplify the continuity equation, and to obtain

$${\partial f\over\partial t}+\sum_{\alpha=1}^6\dot l_\alpha{\partial f\over\partial l_\alpha}=0\,.$$ (8.18)

Rewriting in terms of positions and velocities, we obtain the collisionless Boltzmann equation,

$${\partial f\over\partial t} + \sum_{i=1}^3\left(v_i{\partial f\ov... ...l x_i}-{\partial\Phi\over\partial x_i}{\partial f\over\partial v_i}\right)=0\,.$$ (8.19)

or in vector notation

$${\partial f\over\partial t}+\hbox{$\bf v$}\cdot{\partial f\over\p... ...l \hbox{$\bf r$}}-\nabla\Phi\cdot {\partial f\over \partial\hbox{$\bf v$}}=0\,.$$ (8.20)