# Current density in phase space

I have a question which arises from looking at the impact free Boltzmann equation.

Let $(\vec{x},\vec{v})$ be a vector in our phase space $\Gamma^N = \mathbb{R}^{6N}$. The dynamics of a state are determined by the distribution function $f(\vec{x}, \vec{v}, t)$. Where $f(\vec{x}, \vec{v}, t) d^3x d^3v$ is the amount of particles at time $t$ in the volume element $d^3x d^3v$.

To derive the impact free Boltzmannequation we simply have to equate the time derivative of the volume-integral of $f$ to the flow of particles out of that volume (The amount of particles going out of a certain phasespace volume determine how the state goes on in time).

This means: $\int_V \frac{\partial}{\partial t} f(\vec{x}, \vec{v}, t) dV = - \int_V div_{\vec{x}, \vec{v}}((\vec{v},\vec{a})f(\vec{x}, \vec{v}, t))dV$

This is where my question arises. Why is the right side of the equation the flow of particles out of $dV$? $(\vec{v},\vec{a})$ is the time derivative of $(\vec{x}, \vec{v})$, but I still dont see it. Can somebody give me some pointers what I have to read about to get an intuitive and mathematical feeling for why this is right?

Cheers

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Any fluid mechanics text should help (Kundu & Cohen, Fox & MacDonald, etc) with understanding the problem. –  Kyle Kanos Jul 21 '13 at 17:03
@KyleKanos which chapter in Kundu & Cohen esp.? –  user17574 Jul 21 '13 at 17:44
Chapter 4: Conservation Laws should be the one. –  Kyle Kanos Jul 21 '13 at 17:52