# I Deriving continuity equation of phase space in Statistical Mechanics

1. Dec 7, 2016

### binbagsss

Hi,

So I am aiming to derive the continuity equation using the fact that phase space points are not created/destroyed.

So I am going to use the Leibiniz rule for integration extended to 3-d:

$d/dt \int\limits_{v(t)} F dv = \int\limits_{v(t)} \frac{\partial F}{\partial t} dV + \int\limits_{A(t)} F \vec{u_{A}}.\vec{n} dA$,

$\vec{n}$ the normal unit vector, where $F=F(x_{1},x_{2},x_{3},t)$

$\vec{u_{A}}$ is the surface velocity.

So let $N_{D}$ denote the number of phase points in a volume element $D$, $N$ the total number of phase points.

Conservation of phase points =>
$\frac{d}{dt} N_{D}(t) =0$

$= - N \frac{d}{dt} \int\limits_{D} dp dq \rho (p,q,t)$

where $\rho = \rho (p,q,t)$ is the density of function.

So by Leibniz rule above I get a term describing the change due to the changing surface of $A$ - $\partial D$ here

$= - N \int\limits_{\partial D} \rho \vec{V}.\vec{n} d\sigma$, which is fine ,

where $\vec{V}$ is the velocity

MY QUESTION:

The corresponding term to $\int\limits_{v(t)} \frac{\partial F}{\partial t} dV$ is zero here.
I.e the contribution due to the change of $\rho$ within $V$. I'm just having a little trouble understanding this conceptually... so my book states that:

"Since phase points cannot be created the only way that phase points can be added or subtracted from the volume V is to flow across its boundary, S. "

That's fine, and seems to make sense to me. But, if points are flowing across the boundary then surely the number inside V is changing. So looking at
$\int\limits_{v(t)} \frac{\partial \rho}{\partial t} dV$,

$\frac{\partial \rho}{\partial t} \neq 0$ if there is either a change in the number of phase points or a change in the volume, doesn't saying that the number is changing across the surface => number is changing within the volume and so $\neq 0$ for a given volume due to the number of phase points changing?

( I am fine with the rest of the derivation just stuck in this step)