Help with integral calculation

Question

I am reading this article and I am having trouble understanding a calculation there.

In it, this following equation is obtained:

$$ \frac{\partial}{\partial t} \int p_i n d \tau + \int p_i \left( \frac{\partial n}{\partial \textbf{x}} \frac{\partial \epsilon}{\partial \textbf{p}} - \frac{\partial n}{\partial \textbf{p}} \frac{\partial \epsilon}{\partial \textbf{x}}\right) d \tau = 0 .$$

where $\epsilon$ and p are the energy and momentum of the quasiparticles, and,

$$ d \tau = g \frac{d^3 p}{(2 \pi)^3} .$$

with g the degeneracy of each state.

This equation was next rewritten, by algebraic manipulation and integration by parts, in the form,

$$\frac{\partial}{\partial t} \int p_i n d \tau + \frac{\partial}{\partial \textbf{x}} \int p_i \frac{\partial \epsilon}{\partial \textbf{p}} n d \tau + \frac{\partial}{\partial x_i}\int n \epsilon d \tau - \int \epsilon \frac{\partial n }{\partial x_i} d \tau =0 .$$

My question is how was this second form obtained? I can't follow the calculation.

Answer 1

It's just a plain double integration by parts. Recall the bold vectors dot each other, and since there is no integration over x, its divergence/gradient term survives. $$\left( \frac{\partial n}{\partial \textbf{x}} \frac{\partial \epsilon}{\partial \textbf{p}} - \frac{\partial n}{\partial \textbf{p}} \frac{\partial \epsilon}{\partial \textbf{x}}\right) =\frac{\partial }{\partial \textbf{x}}\left( n \frac{\partial \epsilon}{\partial \textbf{p}}\right ) - \frac{\partial }{\partial \textbf{p}}\left ( n \frac{\partial \epsilon}{\partial \textbf{x}}\right) . $$

Your second starting integral, thus, reduces to $$ \int\!\! d\tau ~~ p_i \left ( \frac{\partial }{\partial \textbf{x}}\left( n \frac{\partial \epsilon}{\partial \textbf{p}}\right ) - \frac{\partial }{\partial \textbf{p}}\left ( n \frac{\partial \epsilon}{\partial \textbf{x}}\right) \right ) \\ = \frac{\partial }{\partial \textbf{x}}\int\!\! d\tau ~~ p_i n \frac{\partial \epsilon}{\partial \textbf{p}} - \int\!\! d\tau ~~ p_i \frac{\partial }{\partial \textbf{p}}\left ( n \frac{\partial \epsilon}{\partial \textbf{x}}\right) \\ = \frac{\partial }{\partial \textbf{x}}\int\!\! d\tau ~~ p_i n \frac{\partial \epsilon}{\partial \textbf{p}} + \int\!\! d\tau ~~ \frac{\partial p_i}{\partial \textbf{p}} n\frac{\partial \epsilon}{\partial \textbf{x}}\\ = \frac{\partial }{\partial \textbf{x}}\int\!\! d\tau ~~ p_i n \frac{\partial \epsilon}{\partial \textbf{p}} + \int\!\! d\tau ~~ n\frac{\partial \epsilon}{\partial x_i} \\ = \frac{\partial }{\partial \textbf{x}}\int\!\! d\tau ~~ p_i n \frac{\partial \epsilon}{\partial \textbf{p}} + \frac{\partial}{\partial x_i}\int n \epsilon d \tau - \int \epsilon \frac{\partial n }{\partial x_i} d \tau . $$