I am working my way through a lecture notes draft (section 5.1.) on kinetic theory of plasmas, and I'm having trouble with the derivation of the law of energy conservation in a Vlasov-Poisson system. The equations I'm working with here are:
Next, I wish to calculate the rate of change of electric energy as follows:
I understand that the first part comes from Poynting's theorem (in Planck units), Leibniz' rule and using the electrostatic approximation to substitute in $E = -\nabla \varphi$. Now what I'm having trouble with is both of the parts where partial integration is used.
So, first, how do I get the result that
$E = \int d^{3} \vec r \frac {\nabla \varphi}{4\pi} \cdot \frac{\partial(\nabla \varphi)}{\partial t} = -\int d^{3} \vec r \frac {\varphi}{4\pi} \frac{\partial(\nabla^{2} \varphi)}{\partial t}$?
My attempt at integrating is this:
$E = \int d^{3} \vec r \frac {\nabla \varphi}{4\pi} \cdot \frac{\partial(\nabla \varphi)}{\partial t} = [{\nabla \varphi \frac {d}{dt} (\nabla \varphi \frac{1}{4\pi}) }]\Biggr|_{-\infty}^{\infty} -\int d^{3} \vec r \frac {\varphi}{4\pi} \frac{\partial}{\partial r}\frac {\partial}{\partial t} \nabla \varphi = -\int d^{3} \vec r \frac {\varphi}{4\pi} \frac{\partial}{\partial t}\frac {\partial}{\partial r} \nabla \varphi = -\int d^{3} \vec r \frac {\varphi}{4\pi} \nabla \frac {\partial}{\partial t} \nabla \varphi = -\int d^{3} \vec r \frac {\varphi}{4\pi} \frac{\partial(\nabla^{2} \varphi)}{\partial t}$,
but I'm unsure how this works out since I'm integrating a dot product, and the derivative is partial. Am I on the right track here? Should I be using the divergence theorem? From what I understand $\nabla$ is taken over the space, so it corresponds to a derivative with respect to $\vec r$.
The second thing I'm not sure of is integrating the first term on the second line of (5.3) by parts, since there's two integrals and a dot product. Do I integrate by parts in $\vec v $ or $\vec r$? If I multiply by $\varphi$, and integrate by parts, I get terms with $\varphi \vec v$ and $\varphi f_{\alpha}$, so I'm not grasping where I would get the ${f_{\alpha} \vec v}$ term in the result.
Any help is greatly appreciated, I'm only familiar with very simple calculations with partial integration and the double integrals, dot products and $\nabla$ are confusing me.
EDIT: 1st part answered, still need help on the second part!
Best Answer
You use the usual rules for integration by parts (from vector calculus). The relevant result, written here with a bunch of parentheses to try and clarify what is operating on what, is: $$ \int \vec \nabla\left(f(\vec x,t)\right) \cdot \vec \nabla \left( g(\vec x, t)\right) =\int \vec \nabla \cdot \left(f\vec \nabla \left(g\right)\right) - \int f\nabla^2g $$
Just use $$ f = \phi $$ and $$ g = \frac{\partial \phi}{\partial t} $$
And apparently "use 5.2" means to use something like: $$ \int \vec \nabla \cdot \left(\phi\frac{\partial }{\partial t}\vec \nabla \phi\right) = \oint_{at\;\infty} \vec dA \cdot (\phi\frac{\partial }{\partial t}\vec \nabla \phi)= 0 $$
Which jives if $\phi$ falls off like $1/r$ or faster (since the surface at $\infty$ grows in area like $r^2$).
You also use the fact that partial derivative wrt $t$ and wrt $\vec x$ commute.