[PhD Quals] Plasma Moments Equations, vector calc

[erin85]

Homework Statement

Derive the moments equations for a plasma. I specifically need help developing the 2nd moment equation (the Energy Balance). I can't get one of the terms...

Homework Equations

The Boltzmann Equation:
$$\frac{\partial f_{\sigma}}{\partial t} + \vec{v}\cdot \nabla f_{\sigma}+\frac{e_{\sigma}}{m_{\sigma}}(\vec{E}+\vec{v}\times\vec{B})\cdot \nabla_{v}f_{\sigma} = C_{\sigma} + S_{\sigma}$$

How to take the moment: Multiply by z_n(v), then integrate over velocity (d³v), where z₂=$$\frac{1}{2}m(\vec{v}\cdot\vec{v})$$.

Other misc. definitions:

$$n_{\sigma}=\int f_{\sigma}(\vec{v})d^3v$$
$$v_{\sigma}=\int f_{\sigma}(\vec{v})\vec{v}d^3v/n_{\sigma}$$
and in general,
$$(A)_{\sigma}=\int f_{\sigma}(\vec{v})A(\vec{v})d^3v$$

Also, $$\vec{Q}_{\sigma} = (1/2)n_{\sigma}m_{\sigma}[(\vec{v}\cdot\vec{v})\vec{v}]_{\sigma} =$$energy flow,
$$TrM_{\sigma}= n_{\sigma}m_{\sigma}(\vec{v}\cdot\vec{v})_{\sigma}$$ = scalar trace of momentum stress tensor.

The 2nd moment equation is:

$$\frac{1}{2}\frac{\partial}{\partial t}(TrM_{\sigma})+\nabla \cdot \vec{Q}_{\sigma} = n_{\sigma}e{\sigma}\vec{v}_{\sigma}\cdot\vec{E}+R^2_{\sigma}+S^2_{\sigma}$$.

Don't worry about the C's, S's, and R's... those definitions are just straightforward, but annoying to type.

One last thing: $$\vec{A}\cdot\nabla f = \nabla \cdot f\vec{A}-f\nabla\cdot\vec{A}.$$

The Attempt at a Solution

I have no trouble getting the first two terms... but the one with the electric field in it is tricky. I get every term going to zero...

Just taking the part of the equation with E (not v cross B yet), and a little simplfication:

$$\int \frac{e_{\sigma}}{2}(\vec{v}\cdot\vec{v})(\vec{E}\cdot\nabla_v f_{\sigma})d^3v$$

Grouping the v's and the E together, and using the vector identity above,
=$$\frac{e_{\sigma}}{2}\int d^3v [\nabla_v\cdot f_{\sigma}((\vec{v}\cdot\vec{v})\vec{E}) - f_{\sigma}\nabla_v\cdot((\vec{v}\cdot\vec{v})\vec{E})]$$

Breaking up the second term, just using the chain rule... I get a term that's del dot a dot product, which should go to zero, and a del dot E, and E is not a function of v, so should also go to zero... Maybe breaking this up via chain rule is not correct.

The first term I also feel goes to zero... the integral and the del basically cancel, so I evaluate what's left at +_ infinity.

Any help please? I am having a lot of trouble with this... and it took a freaking long time to type, so it would be really excellent to get an answer.

 

[olgranpappy]

Homework Statement

Derive the moments equations for a plasma. I specifically need help developing the 2nd moment equation (the Energy Balance). I can't get one of the terms...

Homework Equations

The Boltzmann Equation:
$$\frac{\partial f_{\sigma}}{\partial t} + \vec{v}\cdot \nabla f_{\sigma}+\frac{e_{\sigma}}{m_{\sigma}}(\vec{E}+\vec{v}\times\vec{B})\cdot \nabla_{v}f_{\sigma} = C_{\sigma} + S_{\sigma}$$

How to take the moment: Multiply by z_n(v), then integrate over velocity (d³v), where z₂=$$\frac{1}{2}m(\vec{v}\cdot\vec{v})$$.

Other misc. definitions:

$$n_{\sigma}=\int f_{\sigma}(\vec{v})d^3v$$
$$v_{\sigma}=\int f_{\sigma}(\vec{v})\vec{v}d^3v/n_{\sigma}$$
and in general,
$$(A)_{\sigma}=\int f_{\sigma}(\vec{v})A(\vec{v})d^3v$$

Also, $$\vec{Q}_{\sigma} = (1/2)n_{\sigma}m_{\sigma}[(\vec{v}\cdot\vec{v})\vec{v}]_{\sigma} =$$energy flow,
$$TrM_{\sigma}= n_{\sigma}m_{\sigma}(\vec{v}\cdot\vec{v})_{\sigma}$$ = scalar trace of momentum stress tensor.

The 2nd moment equation is:

$$\frac{1}{2}\frac{\partial}{\partial t}(TrM_{\sigma})+\nabla \cdot \vec{Q}_{\sigma} = n_{\sigma}e{\sigma}\vec{v}_{\sigma}\cdot\vec{E}+R^2_{\sigma}+S^2_{\sigma}$$.

Don't worry about the C's, S's, and R's... those definitions are just straightforward, but annoying to type.

One last thing: $$\vec{A}\cdot\nabla f = \nabla \cdot f\vec{A}-f\nabla\cdot\vec{A}.$$

The Attempt at a Solution

I have no trouble getting the first two terms... but the one with the electric field in it is tricky. I get every term going to zero...

Just taking the part of the equation with E (not v cross B yet), and a little simplfication:

$$\int \frac{e_{\sigma}}{2}(\vec{v}\cdot\vec{v})(\vec{E}\cdot\nabla_v f_{\sigma})d^3v$$

Grouping the v's and the E together, and using the vector identity above,
=$$\frac{e_{\sigma}}{2}\int d^3v [\nabla_v\cdot f_{\sigma}((\vec{v}\cdot\vec{v})\vec{E}) - f_{\sigma}\nabla_v\cdot((\vec{v}\cdot\vec{v})\vec{E})]$$

Breaking up the second term, just using the chain rule... I get a term that's del dot a dot product, which should go to zero, and a del dot E, and E is not a function of v, so should also go to zero... Maybe breaking this up via chain rule is not correct.

The first term I also feel goes to zero... the integral and the del basically cancel, so I evaluate what's left at +_ infinity.

Any help please? I am having a lot of trouble with this... and it took a freaking long time to type, so it would be really excellent to get an answer.

Hmm... let's see...

$$\int d^3 v \frac{m}{2}v^2\frac{e}{m}\vec E \cdot \frac{\partial f}{\partial \vec v}$$

$$= \frac{e}{2}\vec E\cdot\int d^3v v^2 \frac{\partial f}{\partial \vec v}$$

$$= -\frac{e}{2}\vec E \cdot \int d^3 v f \frac{\partial}{\partial \vec v}v^2$$

$$=-{e}\vec E\cdot \int d^3 v f\vec v$$

Also, just wondering... what school are these quals for?

 

[erin85]

Hi, thanks for the response. Sorry, I didn't notice... physicsforum usually e-mails me when I get a response... What rule is that for the transition from the 2nd to 3rd line? Not quite sure I'm getting that step. Although it is almost 3 in the morning, and I've been studying for a while...

Also, this quals is for Georgia Tech, the nuclear engineering department.