Statistical mechanics and problem with integrals

[LCSphysicist]

Homework Statement

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Relevant Equations

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So we have a system of N non interacting particles, on a d-dimensional space, the system is in contact with a bath of temperature T. The hamiltonian is $$H = \sum_{l = 1}^{N} (A_{l}|p_{l}|^{s}+B_{l}|q_{l}|^{s})$$.

What is the avarage energy?

Now, i have some problems with statistical mechanics, and i always find myself surrounded by a lot of different roads, and don't know which one to choose. But my first step was to try to simplify this hamiltonian:

$$(A_{l})^{1/s}p_{i,l} = f_{i,l} \n (B_{l})^{1/t}q_{i,l} = g_{i,l}$$

And calling $$\sum_{i} f_{il}^2 = (f_{l})^2 \n \sum_{i} g_{il}^2 = (g_{l})^2$$
Our Hamiltonian now is $$H = \sum_{l=1}^{N} ((f_{l})^{s}+(g_{l})^{s})$$

Now, $$U = \int e^{-\beta \epsilon} g(\epsilon) d\epsilon$$

How the heck i find the density of states g?
If s were 1 or 2, ok, i could make a guess, but i have no idea what to do with s.

 

[LastScattered1090]

Homework Statement:: .
Relevant Equations:: .

So we have a system of N non interacting particles, on a d-dimensional space, the system is in contact with a bath of temperature T. The hamiltonian is $$H = \sum_{l = 1}^{N} (A_{l}|p_{l}|^{s}+B_{l}|q_{l}|^{s})$$.

What is the avarage energy?

Now, i have some problems with statistical mechanics, and i always find myself surrounded by a lot of different roads, and don't know which one to choose. But my first step was to try to simplify this hamiltonian:

$$(A_{l})^{1/s}p_{i,l} = f_{i,l}~\\ (B_{l})^{1/t}q_{i,l} = g_{i,l}$$

And calling $$\sum_{i} f_{il}^2 = (f_{l})^2~\\ \sum_{i} g_{il}^2 = (g_{l})^2$$
Our Hamiltonian now is $$H = \sum_{l=1}^{N} ((f_{l})^{s}+(g_{l})^{s})$$

Now, $$U = \int e^{-\beta \epsilon} g(\epsilon) d\epsilon$$

How the heck i find the density of states g?
If s were 1 or 2, ok, i could make a guess, but i have no idea what to do with s.

I can't help much with this problem as it's been a long time since looking at the Hamiltonian formulation in classical physics, for me. Just wanted to ask some questions for clarification:

0) Pro-tip that '\n' is not a line break in LaTeX. Try using '\\\' instead.

1) Is the idea here that the terms in the sum would be something like $|p_l|^2/2m_l$ for a familiar scenario, but because we're generalizing this to a d-dimensional space, we have a different exponent and sort of generalized KE and PE terms for each particle without specific regard for the physical interpretation of the coefficients A and B?

2) Why do you write $(B_l)^{1/t}?$ What is $t$? Is this a typo, and should it actually say $(B_l)^{1/s}?$

3) I understand that $l$ indexes particles from 1 to N, but what does subscript $i$ represent? You introduce it in your modifications to $\mathcal{H}$. I had a thought that maybe it could index dimensions from 1 to $d$, but is it necessary to consider these components when they don't appear in the original Hamiltonian?

Thanks for any clarification you can provide. I thought the answers to these questions could help someone more knowledgeable step in.

 

[LCSphysicist]

I can't help much with this problem as it's been a long time since looking at the Hamiltonian formulation in classical physics, for me. Just wanted to ask some questions for clarification:

0) Pro-tip that '\n' is not a line break in LaTeX. Try using '\\\' instead.

1) Is the idea here that the terms in the sum would be something like $p_l^2/2m_l$ for a familiar scenario, but because we're generalizing this to a d-dimensional space, we have a different exponent and sort of generalized KE and PE terms for each particle without specific regard for the physical interpretation of the coefficients A and B?

2) Why do you write $(B_l)^{1/t}?$ What is $t$? Is this a typo, and should it actually say $(B_l)^{1/s}?$

3) I understand that $l$ indexes particles from 1 to N, but what does subscript $i$ represent? You introduce it in your modifications to $\mathcal{H}$. I had a thought that maybe it could index dimensions from 1 to $d$, but is it necessary to consider these components when they don't appear in the original Hamiltonian?

Thanks for any clarification you can provide. I thought the answers to these questions could help someone more knowledgeable step in.

Af, yes, there is a typo here. On the hamiltonian, should be $|q_l|^t$, being t and s integers.

The subscript i represents the dimension. What i have made above is simply expanding $|q_l|,|p_l|$ in its components, $|q_l|^2 = \sum_{i=1}^{d} q_{il}^2$, supposing of course the basis of the d dimensional space is orthogonal, and inserting the A on it, so that we can get rid of it. Just like we can write $p^2/2m + kq^2/2$ as $p'^2 + q'^2$, if we call $p' = p / \sqrt{2m}$ etc..

I am aware of "\\", but it was not working for some reason.