# Statistical mechanics and problem with integrals

• LCSphysicist
Maybe it's because of the environment, i don't know. I'm new to LaTeX.In summary, the conversation discusses a system of N non-interacting particles in a d-dimensional space that is in contact with a bath of temperature T. The system is described by a Hamiltonian with terms involving the coefficients A and B and exponents s and t. The question asks about finding the average energy and the density of states g. The conversation also includes a discussion about simplifying the Hamiltonian by expanding it in terms of its components.

#### LCSphysicist

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:

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

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.

LCSphysicist said:
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.

Last edited:
LCSphysicist
LastScattered1090 said:
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.