- #1

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- 157

- 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.

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.