Canonical Ensemble and potential energy

[Abigale]

Hi guys ,

I regard an ideal diatomic gas which is in a Volume x,z and got a angular $\phi$:
$0 \le x \le L$, $~~~~~~~~~~~~ 0 \le z \le \infty$, $~~~~~~~~~~~~0 \le \phi \le 2 \pi$

The hamiltonian for the single particle is:
$~~~~~~H= \frac{p^{2}_{z} + p^{2}_{x} }{2M} + \frac{P^{2}_{\phi}}{2I}+mgz$
($m$ is the mass of a diatomic particle, $g$ the gravitational constant, $p_{x,y,\phi}$ are momentums and angular momentum.)

I am interessted in the average value of th potential energy!


At first i have callculated the canonical partition function for a single particle: $\text{z}=\frac{-L^{2} N^{2} k^{3} T^{3} I }{\pi \hbar^{5}}$
The partition function for the whole system can be calculated by: $Z=\frac{\text{z}^{N}}{N!}$
The average internal energy of the canonical System can be callculated with the formula: $\bar{E}=(\frac{\partial \ln{Z}}{\partial \beta})$.


But the problem is, I want to callculate the average of the potential energy.
There should be two capabilities of callculating it.

The one solution is with the average internal energy,
the other with the definition of the expected value.


With the definition of expected value, i can callculate the average potential energy of a single particle.

$\bar{E}_{potential}= \frac{\int \int d^{3N} q ~ d^{3N} p ~~~ mgz~~ \exp{- \beta H(q,p) } } { \int \int d^{3N} q ~ d^{3N} p ~~~~~ \exp{- \beta H(q,p) }}$

But i don't know how to callculate the average potential energy for the whole System of $N$ particles.

THX for your help

Bye Abby

 

[eljose79]

Hi Abby,

First of all, great job on calculating the canonical partition function for a single particle! That's an important first step in understanding the thermodynamics of a system.

To calculate the average potential energy for the whole system, you can use the definition of expected value as you mentioned. The only difference is that instead of integrating over a single particle's coordinates and momentums, you will integrate over the coordinates and momentums of all N particles.

So the average potential energy for the whole system can be calculated as:

\bar{E}_{potential}= \frac{\int \int \cdots \int d^{3N} q ~ d^{3N} p ~~~ mgz~~ \exp{- \beta H(q,p) } }

{ \int \int \cdots \int d^{3N} q ~ d^{3N} p ~~~~~ \exp{- \beta H(q,p) }}

Where the integrals are taken over all N particles' coordinates and momentums.

I hope this helps! Let me know if you have any further questions.