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Say your allowed occupancies for an orbital are $0, 1, 2$, and the values of energy associated with these occupancies are $0, \epsilon, 2 \epsilon$.

Give an expression for the ensemble average occupancy $\langle N\rangle$ when the system composed of this orbital is in thermal and diffusive contact with a reservoir at temperature $\tau$ and chemical potential $\mu$.

How does $\langle N\rangle$ behave in the limit:

(a) $\tau \to \infty$

(b) $\tau \to 0$ with $\mu < \epsilon$

(c) $\tau \to 0$ with $\mu > \epsilon$

I found $\langle N\rangle$ to be:

$$\langle N\rangle = \frac{\exp\left(\frac{\mu-\epsilon}{\tau}\right) + 2 \exp\left(\frac{2(\mu-\epsilon)}{\tau}\right)}{1 + \exp\left(\frac{\mu-\epsilon}{\tau}\right) + \exp\left(\frac{2(\mu-\epsilon)}{\tau}\right)}$$

And am reasonably confident it is correct. This means that

as $\tau \to \infty$, $\space \langle N\rangle \to 1$

as $\tau \to 0$, $\space \langle N\rangle \to 0 \space$ for $\mu < \epsilon$

as $\tau \to 0$, $\space \langle N\rangle \to 2 \space$ for $\mu > \epsilon$

But why should I expect these answers to be correct?

I believe that as $\tau \to 0$, the particles will enter the lowest energy state available to them, and as far as I understand chemical potential, it is sort of a measure for how much work must be done to remove a particle from a system.

However, these do not seem align with the answers I got, so perhaps I am thinking about this wrong.

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There is not only thermal contact in this problem but also chemical. Hence the particles will enter the state with lowest $E - \mu N$ value as $\tau \to 0$.

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  • $\begingroup$ If $0 < \mu < \epsilon$, then $\epsilon - \mu < \epsilon$, so I would expect $\langle N\rangle \ge 1$, but this is not the case $\endgroup$ – infinitylord Dec 10 '17 at 15:58
  • $\begingroup$ I don't understand how $\left< N\right> \geq 1$ follows from $\epsilon - \mu < \epsilon$. If $\epsilon - \mu > 0$, than addition of a particle increases value of $E-\mu N$, while this quantity should be minimal at equilibrium as $\tau \to 0$. $\endgroup$ – Gec Dec 10 '17 at 18:30

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