# Average number of particles/subsystems in a state

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1. Oct 15, 2015

### devd

1. The problem statement, all variables and given/known data
A system in thermal equilibrium at temperature T consists of a large number of subsystems, each of which can exist only in two states of energy and , where . In the expressions that follow, k is the Boltzmann constant.

For a system at temperature T, the average number of subsystems in the state of energy is given by

2. Relevant equations
Probability of a system to be in a system-microstate of total energy $E_R$,
$P_R = \frac{e^{-\beta E_R}} {\sum_{R} e^{-\beta E_R}}$

3. The attempt at a solution
We have the constraint $\sum_r n_r= N_0$ and $\sum_r n_r \epsilon_r = E_R$
Where, r labels the single particle states.
Therefore, the average number of particles in the sth 1 particle state,
$\langle n_s\rangle = \frac{\sum_R n_s e^{-\beta (\sum_r n_r \epsilon_r)}}{\sum_{R} e^{-\beta (\sum_r \epsilon_r)}}$

To proceed one needs the nature of the particles.
For example,
$\langle n_s\rangle = \frac{1}{e^{\beta \epsilon_s} -1}$ for Photons
$\langle n_s\rangle = \frac{1}{e^{(\beta \epsilon_s -\mu)}+1}$ for FD statistics etc.

How do i proceed without further info? The question seems to conflate states of the total system with the subsystem states. I think the question is problematic and ambiguous. Anyhow, the supplied 'correct' answer is option (B).

2. Oct 17, 2015

### devd

The given options only makes sense if by 'system' they mean ensemble and by 'subsystem' they mean members of the ensemble.
Then, $P(E_1)= \cfrac{e^{-\beta E_1}}{e^{-\beta E_1}+e^{-\beta E_2}}$
Again, $P(E_1)= \cfrac{N_1}{N_0}.$ Therefore, $N_1 = N_0\left(\cfrac{e^{-\beta E_1}}{e^{-\beta E_1}+e^{-\beta E_2}}\right)$
And hence, $N_1 = N_0\left(\cfrac{1}{1+e^{-\beta \epsilon}}\right)$ , where $E_2-E_1= \epsilon$ .

But, if they're talking about a single system, then the options don't make sense to me. But, this question appeared in the GRE, so they aren't likely to make such errors. So, what am i missing?