# Confusion regarding the use of partition function

Suppose we have a system filled with $$N$$ particles. There are $$k$$ energy levels in this system, labeled by $$\epsilon_i$$, each with a degeneracy of $$g_i$$. Let us imagine $$n_j$$ particles out of these $$N$$ particles occupy the energy level $$\epsilon_j$$.

The single particle partition function $$Z_{sp}$$ is given by : $$Z_{sp}=\sum g_ie^{-\beta\epsilon_i}$$

Let $$Z_N$$ be the multi-particle partition function. Its expression is a little more complicated, and can be given by: $$Z_N=\sum G_ie^{-\beta E_i}$$.

Here $$E_i$$ is the total energy of all the $$N$$ particles and we are checking over all the combinations. Similarly, $$G_i$$ is the degeneracy of these total energies.

Here is my question :

What is the difference between asking how many particles are in the energy level $$\epsilon_j$$, and the probability of the system to have a total energy $$E_m$$?

The second question is simple, and the answer is given by : $$P(E_m)=\frac{G_me^{-\beta E_m}}{Z_N}$$

However, how do I find out the answer to the first question, i.e. find $$n_j$$, the number of particles in the $$\epsilon_j$$ energy level ?

According to Wikipedia, the number of particles in a particular energy level is nothing but the probability of a single particle to occupy that level, multiplied by the total number of particles. Since the probability of a single particle being in energy level $$\epsilon_j$$ is given using the single-particle partition function, our final answer comes to be :

$$n_j=\frac{g_j e^{-\beta \epsilon_j}}{Z_{sp}}N$$

Is this correct?

Like for a system with only one particle, the probability of that one particle to be in a particular energy level is the same as the probability of the entire system having that energy. However, for a system with $$N$$ particles, it seems that the probability of a particle being in a particular state is very different from the probability of the entire system having a certain total energy.

Are the statements all correct ?

• Can you link to the wikipedia article? Nov 20, 2021 at 23:42
• @Andrew here is the link Nov 21, 2021 at 0:39

It sounds like the main thing that would convince you is the check that I recommended in the last question :). I hedged my bets there by saying that the formula I gave for $$N_j$$ would agree with Wikipedia at large $$N$$. But it actually holds for general $$N$$.
The expected number of particles at level $$j$$ is found by summing $$n_j$$ over all microstates with their Boltzmann factor probabilities as weights. So again, that means $$$$N_j = Z_N^{-1} \sum_{n_1 + \dots + n_k = N} n_j G(n_1, \dots, n_k) e^{-\beta (\epsilon_1 n_1 + \dots \epsilon_k n_k)}.$$$$ But notice that this sum looks exactly like the the full partition function (which we know to be $$Z_{\mathrm{sp}}^N$$) differentiated with respect to $$\epsilon_j$$! Therefore, \begin{align} N_j &= -\beta^{-1} Z_N^{-1} \frac{\partial}{\partial \epsilon_j} Z_N \\ &= -\beta^{-1} Z_{\mathrm{sp}}^{-N} \frac{\partial}{\partial \epsilon_j} Z_{\mathrm{sp}}^N \\ &= -N\beta^{-1} Z_{\mathrm{sp}}^{-1} \frac{\partial}{\partial \epsilon_j} Z_{\mathrm{sp}} \\ &= N Z_{\mathrm{sp}}^{-1} g_j e^{-\beta \epsilon_j} \end{align} and everything checks out.