The partition function

The additivity of the energy in formulae , is a consequence of the absence of interactions between the particles. This is a big simplification and makes it particularly simple to write the partition function of the system, because the total probability factorises. The probability $p_{r_1}$ of particle 1 being in single particle microstate $r_1$ with energy $\epsilon_{r_1}$ and $p_{r_2}$ of particle 2 being in single particle microstate $r_2$ with energy $\epsilon_{r_2}$ and $\dots$ and $p_{r_N}$ of particle $N$ being in single particle microstate $r_N$ with energy $\epsilon_{r_N}$ is:

$$p[r_1,r_2,\dots,r_N] = p_{r_1} p_{r_2} \dots p_{r_N}.$$ (4.4)

If the particles are kept in contact with a thermostat at temperature $T$, the system obeys the Boltzmann distribution Eq.  and we have:

$$p[r_1,r_2,\dots,r_N] = \frac{1}{Z} e^{-E(r_1,r_2,\dots,r_N)/k_{\rm B}T} = \prod_{i=1}^N \frac{1}{Z_i} e^{-\epsilon_{r_i}/k_{\rm B}T},$$ (4.5)

where the last equality comes from  . Since the particles of gas are all identical the single particle partition functions $Z_i$ are all identical. In addition, we have $e^{-E(r_1,r_2,\dots,r_N)/k_{\rm B}T} = \prod_{i=1}^N e^{-\epsilon_{r_i}/k_{\rm B}T}$, and so the partition function $Z$ of the whole system is given by:

$$Z = (Z_1)^N,$$ (4.6)

with $Z_1$ the single particle partition functions:

$$Z_1 = \sum_{r} e^{-\epsilon_{r}/k_{\rm B}T},$$ (4.7)

and the sum running over the full list of available single particle microstates.