# Partition function for a bidisperse packing of soft-core particles

My goal is to study the nematic order in bidisperse packings of soft-core spheroidal particles of identical aspect ratio with harmonic repulsion from a statistical mechanics point of view by adapting Onsager's hard rod model. I found out this was performed for monodisperse packings of hard-core spheroidal particles in Physical Review E 96, 022704 (2017), available here for free or on the APS website through paywall.

For a monodisperse packing of $N$ particles, we can consider that these are indistinguishable, therefore with $\mathcal{V}_{ij}(\vec{r}_{i},\vec{r}_{j}\in\mathbb{R}^3,\hat{\textbf{I}}_i,\hat{\textbf{I}}_j\in\mathbb{S}^2)$ the interacting potential between particles $i$ and $j$ — which depends on their respective positions and orientations — we can write the partition function to a multiplicative constant depending on kinetic parameters $$Z = \frac{1}{N!} \int\ldots\int d^{3N}\vec{r}~d^{2N}\hat{\textbf{I}} ~\exp\left(-\frac{1}{k_BT}\sum_{i<j}^N~\mathcal{V}_{ij}(\vec{r}_{i},\vec{r}_{j},\hat{\textbf{I}}_i,\hat{\textbf{I}}_j)\right)$$

If I now consider a packing of $N/2$ "small" and $N/2$ "big" particles, the potential $\mathcal{V}_{ij}$ will depend on the sizes of both interacting particles. How can I then adapt the partition function?

Thank you very much for any idea!

There is no difference, apart from the fact that instead of a factor $N!$ at the denominator you will have a factor $(N/2)! \cdot (N/2)!$.
$$Z \propto \frac 1{\left(\frac N 2!\right)^2} \int \dots \int d^{\frac{3N}2} \mathbf r_a \cdot d^{\frac{3N}2} \mathbf r_b \cdot d^{N} \cdot \mathbf{\hat{I}}_a \cdot d^{N} \mathbf{\hat{I}}_b \cdot\exp\left(-\frac{1}{k_BT}\sum_{i<j}^N~\mathcal{V}_{ij}(\mathbf{r}_{i},\mathbf{r}_{j},\hat{\textbf{I}}_i,\hat{\textbf{I}}_j)\right)$$
where $\mathbf r_i,\mathbf r_i \in \{\mathbf r_a,\mathbf r_b\}$ and $\mathbf{\hat{I}}_i,\mathbf{\hat{I}}_j \in \{\mathbf{\hat{I}}_a,\mathbf{\hat{I}}_b\}$.