I am not sure how to look for references for the following problem. Imagine having a collection of 2 kinds of particles of density $n_1$ and $n_2$ that can arrange themselves in clusters (e.g., something like protons and neutrons that can arrange themselves in nuclei). For a given temperature and chemical composition (i.e., given values of $n_1$ and $n_2$ in a specified fixed volume) there should be a statistically well-defined fraction of particles that are not bound in any cluster and a distribution of clusters (possibly of different sizes and composition): which are the methods or approximations that have been developed to study this kind of systems?

Starting from the Hamiltonian of the system (something like $K_1+K_2+V_{11}+V_{22}+V_{12}$, where $K_i$ is the kinetic term for the $i$-species and $V_{ij}$ is the 2-body potential between $i$-species and $j$-species particles, $i,j=1,2$), one could in principle calculate the configurational probability of a microstate (Gibbs weight) or the partition function. However, this "ab-initio" approach seems too general and difficult to be useful. Is there any method or approximation that may be used to find the distribution of clusters? Are there any criteria to assess if the system will be "homogeneous" (i.e. a gas/liquid of homogeneously mixed particles of the two species) or a collection of clusters?

Note: the cluster expansion is a cool and useful method but it's not directly related to the "clusters" I am considering here... or at least I see no direct connection. Somehow, I have the feeling that this kind of problem is more similar to the problem of ionization leading to the Saha equation (some electrons can be free and others in bound states).

I am not sure how to look for references for the following problem. Imagine having a collection of 2 kinds of particles of density $n_1$ and $n_2$ that can arrange themselves in "clusters" (e.g., something like protons and neutrons that can arrange themselves in nuclei). For a given temperature and chemical composition (i.e., given values of $n_1$ and $n_2$ in a specified fixed volume) there should be a statistically well-defined fraction of particles that are not bound in any cluster and a distribution of clusters (possibly of different sizes and composition): which are the methods or approximations that have been developed to study this kind of systems?

Starting from the Hamiltonian of the system (something like $K_1+K_2+V_{11}+V_{22}+V_{12}$, where $K_i$ is the kinetic term for the $i$-species and $V_{ij}$ is the 2-body potential between $i$-species and $j$-species particles, $i,j=1,2$), one could in principle calculate the configurational probability of a microstate (Gibbs weight) or the partition function. However, this "ab initio" approach seems too general and difficult to be useful. Is there any method or approximation that may be used to find the distribution of clusters? Are there any criteria to assess if the system will be "homogeneous" (i.e. a gas/liquid of homogeneously mixed particles of the two species) or a collection of clusters?

Note: the cluster expansion (see e.g. this question) is a cool and useful method but it's not directly related to the "clusters" I am considering here... or at least I see no direct connection. Somehow, I have the feeling that this kind of problem is more similar to the problem of ionization leading to the Saha equation (some electrons can be free and others in bound states).