Statistical mechanics of a mixture that can form clusters

Question

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).

Answer 1

This sounds like a perfect case for SAFT theory, although its applicability for your specific system will depend on some details (interaction type and strength, density etc).

I am especially familiar with Wertheim's Thermodynamic Perturbation Theory (a specific version of SAFT theory), in which the system's free energy is split in a reference contribution (which should be known before hand) and a bonding one, which accounts for cluster formation. This theory works very well for particles decorated with sites that can attract each other (e.g. patchy particles).

If the reference system is purely repulsive and the attraction is only due to the bonding term, then an output of the theory is the probability that a site is involved in a bond. Under some assumptions (that also depends on the system at hand) you can use this quantity to estimate the cluster size distribution of the system (see for instance this paper).

Answer 2

You can use the same formalism developed for chemistry (which includes the Saha equation). You need to input by hand what are the possible clusters (this includes individual particles which are a trivial example of clusters). You’ll also need to include the possible reactions that can occur between them (how they fuse/dissociate). Since this is a phenomenological approach, this is not deduced ab initio but comes from experience, intuition or measurements.

You then apply statistical physics. The simplifying approximation is to neglect the interaction between the clusters. You therefore reduce the problem to a non interacting one up to the prescribed reactions. Quantitatively, you’ll need to know the energy of each cluster and its entropy (its chemical potential). From there, you can deduce a laws of mass action which will determine the equilibrium distribution of clusters.

Hope this helps.