# Constrained partition function(s) of a multi cluster system

Can a system have multiple partition functions?

I am trying to find the partition function of two cluster system with specific constraints. For the sake of brevity, I am consider a simple case. The problem is based on the decoration of atoms, say a binary system consisting of A and B type atoms.

Case 1 : Single cluster system

Consider an equilateral triangle cluster and trying to decorate it with A and B atoms. This results in 4 possible configurations, namely, AAA(1), AAB(3), ABB(3), BBB(1) clusters. Each cluster would be having different energy. Also consider that each site is definitely occupied by one of the atoms.

The partition function for this system can be written as $$Z=e^-{\frac{\epsilon_{AAA}}{k_B T}}+3e^-{\frac{\epsilon_{AAB}}{k_B T}}+3e^-{\frac{\epsilon_{ABB}}{k_B T}}+e^-{\frac{\epsilon_{BBB}}{k_B T}}$$ where $$\epsilon_j$$ repersents the energy of cluster with the configuration $$j$$. The probability of finding a specific configuration $$j$$ for the given average composition of the alloy at the given temperature can be found as $$y_j=n_j/N=e^-{\frac{\epsilon_{j}}{k_B T}}/Z$$ where $$N=\sum_j n_j$$. Based on the constraints mentioned above, each site of the cluster is occupied by one of the atoms, $$\sum_j y_j=1$$. Thus, the sum of probabilities of the cluster configurations is 1, which is one of the basic conditions to be satisfied for each cluster chosen.

Using these triangle cluster probabilities, the probability of finding a specific pair configuration can be be calculated using super position rules.

Case 2: Two cluster system without common (sub)clusters

Consider the extension of case 1. For a system composed of two clusters say an equilateral triangle and a pair cluster (this pair cluster is not part of triangle cluster we considered. Like equilateral triangle with side of $$a\sqrt{3}/2$$ and the pair cluster with a length of $$a$$). In this case, the energies of the system to be considered for writing the partition function are $$\epsilon_{AAA}, \epsilon_{AAB}, \epsilon_{ABB}, \epsilon_{BBB}$$ for the triangle cluster and $$\epsilon_{AA}, \epsilon_{AB}, \epsilon_{BB}$$ arising from the pair cluster. Since each site of the triangle as well as pair cluster is occupied by one of the atoms, the sum of probabilities of the cluster configurations should be 1

$$\sum_{j\in tri} y_j=1$$ and $$\sum_{k\in pair} y_k=1$$. In addition, the average composition calculated using both the triangle cluster probabilities and pair cluster probabilities should be same. For this specific case, $$n_{AAA}+2n_{AAB}+n_{ABB}=n_{AA}+n_{AB}$$.

Under these constraints, what could be the partition of the system and $$y_j \& y_k$$? What would it be if there are some common subclusters?

This problem appears similar to that of case-1, case-2 and case-3 with a constraint. However, I am not sure how to introduce the constraints on the cluster probabilities!

further details

Based on the comment of @dgamma, I would like to add a few more details. Consider the composition of the alloy to be equi-atomic. i.e. equal number of A and B atoms for a binary system and the system is composed of equilateral and pair cluster as discussed in the case2. For the chosen energy set, based on the temperature, the system tries to have different proportions of atomic configurations on the triangle cluster and pair clusters such that the system average composition of the alloy on the triangle and pair cluster is same and equal to be equi-atomic. Under these conditions, can the partition functions of the clusters be independent? One of the basic properties of the cluster probabilities expressed through partition functions for a single cluster system is normalization. The normalization conditions should be satisfied for both triangle cluster and pair clusters independently. Is it possible with a single partition function? Can a $$\delta$$ function be helpful such that the cluster probabilities can be distinguished between the clusters?

• I'm having trouble totally understanding the question and motivations for this so I might be off -- but if your two systems (triangles and clusters) are independent, then why wouldn't the total partition function just be the product of the partition functions? Commented Nov 29, 2023 at 5:35