Figure 1 . Potential well as a function of radius.

Using the data sets from equations of (EOS) and the Indiana Silicon Sphere (ISiS) multifragmentation experiments, Elliott et al. (2002) showed that it is possible to construct a phase diagram for nuclear matter. Figure 2 shows such a phase diagram for the nucleus of a krypton atom (left), which shows similar behavior to the phase diagram of an ensemble of krypton atoms (right).

Figure 2. Nucleons in a krypton nucleus interacting via the strong nuclear force (EOS experiment at LBNL) (left); and (right) krypton atoms in a gas interacting via the electromagnetic interaction. p and pc are the density and critical density, respectively. Likewise, T and Tc are the temperature and critical temperature, respectively.

In understanding how nuclei exhibit such an uncanny resemblance to that of regular matter, one must investigate the conditions under which such phase transitions occur. In the 1960s, Michael E. Fisher proposed the “Fisher’s Droplet Model,” which provides a general formula for the concentration of clusters of non-interacting constituents near a system’s critical point (Fisher 1967, 1969). Fisher’s model, which is valid for many systems, gives a parameterization for the number of the of clusters n_A of size A in a vapor at temperature T, which depends on bulk and surface energies, as well as a topological term, which is based on the geometry of the clusters on a lattice:

where is the topological part of the formula, τ is a topological exponent, Δμ is the distance from coexistence of liquid and vapor, c₀ is the surface energy coefficient, ε is defined to be (T_c-T)/T_c, where T_c is the critical temperature, and σ is the surface to volume exponent.

Fisher arrived at this expression by approximating the grand partition function of a droplet as:
(1)
where is the configurational partition function of a cluster of size A at temperature T (in natural units), and z is the fugacity of the system: . The partition function of a cluster depends on the free energy of the cluster, , where the free energy is so that .

The energy of a droplet at a fixed configuration is given by:
(2)
where E₀ is the binding energy per constituent, A is the number of constituents, s is the surface area, and ωs is the lost energy by constituents near the surface. The surface area s for the most probable surface can be expressed as:
(3)
Here a₀ is some geometric constant of proportionality and σ is a critical exponent of the surface to volume. At low temperature, where the droplets are compact, σ = 1 – 1/d, where d is the Euclidian dimension of the system. For example, for d = 2, σ=1/2, and, for d = 3, σ=2/3.

Fisher estimates the entropy of a cluster as , a combinatorial factor, which describes the number of ways a cluster of a given size can exist, being proportional to the number of configurations of A identical constituents, and having the following form:
(4)
Here is related to the limiting entropy per unit of cluster surface. Here τ and σ are critical exponents.

One can now write the configurational partition function of a droplet as follows:

(5)

We can re-write the partition function as:

(6)

Now, the grand partition function becomes:

(7)

Here µ is the chemical energy. The pressure of the system can be then obtained as follows:

(8)

The distance from coexistence is given as . At T = T_c, the surface energy vanishes so that ω = ωT_c. We can re-write at T = T_c as , and then call the zero-temperature surface-energy coefficient and .

Assuming an ideal gas, p = TN/V, Equation (8) shows that the concentration of droplets with A constituents is:

(Fisher’s Formula) (9)

In this formula q₀ is a proportionality constant.

In the case of percolation, Δμ= 0. Percolation is a mathematical model that can be used to investigate the spread (or flow) of an entity on a lattice. From studying epidemics to phase transitions, percolation effectively describes various stages of the flow, particularly at a critical point in which the system makes a transition from one phase to another.

When percolation occurs, it is a standard practice to replace T with q, the site occupation probability. The FDM then becomes:

(10)

Where ε= (q_c-q)/q_c, and q_c is the critical probability of the percolation occurring.

While the energetic term in the Fisher's equation is very well understood thermodynamically, the origins of the entropic term are less well understood. Fisher theorizes the entropic term to have a particular functional form based on the combinatorics of clusters. One way to gain insight into this part of the Fisher's formula is to divide the experimental fragment yields by the energetic factor leaving only the entropic portion. For clusters on a two-dimensional lattice, we can then compare the result against well-known enumerations of the combinatorics, e.g. that of the self-avoiding polygons (SAPs). Self-avoiding polygons are a class of geometric polygons none of whose sites can step over itself.

In this study, we attempt to examine Fisher's functional form of the entropic factor by using a standard d = 2 site percolation on a square lattice. The goal is to explain a term whose origin is purely physical using only a geometric model. This can give us some insight into understanding a part of the Fisher's theory, which would otherwise be extremely hard to understand directly.

By taking advantage of the Hoshen-Kopleman algorithm, in which all the sites of a given cluster on the lattice are assigned a unique label, many of the observables, such as the size of the clusters, the number of the clusters of the given size, the number of holes in each clusters and their average sizes, in the simulation were computed using several parallel one-dimensional arrays of integers which took the label of each cluster as their index. For counting the perimeter of each cluster, the program used a neighborhood operation: when visiting each site on the lattice, if the site was occupied, thus belonging to a cluster, the program would store the label of that site to count the perimeter of the cluster with that label. The program would then re-traverse the entire lattice and find all the sites which belonged to the cluster with that label. Each site belonging to the cluster with a given label was assumed to have a perimeter of four. Then by looking at its four neighbors, the perimeter was decremented for each occupied neighbor. The result would then be added to a perimeter array whose index was the label of the visited site.

For counting the number of holes, the program employed a novel algorithm in which the Hoshen-Kopleman algorithm was reused. As with the counting of the perimeter, each site on the lattice was visited and if the site was occupied the label of that site was stored. The program then re-traversed the entire lattice to search for all the sites belonging to the cluster with a particular label. Upon visiting a site that belonged to the cluster in question, that site was turned off on a parallel lattice whose sites were all turned on initially. When all the sites of the cluster in question were examined, the result was a “negative image” of the cluster on the parallel lattice. That lattice would then be processed using the Hoshen-Kopleman algorithm. In this manner one can extract all the information that is accessible for the clusters for the holes inside the clusters, including the average size of the holes in a cluster as well as the internal perimeter of the cluster (the total perimeter of the holes).

At each realization of the lattice with site occupancy probability q, a data file was written that contained for every cluster size A, the perimeter P, the number of holes, and the average size of holes of that cluster. The raw data was then fed to the Physics Analysis Workstation (PAW) to take the averages of the cluster yields n_A(A,P,q) (from Equation 10).

Figure 3. Site percolation on a square lattice at different probabilities for lattice size = 50.

The preliminary analysis on the fragment yields produced the following two graphs: Figure 4 illustrates the fragment yield distribution scaled by the power law pre-factor against the inverse probability scaled by Fisher’s parameterization of the surface energy for clusters of various sizes. The cluster size of each range is noted by a different symbol for the range of probability 0.2 < q`< q<sub>c</sub>` . Here q` = 1 – q, so that the lattice behavior as a function of is reminiscent of a fluid’s behavior as a function of T. In this plot σ, τ and q_c are left to be free parameters.

Figure 4. Clusters of 10≤A≤20 and q ≤ qc collapse to a single curve by plotting nA(q)/q0A-tVs. Ase/q.

Figure 5 is a plot of the combinatorics g(P,A) versus the perimeter of clusters P for various cluster sizes. The dashed line in this figure denotes the ratio of ω=c/q_c, which is obtained from dividing the theoretical value of the surface tension c by the theoretical value of the critical probability for an infinite square lattice.

Figure 5. A plot of the combinatorics vs. the perimeter for various cluster sizes.

One might notice that the small clusters have a better collapse in this figure, particularly for the cluster sizes 10 = A = 20. This may have been caused by the poor statistics for the larger clusters. As can be seen in Figure 3, large clusters are generally formed at probabilities greater than q_c. Also, the data represented here is for only 2000 realizations, which results in poor statistics for the probability regions of q > q_c.

In Figure 5, one can again notice the lack of statistics for the largest clusters. Here, the g(P), which represents the combinatorics summed over all cluster sizes and Fisher’s expression, overlaps up to perimeter of 30 or so and then deviates from the dashed line and finally flattens out. Flattening of the combinatorics curve is indeed an indication of low statistics. Despite the low statistics for large clusters, Figure 5 confirms the form of the combinatorial predicted by Fisher.

The next step in this analysis is to find out what types of combinatorics on a two-dimensional lattice correspond to the clusters generated on a square lattice in site percolation. Once we can understand the entropic form of Fisher, we can apply this understanding to nuclear data to gain some insight to the density of states.