^{*}

The author has declared that no competing interests exist.

Conceived and designed the experiments: TH. Performed the experiments: TH. Analyzed the data: TH. Contributed reagents/materials/analysis tools: TH. Wrote the paper: TH.

We compute the singular value decomposition of the radial distribution function

To date, there have been many numerical calculations of the radial distribution function (RDF) and equations of state for discrete potentials

Since the Fourier transform of

Our approach is similar to the work of di Dio, et. al.

The first is that the decomposition is very clean, in that the singular values are well separated from each other when considering variations in both density and attractive strength. This leads to a reduced representation of the RDF across the parameter range. However, variations of the SW length did not permit separation into a small number of components, consequently this parameter did not lend itself to a reduced representation.

Secondly, we note the dependence of each basis vector as a function of the system parameters was well described by a low-order polynomial. The degree of this polynomial increased linearly with the rank order of the basis vector. This means that a reconstruction of the RDF for any interpolated value of packing fraction or attractive strength can be expressed by a set of polynomial coefficients, a vector of singular values, and a small set of vectors describing the basis functions. This compact representation along with a program to reconstruct the RDF is given in the Supplementary Information.

For a given system, we compute the radial distribution function

The packing fraction ^{4} equal time intervals. Since temperature is irrelevant in a HS DMD simulation, a single time step corresponded approximately 60 collisions per particle at the largest packing fractions and approximately

For hard spheres the system is completely parameterized by the packing fraction

Let

If there is a sharp decay in the spectrum of singular values, in that

The measurement of the RDF in a canonical ensemble becomes problematic in the presence of a phase separation due to finite size boundary effects. We therefore restrict the parameters to regions containing only a single liquid or vapor phase. We note that it should be possible to perform the simulation under a grand canonical ensemble. The so-called Gibbs ensemble method popularized by Panagiotopoulos

We sampled

HS, SW_{1}, SW_{2} refers to the hard sphere, single and double square well systems respectively. The asterisk indicates that the system was run with

In _{10} of the residuals from the polynomial fits, which we report for the more accurate HS

Results are identical, but less accurate, for the smaller

The polynomial fits to the vectors are shown as dashed lines.

In

Note that within the plotting resolution the lines are nearly identical. Left Inset: To illustrate the failure of the basic integrals closures, Percus Yevick (blue), Hypernetted chain (green), and Roger-Young (orange), we plot the predicted RDF calculated from

We explored a variety of parameters for the square well, varying packing fraction SW

The vertical and horizontal dashed lines indicate the variation for SW

avg. error | |||||

HS |
- | - | 2^{12} |
.0064 | |

HS |
- | - | 2^{13} |
.0046 | |

SW |
1.0 | 1.25 | 2^{12} |
.0093 | |

SW |
1.25 | 2^{12} |
.0050 | ||

SW |
1.0 | 2^{12} |
.0350 | ||

SW |
1.25 | 2^{12} |
.0086 | ||

SW |
(−1.0, −1/2) | (1.125, 1.25) | 2^{12} |
.0063 |

The singular values for the SW system, shown in

Since a SW is often a first-order approximation to a continuous potential, a double square well is the next logical step in the approximation. The system SW

The basis vectors, which are given in the Supplementary Information, are similar to the HS except for the presence of discontinuities at the potential boundaries. The coefficient vectors also show the same general reduction to a polynomial of degree

The static structure factor for an isotropic fluid can be calculated by a Fourier transform of the RDF,

While useful because of their compact representation, it is unclear if the exact polynomial representations of the coefficient vectors hold any special significance. Although they bear a resemblance to the radial Zernike polynomials

In theory, the basis functions themselves may yield qualitative insight, especially when the singular values are well separated. In the case of the RDF, it is possible that the basis functions are related to an expansion of the Ornstein-Zernike equation. For example, we find that the largest basis vector

Preliminary work on both binary mixture and polydisperse hard sphere fluids show the same sharp separation of the singular values. These systems and other simple continuous potentials are being currently studied in more detail. For HS and SW systems we have shown that the reduced decomposition allows for the RDF to be reproduced with extremely high accuracy over all interpolated values of

Source code to compute the radial distribution profiles for HS and SW can be found at