Conditional pair distributions in many-body systems: Exact results for Poisson ensembles

We introduce a conditional pair distribution function (CPDF) which characterizes the probability density of finding an object (e.g., a particle in a fluid) to within a certain distance of each other, with each of these two having a nearest neighbor to a fixed but otherwise arbitrary distance. This function describes special four-body configurations, but also contains contributions due to the so-called mutual nearest neighbor (two-body) and shared neighbor (three-body) configurations. The CPDF is introduced to improve a Helmholtz free energy method based on space partitions. We derive exact expressions of the CPDF and various associated quantities for randomly distributed, noninteracting points at Euclidean spaces of one, two, and three dimensions. Results may be of interest in many diverse scientific fields, from fluid physics to social and biological sciences.

Figure 1

Left: Example of a point pattern with a characterization of the evaluation of the pair distribution functions $g\left(\omega \right)$, $g_v\left(\omega \right)$, and $g_{v{v}^{\prime }}\left(\omega \right)$. The point $P^{\prime }$ lies in the surface of the spherical volume $\omega$ centered at the reference point $P$. Nearest neighbors of $P$ and $P^{\prime }$ lie in the surfaces of spherical volumes $v$ and $v^{\prime }$ centered in $P$ and $P^{\prime }$, respectively. Right: Representation of the four classes of ($v,v^{\prime }$) pairs (see text). Here, $v_{+}\equiv \max (v,v^{\prime })$ and $v_{-}\equiv \min (v,v^{\prime })$.

Figure 2

Contact values $g\left(a^{+}\right)$ of the PDF for attractive hard spheres in the TSP formalism based on Eq. (3.11) (solid lines) are compared with exact values in the limit of zero density (triangles), Monte Carlo simulations [52] (circles), results of a perturbative theory [53] (dashed lines), and results derived from Eq. (3.9) (dotted lines). Here $T^{*}=kT/\epsilon$ is the reduced temperature, $\rho =n{d}^3$ the reduced number density, and $\lambda$ the potential range in units of the particle diameter $d$ [i.e., $\xi =a({\lambda }^3-1)$ and $a=4\pi d^3/3$ in Eq. (D1)].

Figure 3

Pairs $(v,v^{\prime })$ showing various geometrical quantities involved in the evaluation of the CPDF $g_{v{v}^{\prime }}\left(\omega \right)$ for Poisson ensembles in one and two dimensions.

Figure 4

Shared neighbor configuration for $D\ge 2$.

Figure 5

Evaluations of $g_{v{v}^{\prime }}\left(\omega \right)$ at dimensions $D=1,2,3,\infty$ for $(nv,n{v}^{\prime })=(0.2,0.2)$ (dotted lines) and $(nv,n{v}^{\prime })=(1,0.5)$ (solid lines).

Figure 6

Contributions $g_{v{v}^{\prime }}^{\left(iii\right)}\left(\omega \right)$ and $g^{\left(iv\right)}\left(\omega \right)$ (dashed and solid lines, respectively) to $g_{v{v}^{\prime }}\left(\omega \right)$ at $v_{+}<\omega <{\omega }_{v{v}^{\prime }}$, for $D=1,2,3,\infty$, and for the two ($v,v^{\prime }$) conditions shown in Fig. 5.

Figure 7

Contributions due to (ii$b$), (iii), and (iv) pair configurations (solid, short-dashed, and long-dashed lines, respectively) to the CPDF $g_v\left(\omega \right)$ as a function of $v$ for one-dimensional Poisson ensembles and different distances as measured by $\omega$ (values $n\omega$ indicated on the plots). The total CPDF is shown by the dotted line.

Figure 8

Same as Fig. 7, for $D=2$.

Figure 9

Same as Fig. 7, for $D=3$.

Figure 10

Same as Fig. 7, for $D=\infty$.

Figure 11

Contributions $g^{\left(\nu \right)}\left(\omega \right)$ due to $\nu =$ (i), (ii), (iii) and (iv) pair configurations (solid, short-dashed, long-dashed, and point-dashed lines, respectively) to the PDF $g\left(\omega \right)$ vs the volume $\omega$ for Poisson ensembles at dimensions $D=1,2,3,$ and $\infty$. The total PDF is shown by a dotted line.