Draft:Path distribution function

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The path distribution function describes the distribution of minimum distances between pairs of strings contained within a given volume. The path distribution function (PthDF) is analogous to the Pair distribution function (PDF), with the exception that it operates on one-dimensional paths as opposed to zero-dimensional points.

Definition

Mathematically, if ${\vec {\gamma }}_{i}(t)$ and ${\vec {\gamma }}_{j}(t')$ are the parametric coordinates of a point on strings with indices $i$ and $j$ respectively, the euclidean distance between two points, one on each string is given by:

$\xi _{ij}=||{\vec {\gamma }}_{i}(t)-{\vec {\gamma }}_{j}(t')||$.

The minimum separation between two strings with indices $i$ and $j$ at a parametric position $t$ on string $i$ is then:

$s_{ij}(t)=\min\{\xi _{ij}|t'\in {\mathbb {R} }\land {\vec {\gamma }}_{j}(t')\in {\mathcal {N}}\}$,

where ${\mathcal {N}}$ are the set of all points within the volume. The Path Distribution Function (PthDF), g(r), which describes the distribution of minimum distances between pairs of strings within the volume can be written as:

$g(r)=\sum _{i}^{N}\sum _{j\neq i}^{N}\langle \delta (s_{ij}(t)-r)\rangle$.

History

The PthDF was introduced by Marcus Newton, a British physicist and author. It was first used to describe the distribution in three-dimensions of topological strings in a multiferroic nanocrystal. Its introduction served as an intuative alternative to the pair correlation function approach previously used to describe the distribution of topological strings.