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What the Wormulator calculates
This online tool calculates end-to-end statistics for polymers -- that is, the likelihood of finding its two ends a given distance apart, and/or pointing in such-and-such direction relative to each other. The model it uses is the so-called 'wormlike chain' model of a polymer, where the polymer bends essentially according to Hooke's Law. Due to molecular bombardment the polymer writhes and twists in solution, and its conformation (configuration) at any given instance can be determined probabilistically. And all of the information about the likely conformations can be determined by these end-to-end statistics.
The way we use this calculator is to 1) pretend that we are at one end of the polymer (how it points is something we specify), and 2) ask what the chances are of finding the other end at a given location, or pointing a given way, or both. In other words, what we calculate is the following probability density:
Here the displacement of the second end is represented by R, the direction the polymer points (following the polymer from beginning to end) is u, and the amount that it twisted is T. L is the length of the polymer. The units of the distribution are given by the three quantities to the left of the vertical bar: it is a probability per volume in R, per solid angle in final tangent uf, per twist angle T. The quantities to the right do not affect the units since they are the 'initial conditions' at the starting end of the polymer.
Other reduced distributions can be obtained by summing or averaging over the quantities in the full distribution. For example, if we sum the distribution over all displacements we get a probability of the tangents and twists having certain values, but there is no dependence on R. This distribution would be written
Likewise, we can sum over the final tangent and relative twists. The initial tangent is not a degree of freedom in the distribution, but rather a condition on the distribution, so we can't integrate over it, but the program does allow us to average the distributions that differ only in their initial tangent.
This online calculator calculates these various P(...) distributions, only at a single point (i.e. at one value of R, uf, and T), unless we use the counter fields at the bottom of the page in which case we can evaluate many points at once. All the parameters of the distribution appear at the top of the page before the Calculate button (most of them in the "Chain geometry" section). The material properties of the polymer are under "Chain parameters". The program has several ways of calculating these statistics, which can be checked and unchecked above the Calculate button.