Threading a Ring or Tube onto a Rod: An Entropically Rare Event.

We calculate the entropy lost when a circular ring or a circular cylinder or tube is threaded onto a long rod in terms of the geometrical parameters, namely rod length and radius, the threaded ring radius, or the radius and length of a threaded tube. These formulas, constructed from a partition function, allow calculation of the fraction of rings/tubes threaded spontaneously onto rods. In all cases of practical interest, this fraction is very small and can be well represented by simple power laws depending strongly upon the geometrical parameters.