Solve the hard-disk partition-function problem

Derive an exact solution for the partition function, and hence the entropic contribution, of a hard-disk system.

Background

The analytical treatment assumes that the partition function of the rough-particle system can be factorized into a hard-disk contribution and a factor associated with orientational degrees of freedom. The authors note that obtaining the hard-disk contribution itself is a longstanding unresolved problem, so their predictions rely on existing approximations and on the geometric calculation of the effective particle size.