On necklaces with hard non-spherical beads

Abstract

In spite of the fact that our world is three-dimensional, one-dimensional statistical models are important for a number of reasons. First, in some cases they are exactly solvable while their higher-dimensional counterparts are not. Consequently, the one-dimensional solutions may provide some insight into basic important mechanisms and useful approximation schemes. Then there are many instances where the description of a physical system as one-dimensional, or even better as quasi-one-dimensional, is quite realistic. In the present article two families of quasi-one-dimensional systems are presented. Although the two families are constructed to describe different physics, their basic mathematical structure is similar. The first family takes into the account the effect of hard core repulsion of non-spherical particles. It is prototyped by a necklace of hard non-spherical beads that are free to move along the necklace and to rotate about it. The second family takes into account the possibility of squeezing of hard objects, because the system is not strictly one-dimensional. This is prototyped by a system of hard objects confined to a narrow pipe with a diameter which is somewhat larger than the size of the objects but not large enough to allow changing the order of the objects along the pipe. The general treatment of such models is discussed and the exact configurational free energy is worked out for a specific example.