Institute of Physics, National Academy of Sciences of Ukraine, Kyiv, Ukraine and Center for Theoretical Physics, Polish Academy of Sciences, Warsaw, Poland
The model of hard spheres (HS) is as fundamental for the physics of simple liquids, as Ising's model is for the magnetism. However, it has not been solved exactly in 2 and 3 dimensions. For this reason,the problem has been approached both from high (even infinite) dimensions (where the mean field is applicable) and from reduced dimensionality such as thin pores. In 1936 Tonks obtained an exact solution of the one-dimensional HS system. Here I present the exact solution of a quasi-one dimensional (q1D) systems of hard disks (HD) in a narrow pore and consider its possible implications for the scenario of melting of the q1D crystalline zigzag. The canonical partition function of a q1D HD system, the longitudinal and transverse pressures, distributions of HD across the pore, and the correlation functions, which have so far been computed from molecular dynamics simulations, are derived analytically. It is found that, as density decreases, melting of the dense crystalline zigzag proceeds via creation of a progressively larger number of windowlike defects through which HDs can exchange their position across the pore thus gaining some entropy. This shows certain similarity to a Kosterlitz-Thouless (KT) transition that proceeds via continuous dissociation of defects pairs into free defects. I briefly review the Kosterlitz-Thouless scenario of melting of 2D crystals and discuss the melting of a q1D HD zigzag in the light of this scenario.