Solvable models of quasi-one dimensional systems of interacting hard spheres and their implication to the quantum ultracold gases

For different reasons, both classical and quantum one- and quasi-one dimensional hard core gases have recently been in the focus of intensive studies. While to incorporate the intermolecular interaction in the case of an ultracold quantum gas is a formidable problem, the recent exact results [1,2] give the tools to do so in a classical gas of hard spheres. The main objective of the proposed project is to analytically study classical quasi-one dimensional hard sphere systems bearing in mind their possible implications for ultracold quantum gases and a possible connection to their hydrodynamic and mean field approaches. The aim of the project is to develop analytically solvable models of interacting classical HSs in narrow quasi-one-dimensional channels of different transverse geometries, solve these models, describe the orientaional and translational ordering, and establish ensuing phase diagrams. We will consider the simplest plane quasi-one-dimensional channel of hard disks with a next neighbor interaction and a channel with a square cross-section with and without interaction. At high densities, hard spheres tend to form an arrangement in the form of a crystalline zigzag and its transformation into a liquid state is the main ordering process. The geometry with the square cross- section offers two equivalent diagonal planes for such a zigzag and the system eventually must spontaneously choose just one. Thus, although phase transitions are not possible in one dimension, such spontaneous symmetry breaking in a quasi-one-dimensional geometry is expected to give rise to a liquid-solid phase transition. The type of this transition will be established. The pair correlation functions will be, when possible, derived analytically and otherwise studied numerically and the process of ordering will be studied in detail. We will especially focus on the dipole-dipole interaction which is known to result in dropletlike structures in the quantum gases near their Bose condensation. A quasi-one-dimensional microcanonical classical model that incorporates the dipole-dipole interaction beyond the mean field level will be developed and studied aiming at finding inhomogeneous dropletlike states. It is expected that the results of these classical systems can be used to construct macroscopic hydrodynamic and mean field approaches to quantum droplets and predict novel states of an ultracold quantum gas.