Main results
Common sense prompts us that if a body is hardly seen, then it is not large. Statistical physics has a less obvious criterion of large bodies: a body is large or macroscopic if it consists of a large number of molecules. According to this view, both air in a football and an ultracold gas of atoms, trapped in a micrometer- sized chamber, are large bodies: the air has 1019 molecules per each cubic centimeter, and the quantum gas in magnetic traps consists of about 105 atoms.
The reason why both bodies are physically large is that both appear as clouds smoothly varying along the spatial coordinate x, so that their state is naturally described in terms of their local density n(x), temperature T(x), and velocity V(x). We see that in this macroscopic view on a body, which is called hydrodynamics, instead of the information related to individual molecules, one is interested only in the statistics resulting from their large number. In the project, this statistical physics approach has been applied to a quantum gas at practically zero temperature and to a classical “liquid” of hard disks.
As classical hydrodynamics (CH) is developed much more comprehensively than quantum hydrodynamics (QH), one task was to use the ideas of the first to advance the second. While CH is derived from the microscopic (molecular) theory and has a universal form, the known QH equations for ultracold gases (T = 0) differ for different interactions and dimensions. Moreover, QH is derived merely from CH by analogy, which for a long time was awaiting “a careful examination.” In this project, I have derived a general QH equation from the microscopic quantum Schrödinger equation. The obtained hydrodynamic equation has a universal form (any interaction and dimension) and in specific settings reduces to all known QH equations, but also has a novel term.
This term describes the interaction between the macroscopic velocity of a quantum gas and its microscopic excitation momentum: thus, the microscopic and macroscopic states can interact! This QH opens new perspectives in the physics of ultracold quantum gases, and its predictions will be investigated in the near future.
Ultracold quantum gases are often obtained in a tube-like, almost one-dimensional (1D), so-called quasi-1D (q1D) geometry. As the simplest but very important interaction of atoms is close to that of tiny hard spheres, the idea was to find a possible connection between this quantum gas and a classical q1D hard sphere system. However, it turned out that at a dense hard sphere packing, at which a classical q1D system becomes interesting, the quantum gas turns into a crystal.
Nevertheless, in the course of the project, I developed the thermodynamics of a 2D hard disk liquid model, which is conceptually similar to the 3D hard sphere model, but much simpler (plane against volume!). These models have been fundamental for molecular physics since 1873, when van der Waals discovered that molecules have a finite size. This means that molecules can move only in the free volume Vf, which is the total volume minus the excluded volume they occupy altogether.
This Vf determines the system’s entropy but has a very complex shape (imagine empty space left between disks densely covering a plane!) and its determination remains a challenge even in numerical simulations. In the project, I derived an exact analytical formula that determines the Vf for given disks’ coordinates and found the entropy and pressure as functions of the disk density. Along with my Ukrainian co-workers from the L’viv Institute for Condensed Matter Physics, we showed that the theory very well describes the known numerical data.
The presented theory can be generalized for more realistic systems, which, in addition to a hard core repulsion, also have a long-range interaction. As the main problem in liquid state theory was the incorporation of molecules’ hard cores, the presented result can considerably advance the statistical physics of dense molecular systems.
