Application of geometrothermodynamics to the two-dimensional systems: Ideal Bose-Gas and system with strong interaction

Abstract

In the framework of the method of geometrothermodynamics we studied the properties of equilibrium manifolds of the following thermodynamic systems: a two-dimensional Bose gas, a Berezinsky-Kosterlitz-Thouless system. The results are invariant under the Legendre transformations, i.e. independent of the choice of thermodynamic potential. For the systems under consideration, the corresponding metrics and scalar curvatures are calculated, and their properties are also described. Research of two-dimensional quantum thermodynamic systems is becoming more urgent. It is sufficiently to mention that such systems are related to, for example, topological insulators, graphene, systems with quantum Hall effect, etc. Two-dimensional quantum systems may have a statistical distribution different from distributions of Fermi-Dirac and Bose-Einstein. Geometric approaches in research of these thermodynamic systems certainly open the new perspective. In this paper, the thermodynamic properties of two-dimensional Bose-Gas and Berezinsky-Kosterlitz-Thouless system have been studied with the help of geometrothermodynamics. The main objective was to reproduce the Bose-Einstein condensation for the first system and find possible new phase transitions for the second. In order to study the above mentioned thermodynamic systems, we have consequently calculated the covariant metric tensors of corresponding equilibrium manifolds and their determinants, then counter-variant metric tensors, Christoffel symbols, curvature tensors and corresponding scalar curvatures. Using the thermodynamic potential, we obtained (using the MatLab system) the corresponding geometric values in a wide range of temperature and area. Explicit formulas were also obtained for each geometric quantity but due to their bulkiness we do not present them in this paper. Examples of calculated scalar curvatures for a certain range of parameters T and S are shown in the figures. The figures also show that despite the significantly different behavior of the curvatures depending on the parameters T and S, both metrics lead to the same General result regarding the location of singularities for the corresponding curvatures. Keywords: geometrothermodynamics, Legendre transformations, metric tensor, scalar curvature, two-dimensional Bose gas, Berezinsky-Kosterlitz-Thouless system