APPLICATION OF GEOMETROTHERMODYNAMICS TO THE TWO-DIMENSIONAL SYSTEMS: IDEAL BOSE-GAS AND SYSTEM WITH STRONG INTERACTION

Аннотация

In the framework of the method of geometrothermodynamics, in present work, 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.
Next, we used geometric thermodynamics for the system of the Berezinsky-Kosterlitz-Thoules. This is a twodimensional
system of Bose particles with a strong interaction (strong in the sense that topological defects - point
vortices-contribute to the thermodynamics of the system) with a complex, not fully studied system of phase
transitions. An ideal two-dimensional Bose gas with a finite number of particles and a Berezinsky-Kosterlitz-Thoules
system are considered. As thermodynamic potentials for these thermodynamic systems, the chemical potential
depending on temperature and area and the Free energy depending on the temperature and size of the system were
taken, respectively. The paper also presents 3-dimensional drawings that clearly show at which values of
thermodynamic variables scalar curvatures tend to infinity or to zero, which indicates possible phase transitions and
possible compensation of interactions by quantum effects, respectively. It is shown that both variants of metrics for
an ideal two-dimensional Bose gas lead to the same arrangement of lines, where scalar curvatures become singular.
This arrangement of lines is consistent with the region where the phase transition occurs - Bose condensation in a
two-dimensional Bose gas. It is also shown that for large values of temperature and area parameters, the curvature is
close to zero and this corresponds to a classical ideal two-dimensional gas. When considering the Berezinsky-
Kosterlitz-taules system, possible new phase transitions were discovered by the method of geometric thermodynamics. The metric calculation leads to a possible phase transition located below the Berezinsky-Kosterlitz-
Taules transition, and the calculation leads to a possible phase transition located above.