It is a long held conjecture in the connection between information geometry (IG) and thermodynamics that the curvature endowed by IG diverges at phase transitions. Recent work on the IG of Bose-Einstein (BE) gases challenged this conjecture by saying that in the limit of fugacity approaching unit-where BE condensation is expected-the curvature does not diverge; rather, it converges to zero. However, as the discontinuous behavior that identifies condensation is only observed at the thermodynamic limit, a study of the IG curvature at a finite number of particles, N, is in order from which the thermodynamic behavior can be observed by taking the thermodynamic limit ( N→∞) posteriorly. This article presents such a study. We find that for a trapped gas, as N increases, the values of curvature decrease proportionally to a power of N, while the temperature at which the maximum value of curvature occurs approaches the usually defined critical temperature. This means that, in the thermodynamic limit, the curvature has a limited value where a phase transition is observed, contradicting the forementioned conjecture.
It is a long held conjecture in the connection between information geometry (IG) and thermodynamics that the curvature endowed by IG diverges at phase transitions. Recent work on the IG of Bose-Einstein (BE) gases challenged this conjecture by saying that in the limit of fugacity approaching unit-where BE condensation is expected-the curvature does not diverge; rather, it converges to zero. However, as the discontinuous behavior that identifies condensation is only observed at the thermodynamic limit, a study of the IG curvature at a finite number of particles, N, is in order from which the thermodynamic behavior can be observed by taking the thermodynamic limit ( N→∞) posteriorly. This article presents such a study. We find that for a trapped gas, as N increases, the values of curvature decrease proportionally to a power of N, while the temperature at which the maximum value of curvature occurs approaches the usually defined critical temperature. This means that, in the thermodynamic limit, the curvature has a limited value where a phase transition is observed, contradicting the forementioned conjecture.
We investigate the thermodynamic geometry of an ideal quantum gas obeying inclusion statistics, characterized by a negative statistical parameter g<0. In this framework the grand-canonical partition function admits a finite maximum fugacity, and the thermodynamic scalar curvature R is strictly positive for all g<0, reflecting effective attractive statistical interactions analogous to those of a bosonic system. As the fugacity approaches its maximum value, R diverges, signaling a phase transition of the Bose-Einstein condensation type. A key distinction from the ordinary ideal Bose gas is that the condensation temperature is elevated relative to the bosonic case, and finite-temperature condensation occurs even in the dimensional regime 1/2<D/σ≤1 where standard bosons do not condense, while for D/σ≤1/2 the transition temperature vanishes. Three independent criteria; divergence of R, the maximum fugacity singularity, and the non-analytic cusp in the specific heat, coincide at the same condensation point, confirming the thermodynamic consistency of the transition.
This paper presents a study of the grand canonical Bose-Einstein (BE) statistics for a finite number of particles in an arbitrary quantum system. The thermodynamical quantities that identify BE condensation---namely, the fraction of particles in the ground state and the specific heat---are calculated here exactly in terms of temperature and fugacity. These calculations are complemented by a numerical calculation of fugacity in terms of the number of particles, without taking the thermodynamic limit. The main advantage of this approach is that it does not rely on approximations made in the vicinity of the usually defined critical temperature, rather it makes calculations with arbitrary precision possible, irrespective of temperature. Graphs for the calculated thermodynamical quantities are presented in comparison to the results previously obtained in the thermodynamic limit. In particular, it is observed that for the gas trapped in a three-dimensional box, the derivative of specific heat reaches smaller values than what was expected in the thermodynamic limit---here, this result is also verified with analytical calculations. This is an important result for understanding the role of the thermodynamic limit in phase transitions and makes possible to further study BE statistics without relying neither on the thermodynamic limit nor on approximations near critical temperature.
Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number N of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature Tc in terms of a power series in ${x}_{0}={\ensuremath{\varepsilon}}_{0}/{k}_{B}{T}_{c}$, where ${\ensuremath{\varepsilon}}_{0}$ denotes the zero-point energy of the trapping potential. This expression, which applies in Cartesian, cylindrical, and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of x0 up to the second order. We show that, for a harmonic trap, the well-known first-order shift of the critical temperature $\ensuremath{\Delta}{T}_{c}/{T}_{c}\ensuremath{\propto}{N}^{\ensuremath{-}1/3}$ is inaccurate when $N\ensuremath{\leqslant}{10}^{5}$, the next order (proportional to ${N}^{\ensuremath{-}1/2}$) being significant. We also show that finite-size effects on the condensation temperature cancel out in a cubic trapping potential, e.g., $V(r)\ensuremath{\propto}{r}^{3}$. Finally, we show that in a generic power-law potential of higher order, e.g., $V(r)\ensuremath{\propto}{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>3$, the shift of the critical temperature becomes positive. This effect provides a large increase of Tc for relatively small atom numbers. For instance, an increase of about $+$40% is expected with 104 atoms in a $V(r)\ensuremath{\propto}{r}^{12}$ trapping potential.
In this paper, we consider the thermodynamic geometry of harmonic trapped Bose and Fermi gases. We construct a thermodynamic geometry for these systems and evaluate the corresponding metric tensor and thermodynamic curvature. By observing that the thermodynamic curvature of the trapped Bose (Fermi) gas is always positive (negative), we conclude that the trapping potential does not affect the intrinsic statistical interactions. By inspecting the singular points of the thermodynamic curvature, we argue that the confinement dimension and the frequency of the trapping potential change the critical value of fugacity for the Bose gas.
Evading the Mermin–Wagner–Hohenberg no-go theorem and revisiting with rigor the ideal Bose gas confined in a square box, we explore a discrete phase transition in two spatial dimensions. Through both analytic and numerical methods, we verify that thermodynamic instability emerges if the number of particles is sufficiently yet finitely large: specifically, N ⩾ 35131 . The instability implies that the isobar of the gas zigzags on the temperature–volume plane, featuring supercooling and superheating phenomena. The Bose–Einstein condensation can then persist from absolute zero to the superheating temperature. Without necessarily taking the large N limit, under a constant pressure condition, the condensation takes place discretely both in the momentum and in the position spaces. Our result is applicable to a harmonic trap. We assert that experimentally observed Bose–Einstein condensations of harmonically trapped atomic gases are a first-order phase transition that involves a discrete change of the density at the center of the trap.
Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N 0 by construction, leads to BEC in a straightforward way. For a density of states of the form G ( )∝ γ , the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature T B , determined by an appropriate critical exponent. The condensate population N 0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.