It is shown that the stability of ideal quantum gases can be measured by means of the Riemann scalar curvature R of the parameter space. The components of the metric tensor were assumed to be the second moments of energy and the number of particle fluctuations. As a result, R is a function of the second and third moments of those quantities. For bosons R is positive and increases monotonically from zero at the classical limit to positive infinity in the condensation region. A system is less stable if R is bigger and vice versa. For fermions R is negative and this means that Fermi gases are more stable than the ideal Bose and ideal classical systems.
It is shown that the stability of ideal quantum gases can be measured by means of the Riemann scalar curvature R of the parameter space. The components of the metric tensor were assumed to be the second moments of energy and the number of particle fluctuations. As a result, R is a function of the second and third moments of those quantities. For bosons R is positive and increases monotonically from zero at the classical limit to positive infinity in the condensation region. A system is less stable if R is bigger and vice versa. For fermions R is negative and this means that Fermi gases are more stable than the ideal Bose and ideal classical systems.
The scalar curvature ( R ) of ideal quantum gases obeying Gentile's statistics is investigated by the method of information geometrical theory. The R value is specified by the fugacity and the maximum number, p , of particles in a state. The lowest case p = 1, corresponds to Fermi-Dirac statistics and the unbounded case, p , to Bose-Einstein statistics. In contrast to R = 0 for ideal classical gases obeying Boltzmann statistics, we find R = (2) 1/2 /32 for p 2 and R = -(2) 1/2 /32 for p = 1, in 0 which is the classical limit. This means that a quantum statistical character is left in R , in the classical limit. Also, a correlation between the sign of R and a quantum mechanical exchange effect is recognized for 0 and >>1. Furthermore, we obtain results that support the instability interpretation of R proposed by Janyszek and Mrugala.
A Riemannian metrical structure of the parameter space has been introduced and investigated for magnetic systems described in the framework of quantum statistics. The introduced metric is based on the conception of the relative information. Two contrasting models have been investigated in detail: the one-dimensional Ising model, with short-range interactions, and the mean-field model of Kac, with long-range interactions. In the second case the metric tensor degenerates. The degeneration has been removed by adding the lattice energy to the original magnetic Hamiltonian. It turns out that in both cases the scalar curvature of parameter space tends toward plus infinity while approaching the critical points. The inverse of the scalar curvature, given by the second and third moments of stochastic variables, has been interpreted as a measure of the stability of the considered magnetic systems. The scalar curvature represents a joint part of fluctuations caused by the interactions of spins.
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.
In this work, improvements are introduced to the current models of the ideal Fermi gas and the ideal Bose gas by incorporating the quantum nature of phase space, which is directly linked to the uncertainty principle. These improved models build upon the recently developed concepts of quantum phase space (QPS) and the QPS representation of quantum mechanics. The Hamiltonian operator for a gas particle and its eigenstates are first determined, and quantum statistical mechanics is used to derive the thermodynamic properties of the ideal gas. Analytic expressions for thermodynamic quantities—including the grand canonical potential, particle number, internal energy, von Neumann entropy, and pressure—are derived, along with the corresponding thermodynamic equations of state for both bosons and fermions. These corrections are particularly significant at low temperatures and in confined volumes, where quantum effects related to system geometry, such as shape and size, become significant. The results also establish a direct link between thermodynamic functions and the quantum statistical variances of momentum. Importantly, the improved models recover well-known classical relations of the ideal gas in the high-temperature and large-volume limits, ensuring consistency with classical physics. By addressing quantum corrections and their thermodynamic implications, this work provides a foundation for further applications in nanoscale systems, quantum gases, low-temperature physics, ultracold physics, and astrophysics.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
If the N bosons that compose an ideal Bose-Einstein gas with energy E and volume V are each assumed to have the average energy of the system E/N, the entropy is easily expressed in terms of the number of bosons N and the number of single-particle microstates n they can occupy. Because the entropy derived is a function of only N and n, and the latter is a function of the extensive variables, E, V, and N, this entropy describes all that can be known of the thermodynamics of this system. In particular, the entropy very simply recovers the Sakur-Tetrode entropy in the classical limit and at sufficiently low temperature describes an unstable system. A thermodynamic stability analysis recovers the Bose-Einstein condensate and a two-phase region. Apart from numerical factors of order one, results are identical with those derived via standard, probabilistic methods.