Abstract This study aims to investigate local density variations of Fermi gases confined in various geometries at the nano-scale, revealing irregularities even in thermodynamic equilibrium. Calculations show that density of the confined gas in a nano scale domain, decreases towards zero near domain boundaries due to a quantum boundary layer linked to the Planck constant. Additionally, Friedel-like density oscillations are observed in nano-confined Fermi gases. Analytical expressions for the local density distribution are derived for degenerate and one-dimensional scenarios, while numerical computations are conducted for complex geometries and weak degeneracy conditions. In line with existing literature, it is understood that the quantum-size effects observed in the global thermodynamic properties of confined gases are attributed to these local irregularities. It becomes evident that in areas smaller than the threshold determined by the quantum boundary layer, the gas empties the part of the domain or reaches lower densities in the considered part relative to other parts of the domain, despite being in thermodynamic equilibrium. The results indicate the potential for gases to benefit from these behaviors and suggest the possibility of designing and manufacturing new nano-scale machines that are not feasible on a macro-scale.
Abstract This study aims to investigate local density variations of Fermi gases confined in various geometries at the nano-scale, revealing irregularities even in thermodynamic equilibrium. Calculations show that density of the confined gas in a nano scale domain, decreases towards zero near domain boundaries due to a quantum boundary layer linked to the Planck constant. Additionally, Friedel-like density oscillations are observed in nano-confined Fermi gases. Analytical expressions for the local density distribution are derived for degenerate and one-dimensional scenarios, while numerical computations are conducted for complex geometries and weak degeneracy conditions. In line with existing literature, it is understood that the quantum-size effects observed in the global thermodynamic properties of confined gases are attributed to these local irregularities. It becomes evident that in areas smaller than the threshold determined by the quantum boundary layer, the gas empties the part of the domain or reaches lower densities in the considered part relative to other parts of the domain, despite being in thermodynamic equilibrium. The results indicate the potential for gases to benefit from these behaviors and suggest the possibility of designing and manufacturing new nano-scale machines that are not feasible on a macro-scale.
Friedel oscillations appear in density of Fermi gases due to Pauli exclusion principle and translational symmetry breaking nearby a defect or impurity. In confined Fermi gases, this symmetry breaking occurs also near to boundaries. Here, density oscillations of a degenerate and confined Fermi gas are considered and characterized. True nature of density oscillations are represented by analytical formulas for degenerate conditions. Analytical characterization is first done for completely degenerate case, then temperature effects are also incorporated with a finer approximation. Envelope functions defining the upper and lower bounds of these oscillations are determined. It is shown that the errors of obtained expressions are negligible as long as the system is degenerate. Numbers, amplitudes, averages and spatial coordinates of oscillations are also given by analytical expressions. The results may be helpful to efficiently predict and easily calculate the oscillations in density and density-dependent properties of confined electrons at nanoscale.
We propose an analytical model for the accurate calculation of size and density dependent quantum oscillations in thermodynamic and transport properties of confined and degenerate non-interacting Fermi gases. We provide a universal, material independent, recipe that explicitly separates oscillatory quantum regime from stationary classical regime. Our model quite accurately estimates quantum oscillations depending on confinement and degeneracy. We construct a phase diagram representing stationary and oscillatory regimes on degeneracy-confinement space. Analytical expressions of phase transition interfaces are derived for different dimensions. The critical point on the phase diagram, which separates entirely stationary and entirely oscillatory regions, is determined and their aspect ratio dependencies are examined. Quantum oscillations as well as their periods are analytically expressed for one-dimensional case. Accuracy of our model is verified through quantum oscillations in electronic specific heat capacity. We also compare the predictions of our half-vicinity model, based on bounded sums, with those of infinite sums, for the oscillatory violation of entropy-heat capacity equivalence in degenerate limit to show the accuracy of our model. Furthermore, similarities between functional behaviors of total occupancy variance and conventional density of states functions at Fermi level are discussed.
A theoretical framework is developed to decompose the total pressure of a quantum-confined Fermi gas into degeneracy pressure, arising from Fermi–Dirac statistics, and thermal pressure, associated with finite-temperature excitations. Using Weyl’s conjecture for the confined density of states, geometric confinement effects are incorporated through surface- and edge-to-volume parameters ( and ). Exact analytical expressions for both components are derived using polylogarithmic functions, and asymptotic forms are obtained for the strongly degenerate ( ) and weakly degenerate ( ) regimes. The analysis shows that quantum confinement enhances degeneracy pressure by a factor of ( ), while suppressing thermal pressure, with the contrast most significant under strong degeneracy. The maximum relative error in the total pressure, occurring at the crossover temperature , is about 22%, while that for thermal pressure reaches 52%. The asymptotic expressions remain valid across and . Separating thermal and degeneracy contributions enables improved interpretation of thermodynamic measurements in confined quantum systems where only total pressure is accessible. The framework is relevant for nanoelectronic devices, ultracold atomic gases, and quantum dots, offering insight into how spatial confinement differentially shapes thermal and quantum contributions to observable thermodynamic behavior.
Temperature-Induced Quantum Migration (TIQM) is a macroscopic quantum phenomenon that fundamentally reshapes the local thermodynamics of confined systems. While previously explored in Maxwell-Boltzmann and Bose gases, we here extend the TIQM framework to noninteracting Fermi gases confined in anisometric rectangular domains. These domains are characterized by D_{I}, representing the number of dimensions that remain unconfined. We derive the local heat capacity and demonstrate that the standard temperature derivative of local energy yields unphysical negative regions, a discrepancy resolved by the TIQM correction. We show that quantum degeneracy in Fermi gases generally suppresses the TIQM contribution to global heat capacity compared to other statistics, yet it drives a distinct redistribution of local heat capacity. Notably, for the most constrained case (D_{I}=0), a Schottky-type anomaly emerges at low densities, which is progressively "washed out" by increasing dimensionality (D_{I}=1,2). Based on an idealized noninteracting model, our calculations suggest that a measurable (22%) enhancement in the local-to-global electronic heat capacity ratio could emerge at approximately 84 nm from the longitudinal boundaries of a 12×12×5600 nm^{3} bismuth nanowire at 30 K, offering a potential experimental signature for TIQM. These results complete a systematic trilogy of studies across all fundamental quantum statistics, establishing TIQM as a necessary correction for the local thermodynamics of nanoscale devices.
In this paper, we calculate the equations of state and the thermodynamic quantities for two- and three-dimensional hard-sphere Bose and Fermi gases in finite-size containers. The approach we used to deal with interacting gases is to convert the effect of interparticle hard-sphere interaction to a kind of boundary effect, and then the problem of a confined hard-sphere quantum gas is converted to the problem of a confined ideal quantum gas with a complex boundary. For this purpose, we first develop an approach for calculating the boundary effect on d-dimensional ideal quantum gases and then calculate the equation of state for confined quantum hard-sphere gases. The thermodynamic quantities and their low-temperature and high-density expansions are also given. In higher-order contributions, there are cross terms involving both the influences of the boundary and of the interparticle interaction. We compare the effect of the boundary and the effect of the interparticle interaction. Our result shows that, at low temperatures and high densities, the ratios of the effect of the boundary to the effect of the interparticle interaction in two dimensions are essentially different to those in three dimensions: in two dimensions, the ratios for Bose systems and for Fermi systems are the same and are independent of temperatures, while in three dimensions, the ratio for Bose systems depends on temperatures, but the ratio for Fermi systems is independent of temperatures. Moreover, for three-dimensional Fermi cases, compared with the contributions from the boundary, the contributions from the interparticle interaction to entropies and specific heats are negligible.
We consider an ultracold Bose gas in a half-open potential well, otherwise confined to a quasi-one-dimensional geometry. The Bogoliubov equations for its elementary excitations are solved in the continuous spectrum to explore the particle and energy content near the edge of the gas, beyond the often applied local density approximation. In particular, gradients in the condensate density enhance density-dominated excitations in the border region. We discuss excess (missing) particles and their energy by comparing to suitable reference solutions for quasi-homogeneous systems. The density profile near the edge shows no Friedel oscillations, but a dipolar feature from the spill-out of thermally excited particles. The calculations are performed in the grand-canonical ensemble and in the thermodynamic limit.