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Disentangling Thermal and Degeneracy Pressure in Quantum-Confined Fermi Gases

Coskun Firat

Published Oct 14, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.57647/jtap.2026.2002.15

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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.
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.
By using the Euler–MacLaurin formula, this paper studies the thermodynamic properties of an ideal Fermi gas confined in a D -dimensional rectangular container. The general expressions of the thermodynamic quantities with the finite-size corrections are given explicitly and the effects of the size and shape of the container on the properties of the system are discussed. It is shown that the corrections of the thermodynamic quantities due to the finite-size effects are significant to be considered for the case of strong degeneracy but negligible for the case of weak degeneracy or non-degeneracy. It is important to find that some familiar conclusions under the thermodynamic limit are no longer valid for the finite-size systems and there are some novel characteristics resulting from the finite-size effects, such as the nonextensivity of the system, the anisotropy of the pressure, and so on.
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.
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.
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.
After a brief historical introduction to Bose-Einstein condensation and Fermi degeneracy, we discuss theoretical results we have recentely obtained for trapped degenerate quantum gases by means of a thermal field theory approach. In particular, by using Gross-Pitaevskii and Bogoliubov-Popov equations, we consider thermodynamical properties of two Bosonic systems: a gas of Lithium atoms and a gas of Hydrogen atoms. Finally, we investigate finite-temperature density profiles of a dilute Fermi gas of Potassium atoms confined in a harmonic potential.