Degeneracy Pressure of a Fermi Gas Confined in Nano Scale Domains

Guozhen Su, Liwei Chen, Tongling Lin +1 more

Kamran Ullah, Farhan Saif

Coşkun Fırat

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.

D. S. Jin

Coskun Firat

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.

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