Geometry-Driven Thermodynamics: Shape Effects and Anisotropy in Quantum-Confined Ideal Fermi and Bose Gases

TL;DR

This paper introduces a unified quantum phase space formalism to describe the thermodynamics of confined ideal Fermi and Bose gases, revealing shape-dependent anisotropic pressures and phase transition control at nanoscale confinement.

Contribution

It develops an analytical framework that captures quantum degeneracy and anisotropic thermodynamics in confined gases, unifying Fermi-Dirac and Bose-Einstein statistics within a single approach.

Findings

Thermodynamics becomes intrinsically anisotropic under nanoscale confinement.

Shape parameters can control phase transitions without changing size, temperature, or density.

Quantum effects are significant at accessible temperatures for confinement scales of 5-50 nm.

Abstract

This study presents a unified description of the thermodynamics of ideal quantum gases under nanoscale confinement using a Quantum Phase Space (QPS) formalism. We show that the statistical momentum variances B_ll capture quantum degeneracy: for fermions, they incorporate the Fermi energy, and for bosons, the condensate energy scale. This bridges our formalism with established results and allows both Fermi-Dirac and Bose-Einstein statistics to be treated within a single framework. From this, we derive exact analytical expressions for key properties - internal energy, anisotropic pressure tensor, and heat capacity - seamlessly describing the transition from classical to quantum regimes. Our results reveal that nanoscale thermodynamics is intrinsically anisotropic: pressure becomes direction-dependent, with fractional anisotropy reaching unity under extreme confinement. Notably, pure shape effects, controlled via geometric parameters in B_ll, enable manipulation of phase transitions without altering system size, temperature, or density. Numerical simulations for confined electron and helium-4 gases show significant quantum effects at accessible temperatures (mK to K) for confinement scales of 5-50 nm. This work provides a theoretical toolkit for nanosystems, with direct implications for nanofluidic devices and quantum sensors.