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

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.