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Exact Density Profiles of 1D Quantum Fluids in the Thomas-Fermi Limit: Geometric Hierarchy to the Tonks-Girardeau Gas

Hiroki Suyari

Abstract

We present a geometric framework for 1D quantum fluids across interaction regimes in the Thomas-Fermi limit. Based on the Linearization Principle via the $q$-logarithm, macroscopic density profiles form a discrete hierarchy: the ideal Bose gas ($q=1$), the mean-field Gross-Pitaevskii condensate ($q=-1$), and the strongly correlated Tonks-Girardeau gas ($q=-3$). We further derive a universal sound velocity scaling, $c \propto ρ^{(1-q)/4}$. This establishes a non-perturbative link between static geometry and dynamical excitations in many-body systems.

Exact Density Profiles of 1D Quantum Fluids in the Thomas-Fermi Limit: Geometric Hierarchy to the Tonks-Girardeau Gas

Abstract

We present a geometric framework for 1D quantum fluids across interaction regimes in the Thomas-Fermi limit. Based on the Linearization Principle via the -logarithm, macroscopic density profiles form a discrete hierarchy: the ideal Bose gas (), the mean-field Gross-Pitaevskii condensate (), and the strongly correlated Tonks-Girardeau gas (). We further derive a universal sound velocity scaling, . This establishes a non-perturbative link between static geometry and dynamical excitations in many-body systems.
Paper Structure (15 sections, 11 equations, 1 figure)

This paper contains 15 sections, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Theory
  3. The Nonlinear Schrödinger Equation and the Thomas-Fermi Limit
  4. The Linearization Principle
  5. Geometric Wavefunction and Density Profile
  6. Derivation of the Standard GPE ($q=-1$)
  7. Geometric Hierarchy to the Tonks-Girardeau Limit
  8. Derivation of the Semicircle Law ($q=-3$)
  9. Discrete Physical Hierarchy
  10. Theoretical Consistency: Connection to Nonlinear Diffusion and Geometry
  11. Correspondence with the Porous Medium Equation
  12. Equations of State and Thermodynamics
  13. Geometric Scaling of the Sound Velocity
  14. Experimental Signatures and the Crossover Regime
  15. Conclusion

Figures (1)

  • Figure 1: Normalized density profiles $n(x)/n(0)$ in a harmonic trap derived from the Linearization Principle for different values of the geometric index $q$. The profiles demonstrate the discrete hierarchy from the ideal Bose gas ($q=1$, dashed line), to the mean-field BEC in the Thomas-Fermi limit ($q=-1$, solid line), and the strongly correlated Tonks-Girardeau gas ($q=-3$, dash-dotted line).