A Geometrodynamic Interpretation of the β-Parameter in Ginzburg-Landau Theory: Centrifugal Saturation of the Superconducting Condensate

Abstract

The Ginzburg-Landau (GL) parameter β, with dimensions of energy × volume, quantifies the condensation saturation that limits the growth of the superconducting order parameter. Established interpretations view β through a microscopic lens (as a measure of pair-pair repulsion) or a thermodynamic lens (as an entropic cost). This work introduces a complementary geometrodynamic interpretation, framing β as the integrated energy of an effective centrifugal potential arising within the coherent condensate. This potential originates from an effective centrifugal force, F_centrifugal = KΓ/(ε_0 μ_0), where Γ is the Cooper pair's charge-to-mass ratio (or gyromagnetic ratio under rotation), and K is a coupling constant that encapsulates the specific dynamical configuration and interaction details. The condensation energy density term (β/2)|ψ|^4 is shown to be equivalent to this centrifugal potential energy density, providing a vivid physical picture: saturation occurs when the cohesive energy gain from pairing is balanced by the centrifugal energy cost of confining charged, massive entities (Cooper pairs) within a phase-coherent volume. The new interpretation provides a concrete geometrodynamic instantiation of the “saturation” process within the tripartite process framework of GL theory (energy competition, saturation, and inertia) as outlined in <cit.>. It directly links the abstract parameter β to a physical picture of geometric confinement and dynamical response, thereby complementing the energy-competition (α) and inertia (m^*) pictures. This interpretation is logically self-consistent, dimensionally sound, and complements—rather than contradicts—existing microscopic and thermodynamic understandings, offering a unified dynamical perspective on condensation saturation.