A Two-Field Lagrangian Reformulation of Coupled Condensate-Thermal Dynamics in Bose-Einstein Condensates

This manuscript proposes a dual-field Lagrangian framework where particle mechanics emerges from the interaction between a real scalar field $\Phi(x,t)$ and a dynamical entropy field $\epsilon(x,t)$. By treating entropy as a geometric degree of freedom, we derive a unified action that accounts for both conservative and dissipative dynamics, recovering Newtonian mechanics as a special decoherent limit. The framework establishes a thermodynamic metric $g_{ij} = \delta_{ij}/\epsilon(x)$, where friction emerges as a consequence of Ricci curvature. We provide explicit, falsifiable predictions for Bose-Einstein condensates (BECs) in temperature gradients, including a relative center-of-mass displacement of $8 \pm 3~\mu\text{m}$ and a unique $\omega^{-1}$ frequency scaling signature. With a calculated Signal-to-Noise Ratio (SNR) of 6.4, the framework offers a clear experimental pathway for distinguishing entropy-modulated dynamics from classical backgrounds such as gravity and thermophoresis.