Entropy-Modulated Real Scalar Field Theory: A Unified Framework for Conservative and Dissipative Dynamics

This manuscript proposes a dual-field Lagrangian framework where particle mechanics emerges from the interaction between a real scalar field Φ(x,t) and a dynamical entropy field ε(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 gᵢⱼ = δᵢⱼ/ε(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 ± 3~ and a unique ω⁻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.