Theoretical Modeling of Dynamic Dimensional Phase Transitions and Cross-Scale Experimental Validation

This paper proposes and validates a theoretical framework describing the dy?namic evolution of geometric dimensions under external field driving. Based onconservation laws and the Landau-Ginzburg variational principle, the dimensionevolution equation is established:∂tDH = ∇ · (D2H∇V ) + α(DH − D0) (V is the external field potential)Through numerical simulations and comparison with existing experimental data,the key mechanisms of dimensional phase transitions are revealed:1. Nonlinear Diffusion: The dimension flux J = −D2H∇V dominates the poten?tial field response, with the diffusion coefficient proportional to D2H.2. Bifurcation Behavior: When the external field gradient exceeds the criticalvalue ∇Vc =ακ, the system undergoes dimensional instability (phase transi?tion).3. Cross-Scale Coupling: Qualitative agreement between theory and experimentis observed in quantum materials (e.g., magic-angle graphene).This study provides theoretical support for dynamic dimension control and offersnew insights for the design of future smart materials and quantum devices.