Abstract
We establish a rigorous mathematical framework for the spontaneous formation of stable structures through the universal principle of energy minimization in multi-interaction systems. Within this formulation, each subsystem contributes to a collective energy function E(x) composed of bounded, weakly nonlinear interaction terms. By proving the strictly positive definiteness of the Hessian and the existence of a unique critical point Q_c satisfying ∇E(Q_c)=0, we demonstrate that the system converges globally to this equilibrium via gradient-flow dynamics. Local and global stability are verified analytically through Lyapunov functions and LaSalle’s Invariance Principle under the assumptions of uniform strong convexity and coercivity. The resulting equilibrium represents a spontaneously formed structure that minimizes the total energy of the interacting fields. Extending beyond reaction–diffusion theory, the proposed framework unifies the energetic mechanisms of self-organization observed across physics, chemistry, and biology. As an illustrative application, the spontaneous assembly of lipid bilayers is interpreted as the natural descent toward a unique global energy minimum, providing a quantitative bridge between thermodynamic stability and dynamic self-organization. This study thus establishes energy minimization as a universal law governing the emergence, persistence, and stability of complex structures in nonlinear systems.
Keywords
Morphogenesis, Reaction-diffusion systems, Pattern formation, Self-organization, Interacting forces