Field-Theoretic Derivation of the Constructal Law from Non-Equilibrium Thermodynamics

Traditional analyses of transport phenomena rely on prescribed geometric boundaries, yet natural flow systems dynamically evolve their architecture to maximize access to currents. To address this disparity, we propose a field-theoretic framework for the constructal law that treats physical geometry as a dynamic state variable, represented by a time-dependent conductivity tensor. Using a variational approach grounded in non-equilibrium thermodynamics, we derive a general tensor evolution equation. Within this framework, macroscopic flow architecture emerges deterministically from the continuous competition between non-linear flux-induced accretion, linear entropic relaxation, and spatial smoothing. Scaling analysis reduces this dynamic to a tri-parameter dimensionless phase space: a morphogenic number driving structural growth, a structural diffusion number governing spatial coherence, and a stochastic intensity number providing the microscopic seeds for symmetry breaking. Our principal result is the analytical prediction of a critical bifurcation. When the local morphogenic number strictly exceeds unity, the system escapes its stable, isotropic configuration and branches into highly conductive, anisotropic architectures. We demonstrate the predictive validity and trans-scalar applicability of this continuum theory by mapping it to highly diverse phase transitions, successfully capturing phenomena ranging from microscopic aerosol agglomeration and microbial resistance, to macroscopic coral plasticity and crystal growth instabilities, and finally to the astrophysical launching of relativistic jets from black holes.