Gradient Systems and Scale‐Aware Heat Equations

ABSTRACT The extreme miniaturization of modern microelectronic architectures exposes the limitations of classical continuum mechanics, as Fourier's law breaks down at sub‐micron scales (100–0.1 ). While molecular models can accurately capture these thermal flow regimes, they remain computationally expensive. Previous continuum‐based alternatives, such as micro‐force balance formulations, provide modeling flexibility but often yield complex nonlinearities that require simplifying assumptions. In this work, we present an approach demonstrating that the extended theory of classical thermomechanics of solids, incorporating a non‐trivial dependence of the gradient of entropy into the energy density, can be derived directly from a gradient system. By selecting an appropriate Onsager operator, the proposed framework provides a structured mechanism for accommodating complex nonlinearities. Moreover, this gradient flow approach inherently satisfies Fourier's law and, therefore, the dissipation inequality. The resulting gradient flow equation is a scale‐aware heat equation (or enhanced heat equation) for rigid conductors. This formulation entirely bypasses specific assumptions previously required in periodic homogenization frameworks and enables the modeling of scale‐size effects through a length‐scale tensor, offering a rigorously consistent mathematical bridge between classical macroscopic models and molecular simulations.