A physics-based Lagrangian for the heat-diffusion equation

The development of finite element computer codes is facilitated by the use of a minimum principle. Unfortunately, the diffusion equation lacks a principle that will produce the term in the first derivative with respect to time. We have developed an action (Lagrangian density) based on the locally random motion in thermal, or diffusion processes by decomposing the material response into global and local fields, which naturally leads to the time dependent diffusion equation by minimizing the action. The result yields the correct governing equation, including the time dependence. We provide a physical interpretation of the results and generalize the approach to the phase-field method.