🎬Fick's second law is used to Chemical Mixing and Heat Transfer and Semiconductor Doping

The crucial distinction between the infinite-domain models (Chemical Mixing, Doping) and the finite-domain Heat Transfer models with boundary conditions lies in the long-term fate of the mass or energy and the existence of a steady state. When diffusion occurs in an infinite domain, the energy/mass is conserved but spreads indefinitely, meaning the concentration peak perpetually drops toward zero in an endless process of dilution, never reaching a true, stable equilibrium. Conversely, the presence of finite boundaries forces the system toward a predictable steady state ( $\partial_t u=0$ ). A Dirichlet (fixed) boundary causes the energy to be lost, resulting in a final uniform concentration of zero (e.g., the cooled rod), whereas a Neumann (insulated) boundary traps and conserves the total energy, leading to a final uniform concentration equal to the initial average value (e.g., the heat is simply redistributed evenly across the insulated rod). Boundaries are therefore the essential factor that governs whether a diffusion process results in perpetual spreading or a final, controlled equilibrium.

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Fick's second law is used to Chemical Mixing and Heat Transfer and Semiconductor Doping | Animated Resultsviadean on Notion