Diffusion Boundaries Decide Uniformity or Zero

0$) increase. This process inevitably causes any localized concentration peak to flatten out and spread, ultimately leading to a uniform distribution where the concentration is constant and the Laplacian is zero. The long-term outcome of this spreading depends critically on the physical boundaries of the system. In an infinite domain, mass or energy spreads indefinitely, and the concentration perpetually drops toward zero, never reaching a true stable equilibrium. However, in a finite domain, boundaries force the system toward a predictable steady state; for instance, a Dirichle"> 0$) increase. This process inevitably causes any localized concentration peak to flatten out and spread, ultimately leading to a uniform distribution where the concentration is constant and the Laplacian is zero. The long-term outcome of this spreading depends critically on the physical boundaries of the system. In an infinite domain, mass or energy spreads indefinitely, and the concentration perpetually drops toward zero, never reaching a true stable equilibrium. However, in a finite domain, boundaries force the system toward a predictable steady state; for instance, a Dirichle"> 0$) increase. This process inevitably causes any localized concentration peak to flatten out and spread, ultimately leading to a uniform distribution where the concentration is constant and the Laplacian is zero. The long-term outcome of this spreading depends critically on the physical boundaries of the system. In an infinite domain, mass or energy spreads indefinitely, and the concentration perpetually drops toward zero, never reaching a true stable equilibrium. However, in a finite domain, boundaries force the system toward a predictable steady state; for instance, a Dirichle">

Fick's Second Law describes diffusion by asserting that the rate of change of concentration ($\partial_t u$) is directly proportional to the Laplacian ($\nabla^2 u$) of the concentration field, which measures the local curvature. This mathematical relationship means that areas of high concentration ("peaks," where $\nabla^2 u<0$) decrease over time, while areas of low concentration ("troughs," where $\nabla^2 u>0$) increase. This process inevitably causes any localized concentration peak to flatten out and spread, ultimately leading to a uniform distribution where the concentration is constant and the Laplacian is zero. The long-term outcome of this spreading depends critically on the physical boundaries of the system. In an infinite domain, mass or energy spreads indefinitely, and the concentration perpetually drops toward zero, never reaching a true stable equilibrium. However, in a finite domain, boundaries force the system toward a predictable steady state; for instance, a Dirichlet boundary leads to a final uniform concentration of zero (energy is lost), while a Neumann boundary conserves the total energy, resulting in a final uniform concentration equal to the initial average value. Fick's Second Law models diverse phenomena such as Chemical Mixing, Heat Transfer, and Semiconductor Doping.

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