# Zero Flux Conservation and Uniform Equilibrium

The no-flux boundary condition mathematically dictates the behavior of diffusion in a completely sealed volume by ensuring the system is mass-conserving. Physically, the requirement that none of the substance may pass the boundary surface leads to the condition that the flow out of the surface is zero ($$\vec{n} \cdot \vec{\jmath}=0$$). When applied to diffusion via Fick's first law, this zero-flow translates into the boundary condition where the normal derivative of the concentration ($$u$$) is zero ($$\vec{n} \cdot \nabla u=0$$). This zero slope/zero gradient condition at the boundaries ensures that no substance enters or leaves the system. Over time, this sealed environment evolves toward a state of thermal or chemical equilibrium, where the concentration becomes uniform throughout the volume, equaling the initial average concentration.

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