Based on the derivation, the diffusion equation and the heat equation are fundamentally the same partial differential equation, both arising from applying a constitutive relationship to the continuity equation for the conservation of an intensive quantity $u$ (concentration or heat concentration). This relationship, known as Fick's first law (for diffusion) or Fourier's law (for heat), posits that the current density $\jmath$ is proportional to the negative gradient of the quantity ( $-\nabla u$ or $-\nabla T$ ), causing flow from regions of high concentration/temperature to low concentration/temperature. For homogeneous and isotropic materials, this yields the standard form $\partial_t u-D \nabla^2 u=\kappa$, where $D$ is the diffusivity or thermal diffusivity, and $\nabla^2$ is the Laplacian, modeling how a substance or heat spreads out over time. When the medium is in motion, the resulting current must also include a convective term, $J_{\text {convection }}=u v$, in addition to the diffusive/conductive current.
The Diffusion Equation Derivation
Foundation: The diffusion equation is derived from the continuity equation ( $\partial_t u+\nabla$ $\jmath=\kappa$ ), which expresses the conservation of a property $U$ with concentration $u$.
Modeling the Current ( $\jmath$ ): For diffusion, the current density $j$ is modeled as being proportional to the negative gradient of the concentration ( $-\nabla u$ ), reflecting movement from high to low concentration.
Fick's First Law: This relationship is simplified for isotropic materials where the current is:
$$ \jmath=-D \nabla u $$
where $D$ is the scalar diffusivity.
The Final Equation: For homogeneous and isotropic materials, the continuity equation combined with Fick's Law yields the standard diffusion equation:
$$ \partial_t u-D \nabla^2 u=\kappa $$
where $\nabla^2$ is the Laplacian operator. The case with no source term ( $\kappa=0$ ) is sometimes called Fick's second law.
Equivalence to the Heat Equation
Mathematical Analogy: The mathematics of heat conduction is equivalent to diffusion. The intensive quantity is temperature ( $T$ ), and the current is the heat current.
Fourier's Law: The heat current is proportional to the negative temperature gradient:
$$ \jmath=-\lambda \nabla T $$
where $\lambda$ is the heat conductivity.
The Heat Equation: For homogeneous and isotropic materials, the equation describing temperature change is:
$$ c_V \rho \partial_t T-\lambda \nabla^2 T=\kappa $$
where $c_V \rho$ is a factor related to the specific heat capacity and mass density, linking heat concentration $(u)$ to temperature $(T)$.
Generalization and Complexity
Anisotropy: In the most general case, the material's properties are described by a rank two tensor (the diffusivity tensor $D_{i j}$ or heat conductivity tensor $\lambda_{i j}$ ), leading to a more complex partial differential equation (PDE).
Non-Homogeneity: If the diffusivity $D$ or $\lambda$ is a function of space (non-homogeneous), the resulting PDE includes a term involving the gradient of $D$ or $\lambda$, such as $(\nabla D) \cdot(\nabla u)$.
Convection: In moving media, the total current is the sum of diffusion/conduction and convection:
$$ \jmath=\jmath_{\text {conduction }}+\jmath_{\text {convection }}=-\lambda \nabla T+u v $$
where $u v$ represents the bulk transport of the quantity $u$ at velocity $v$.
Dissecting the Deterministic Roles of Diffusivity Decay and Dimensionality in PDEs
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