I know that the diffusion equation is a more general version of the heat equation. But what is the exact difference informally?
[Physics] the difference between the diffusion equation and the heat equation
definitiondiffusionterminology
Related Question
- Fluid Dynamics – What’s the Exact Difference Between Diffusion, Convection, and Advection?
- [Physics] the correct form of heat diffusion equation taking into account temperature dependence of specific heat
- Thermodynamics – Difference Between ‘Heat’ and ‘Thermal Energy’
- Schroedinger Equation – Difference Between Solutions of the Diffusion Equation with an Imaginary Diffusion Coefficient and the Wave Equation
Best Answer
The difference is typically the diffusion coefficient: \begin{align} \frac{\partial \psi}{\partial t}&=\nabla\cdot\left(\kappa\nabla\psi\right)\tag{diffusion}\\ \frac{\partial \psi}{\partial t}&=\kappa\nabla^2\psi\tag{heat} \end{align} Under the diffusion equation, we typically take $\kappa$ to be a spatially-dependent variable whereas in the heat equation it is a uniform constant (allowing us to use the Laplacian on $\psi$).
However, the heat equation can have a spatially-dependent diffusion coefficient (consider the transfer of heat between two bars of different material adjacent to each other), in which case you need to solve the general diffusion equation.
There is no relation between the two equations and dimensionality. Both equations can be solved in one dimension, with a straight-forward substitution $\nabla\to\frac{\partial}{\partial x}$, or left in the multiple dimensions as I give above (which would likely require a numerical solver).