where q is the heat transfer rate or heat flux density with units of W/m<sup>2</sup>, T is the temperature expressed in °C or °K, ∇ is the gradient operator and k is the thermal conductivity with units of W/(m.K). It can be shown that the distribution of temperature is governed by the heat diffusion equation subject to the appropriate boundary conditions:
<math> \nabla^2 T(\mathbf{r}) - \frac{1} {\alpha}\frac{\partial T}{\partial t} = \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) - \frac{1} {\alpha}\frac{\partial T}{\partial t} = -\frac{w(\mathbf{r}) / }{k } </math>
where α is the thermal diffusivity with units of W/m and w(r) is the volume heat source with units of W/m<sup>3</sup>.
In the steady-state regime, the time derivative vanishes and the diffusion equation reduces to the Poisson equation:
<math> \nabla^2 T(\mathbf{r}) = -\frac{w(\mathbf{r}) / }{k } </math>