= Thermal diffusion length
{title2=$\ell(t)=\sqrt{\alpha t}$}
The <thermal diffusion length> $\ell(t)=\sqrt{\alpha t}$ is the characteristic <length> over which <thermal conduction> changes a <temperature> distribution during time $t$ in the constant-<thermal diffusivity> model. It is a scale, not a sharp front or distance travelled at a constant speed: the <heat equation> has <infinite propagation speed> mathematically. Different definitions of penetration depth introduce numerical factors. In particular, the normalized one-dimensional <heat kernel> has spatial <variance> $2\alpha t$, so its root-mean-square width is $\sqrt{2\alpha t}$.
\Image[/thermal-diffusivity-length-time-scaling.png]
{title=<Thermal diffusion length> grows as the square root of time; <thermal diffusion time> grows as <length> squared}
{description=Original predictions for the two stipulated materials in <thermal diffusivity>. Left: $\ell=\sqrt{\alpha t}$, with four times the time needed to double <length>. Right: $t_D=L^2/\alpha$ on logarithmic axes. Equal <thermal conductivity> with four times the <volumetric heat capacity> halves the reached <length> at a given time and quadruples the time at a given <length>. The scale $t_D$ differs from the insulated-bar decay time by the factor $\pi^2$.}
At $t=10\,\mathrm s$, the model materials have <thermal diffusion lengths> $4.47$ and $2.24\,\mathrm{mm}$. For $L=10\,\mathrm{mm}$ their <thermal diffusion times> are $50$ and $200\,\mathrm s$. These values describe the same <heat equation> as the <thermal diffusion rate of a Fourier mode>; the different numerical factors come from the definition of the scale.
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