= Thermal diffusion time
{title2=$L^2/\kappa$}
The <thermal diffusion time> $t_D=L^2/\alpha$, with $\alpha=\kappa$ the <thermal diffusivity>, is a characteristic scale for smoothing a <temperature> variation over <length> $L$. It is the thermal instance of the <diffusive time scale>. Rescaling $x=L\widehat x$ and $t=(L^2/\alpha)\widehat t$ removes $L$ and $\alpha$ from the constant-coefficient <heat equation>. Doubling <length> therefore multiplies comparable internal equilibration times by four; increasing <thermal diffusivity> by four divides them by four.
This scale is not a universal time of complete <thermal equilibrium>. A nonzero <Fourier mode> decays exponentially and never becomes exactly zero at finite time. The geometry, <boundary conditions>, initial pattern and accepted residual determine the numerical factor. For example, an insulated bar's slowest nonuniform <Fourier mode> has decay time $L^2/(\pi^2\alpha)$, and reducing its amplitude to a fraction $\varepsilon$ requires
$$
t_\varepsilon=\frac{L^2}{\pi^2\alpha}\log\frac1\varepsilon,
\qquad 0<\varepsilon<1.
$$
External <heat transfer> can impose a longer cooling time even when internal <temperature> differences smooth quickly. <Thermal diffusivity> alone does not determine cooling to the room's <temperature>.
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