= Photon diffusion coefficient for isotropic scattering
{title2=$D=v\ell/3$}
For conservative <photon> transport at constant speed $v$, suppose successive scattering directions are independent and isotropically distributed, and free-path lengths follow an <exponential distribution> of mean $\ell$. In three spatial dimensions, the resulting long-time <diffusion coefficient> is
$$
\boxed{D=\frac{v\ell}{3}.}
$$
Here $\ell$ is the scattering <mean free path>. These assumptions describe homogeneous, memoryless interactions without <absorption>.
The <velocity autocorrelation function> gives a direct derivation. An initially isotropic ensemble has zero mean <velocity>. During a lag $s$, the probability of no scattering is $e^{-vs/\ell}$. No scattering preserves the direction; one or more independent isotropic scattering events give zero expected directional correlation. Consequently
$$
C_v(s)=\mathbb E[\mathbf v(0)\cdot\mathbf v(s)]=v^2e^{-vs/\ell}.
$$
The three-dimensional <Green-Kubo formula for diffusion> now yields
$$
D=\frac13\int_0^\infty C_v(s)\,ds=\frac{v\ell}{3}.
$$
The factor $1/3$ converts the dot-product correlation into one Cartesian component under isotropy.
The same correlation gives the full <mean squared displacement>:
$$
\mathbb E|\mathbf r(t)-\mathbf r(0)|^2=2\int_0^t(t-s)C_v(s)\,ds=2v\ell t-2\ell^2\left(1-e^{-vt/\ell}\right).
$$
At short times this is $v^2t^2$; at long times its leading term is $6Dt$. The model therefore connects direct flights to the diffusive many-scattering limit.
The free-path distribution matters. A <fixed-step isotropic random flight> with every length equal to $\ell$ has mean-square displacement $n\ell^2$ after $n$ steps and elapsed time $n\ell/v$, giving $D=v\ell/6$. Replacing exponential flights by fixed-length flights thus changes the coefficient by a factor of two. For anisotropic <scattering>, directional persistence instead leads to a <transport mean free path>; strong <absorption> and boundaries require additional modeling.
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