Hi,

the first formula ((16) of DiffuDict 2023 User Guide) is used to compute the 3x3 diffusivity matrix from a set of tracked molecules. For every molecule we have a start point \( x_0 \) and an end point \( x_t \) and we can use this set to compute the diffusivity by:

\[ D=\frac{E((x_t-x_0)(x_t-x_0)^T)}{2t} \]

The 3D geometry and the velocity are an input for the simulation in this case.

Your second formula is actually formula (17) of the DiffuDict 2023 User Guide:

\[ d_0=\frac{1}{3}L\bar{v} \]

where our char. Length L is your d, and the mean thermal velocity is

\[ \bar{v}=\sqrt{\frac{8RT}{\pi M_A}} \]

(which is also formula (26) of the User Guide). This computes the one-dimensional diffusivity of a cylindrical pore and neglects all 3D effects that a more complex structure may show. However, by assuming that there is some characteristic length L of the 3D structure, one can use the value \( d_0 \) as a scaling factor for the diffusivity computed with Einsteins formula, and then one gets a dimensionless 3x3 diffusivity \( D^{\ast} \) that is independent from the diffusing species. This is then formula (18) of the User Guide:

\[ D=d_0\cdot D^{\ast} \]

The \( D^{\ast} \) could be understood as the three-dimensional pore shape factor that is missing in your cylindrical pore formula.

However, if you have a look at page 38 of the DiffuDict 2023 User Guide, you can see that if you enter GeoDict with a single cylindrical pore structure, you get exactly the result from your second formula in the pore direction (and 0 in all other directions).

Best regards.