Is measuring mean-squared-displacement in Monte Carlo simulations uncommon? I'm very interested to find out if this has actually ever been tried. For instance, in the context of spheres, or cylindrical particles in a periodic simulation box, where we have our usual Monte Carlo moves of translation and rotation. So a particle randomly selected and moved (either translation or rotation), this is repeated $N$ times in one Monte Carlo cycle, $N$ being the number of particles.
So we can after each cycle, or even each displacement move, also calculate the MSD with respect to the initial configuration of the system. For example having the centre of masses $\vec{R}$ of each cylinder, and their respective orientation vectors $\vec{O},$ after each move we can update the MSD along the direction of the cylinder by: $$ \begin{align} \Delta \vec{R}(t) &= \vec{R}(t)-\vec{R}(0) \\ \Delta \vec{R_{||}} (t) &= (\Delta\vec{R}(t)\cdot \vec{O}(0))\vec{O}(0)/l^2 \end{align} $$
and the total msd along the parallel direction gets updated (i.e., incremented) by $\Delta \vec{R_{||}}\cdot \Delta \vec{R_{||}}/N.$ Where $\vec{O}(0)$ is the orientation vector of the chosen cylinder at time 0 ( so initial condition). From this, I guess we can write the perpendicular to long axis contribution to msd as, $\Delta \vec{R}_{\perp} = \Delta \vec{R} - \Delta \vec{R}_{||}$.
Ultimately, from the MSD curves calculated during the MC simulation, we'd extract diffusion properties among other things. I wonder, is this at all meaningful to do in Monte Carlo simulation? Has the validity of this been discussed somewhere? I so far haven't found anything work on this, curious to find out if there are in fact any.