# Mean-squared displacement in Monte Carlo studies

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.

This is possible (see [1]) but uncommon, as it requires Monte Carlo moves that alter the current conformations by a very small perturbation. In that setting of "small" Metropolis MC moves, it is usually easier (both in theory and in practice) to just use Molecular Dynamics instead.

[1] Kikuchi, K., Yoshida, M., Maekawa, T., & Watanabe, H. (1991). Metropolis Monte Carlo method as a numerical technique to solve the Fokker—Planck equation. Chemical Physics Letters, 185(3–4), 335–338. https://doi.org/10.1016/S0009-2614(91)85070-D

• Very very interesting! Unfortunately, I cannot access the article and there does not seem to be an arXiv pre-print :( I wonder, how is the sampling expected to be done? As in, updating the msd after each cycle, or stepwise (this, after each displacement)? Does the paper provide a rule for determining how small the steps should be? Or is it simply a trial and error type of approach? Many thanks. Mar 4 '18 at 16:58
• The authors of the paper write a Fokker-Planck equation for the stochastic process associated to the Metropolis MC sampler. They extract the local drift and diffusion (via $\langle \Delta R \rangle$ and $\langle (\Delta R)^2 \rangle$) and show that it coincides with the Fokker-Planck equation associated to the corresponding stochastic differential equation. The example they use for illustration is a one-dimensional Ornstein-Uhlenbeck process where they compare their results to the analytic solution and they take steps of at most $10^{-2}$ units of length. Mar 4 '18 at 18:39

It is in some cases possible to map the dynamics obtained in MC simulations to other (more realistic) dynamics, especially for the case of dense colloidal suspensions. The following two papers talk about the problems and caveats of performing such a mapping:

Even though both papers are behind paywalls, it should be possible to find reprints online.

• Thank you very much, really the kinds of work I was looking for. These papers do not really go down to the nitty-gritty details, but do you reckon it matters how the msd is sampled during the simulation run? As in, move-wise, cycle-wise etc. Assuming the way I intend to decompose the msd into parallel (to long cylinder axis) and perpendicular contributions (a bit shown in the post) is correct :) Mar 5 '18 at 0:06
• I would definitely compute it cycle-wise to mimic what is done in MD simulations, but I'm not sure it would have a big overall impact. Just a word of warning: the rotational and translational diffusion coefficients in the ideal gas limit are linked by the Stokes-Einstein relation. For example, for spheres with no-slip boundary conditions $D_r = 3 D_t / \sigma^2$, where $\sigma$ is the sphere diameter (see e.g. here or here). Mar 5 '18 at 9:07
• This means that the trial displacements of the two types of MC moves should be chosen carefully. Mar 5 '18 at 9:08
• Dear Lorenzo, I am struggling on how to deal with periodic moves when computing MSD cycle-wise... I try to count the number of times periodic moves are made during the cycle, but I think I'm accounting for them still wrongly. Should I just then do things move-wise? Additionally, I'm recomputing MSD after each move attempt, i.e., irrespective of whether it is an accepted MC move or not. That makes sense right? Mar 19 '18 at 13:38
• The best course of action would be to store the absolute position of the particles rather than their box-wrapped one. This might cause some issues if your particles diffuse too much and the absolute positions become too large though. I thus advise you to store both sets of coordinates if you can't manage to just store the number of times each particle has crossed the box edge along each direction. Mar 22 '18 at 8:08