I am involved with a project to simulate the ionic conductivity of a system using a machine learning based force field which has been integrated with LAMMPS. The literature encourages very long runs on large simulation cells. However, some trial runs (using fix nve and the verlet integrator) have made it clear that the Lyapunov instability is very much in control of the output. The textbook “Understanding Molecular Simulation” by Frenkel and Smit suggests that it is nevertheless possible to put this “skeleton in the closet” and get reasonable (and approximately reproducible???) estimates of average correlation functions from molecular dynamics simulations. The literature mentions a number of “recipes” for performing the averaging of the correlation functions, but I wonder if the LAMMPS software developers have some preferred averaging methods or perhaps some documentation on this issue. Any advice is greatly appreciated.
Thanks, Natalie Holzwarth
Hi Natalie,
Speaking with my “academic” hat on rather than my “LAMMPS dev” hat on – you should look into Evans and Morriss as the starting textbook for nonequilibrium molecular dynamics, on top of Frenkel and Smit. Ionic conduction is that nastiest of beasts, a low-field non-equilibrium steady state, and you are absolutely right that a straightforward sampling approach requires very off-putting simulation sizes and times.
In terms of recent publications I recommend https://pubs.acs.org/doi/10.1021/acs.jpcb.2c08047 – I believe @Oystein is occasionally here, and I am a huge fan of the reproducibility that went into making the inputs available freely online, so I am always happy to recommend this paper as a good recent study.
There is also Nonequilibrium molecular dynamics for accelerated computation of ion–ion correlated conductivity beyond Nernst–Einstein limitation | npj Computational Materials for which the first author recently visited my prior supervisor (Prof Debra Bernhardt – who’s done a lot of non-equilibrium simulation and works on ionic conductivities too). I’ve worked on and off with trying to re-formalize their proof in deterministic instead of Langevin dynamics – it’s a very powerful and underrated approach that deserves more attention. Happy to work with you on the theory and coding if needed (I have a neat little fix for doing what this paper does in LAMMPS, but I need to be more rigorous about the best place in the timestep to apply it).
Note that the Lyapunov instability is a feature, not a bug – ionic conductivity is hard precisely because the ionic interactions induce dynamic correlations that are long-ranged (relative to the immediate g(r) peaks) that make the dynamics significantly more chaotic.
You should also consult this paper, and the papers that cite it, if you haven’t done so yet: https://pubs.acs.org/doi/10.1021/acs.jpcb.0c07704
… except I’ve just noticed they used NpT integration but did not compensate their MSD measurements for changing box size as far as I can tell (see https://pubs.aip.org/aip/jcp/article/153/2/021101/76127/Systematic-errors-in-diffusion-coefficients-from ) so their numerical results are unusable. They’re asking the right questions though! 
Thanks @srtee for the mention!
From what I understand, the order-n algorithm for calculating correlation functions described in “Understanding Molecular Simulation” is still one of the most efficient methods. We used the modified order-n algorithm by Dubbeldam et al., which is an improvement over the original, and described here: https://doi.org/10.1080/08927020902818039
There aren’t that many implementations of it yet (as far as I know), but the OCTP module (https://pubs.acs.org/doi/10.1021/acs.jcim.8b00939) uses it and we used it in our own implementation/code for the “Charge transport …” paper. I can send it to you if you want. A great benefit of using this is that you can calculate the MSDs on-the-fly, i.e. during the simulation, which means you don’t have to save large dump files from the simulation for post-processing.
Cheers – I did find the OCTP stuff very interesting too. Let me know if it’s something you or your group are working on, I’m happy to help.
Thanks very much for all of these helpful comments and particularly for the references which I am still reading and digesting. It is such an important topic and your help is greatly appreciated. Sincerely, Natalie Holzwarth