Coles, Samuel W

Dunn, Josh

McCluskey, Andrew R

Morgan, Benjamin J

Squires, Alexander G

English

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                          McCluskey, A. R., Squires, A. G., Dunn, J., Coles, S. W., & Morgan,B. J. (2023). Accurate Estimation of Diffusion Coefficients and theirUncertainties from Computer Simulation. Poster session presented atRoyal Society of Chemistry Solid State Chemistry Group ChristmasMeeting 2023, Edinburgh, United Kingdom.Peer reviewed versionLicense (if available):CC BY-SALink to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/Accurate Estimation of Diffusion Coefficients and their Uncertainties from Computer SimulationAndrew R. McCluskey1,2, Alexander G. Squires3, Josh Dunn1, Samuel W. Coles4,5, Benjamin J. Morgan4,51. School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, UK. 2. European Spallation Source ERIC, Ole Maaløes vej 3, 2200 København N, DK. 3. School of Chemistry, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK. 4. Department of Chemistry, University of Bath, Claverton Down, Bath, BA2 7AY, UK. 5. The Faraday Institution, Quad One, Harwell Science and Innovation Campus, Didcot, OX11 0RA, UK. email: andrew.mccluskey@bristol.ac.ukThe optimal scheme to estimate the diffusion coefficient requires bothvariance and covariance to be considered.The Mean-squared displacement over a given number of steps is normally distributed...2D Random Walk for 16 StepsFrom the central limit theorem.12 16 20MSD(16)...However, as the number of steps increases so does the variance*...2D Random Walk for 128 Steps*Called Heteroscedasticity 96 128 160MSD(128)A particle can reach a greater varietyof displacement values given more steps....And The displacements are correlated.MSD(128)ρ=0.290MSD(32)MSD(128)ρ=0.499MSD(64)MSD(128)ρ=0.992MSD(127)the distance a particle covers in n steps affects the distance that it can cover in N+1, N+2, N+3, etc steps.a Covariant Multi-dimensional Normal Distribution can be used as a generative model for mean-squared displacement......where each dimension is a different number of steps or time-interval. More than two dimensions, make me feel funny...Heteroscedasticity and correlation must be accounted for to efficiently Estimate the diffusion coefficient and accurately estimate the uncertainty. Heteroscedastic and uncorrelatedHeteroscedastic and correlatedPopulation standard DeviationEstimated Uncertainty...any single estimate of D * is more likely to be closer to the true value......And the uncertainty can be accurately estimated from a single analysis. by Including heteroscedasticity and correlation... 1.20.81.0D *Homoscedasticdata from molecular dynamics simulation can be used To Parameterise the model. the Full covariance matrix can be modelled based on freely diffusing particles*Check out this link for the derivation of the covariance matrix.MSD(steps)MSD(steps)Modellednumericalratiolooks like, The disagreement is minimal and due to the large-step limit assumption.*In the large-step/long-time limit...so we must rescale the observed variance by the number of independent samples.σi =σNiMSD(steps)σOverlapping samples are drawn to maximise the number of observations.This ensures that we have a good estimate of the mean of the distribution...0 1 2 3 4 5 6Simulation Stepsnumber of steps = 2...but now our samples are notstatistically independent...With support for pymatgen, ASE, and MDAnalysis objectspip install kinisiAvailable Now! Open-Source!We can sample the parameterised distribution Likelihood with a Heaviside Prior to get an estimate of the diffusion coefficient distribution, which would typically require many 1000s of simulations....And using this method, on 4096 individual simulations, gives an unbiased, efficient  Estimate of D * with good variance estimate.This is a Bayesian sampling approach that uses Markov chain Monte Carlo...tourMSDp (m | MSD)MSD(time)80040000 64 128time/s1.11.00.9(D *|MSD) σ(D *)D *P (D *|MSD) Here, you will see the posterior distribution of linear models, and, up ahead, the marginal posterior for the diffusion coefficient. the estimated distribution is a bit broader than the numerical and the variance is slightly over estimated...p [σ2(D *)]σ2(D *)σ2(D *)10-30σ(D *)P (D *|MSD) P (D *) D *1.11.00.9...due to that pesky assumption......but the estimates are generally very good.thanks for reading!

Accurate Estimation of Diffusion Coefficients and their Uncertainties from Computer Simulation

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Accurate Estimation of Diffusion Coefficients and their Uncertainties from Computer Simulation

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