Diffusion problems or: How I Learned to Stop Worrying and Love Correlated Data
ISIS ML Discussion Group — 2023/02/02
Acknowledgements
Ben Morgan & Sam Coles
University of Bath/Faraday Institute
Diffusion
- How atoms and molecules move in materials is of fundamental interest.
- We can study diffusion with molecular dynamics (MD) simulations.
A particle on a 2D random walk.
Diffusion quantification
- In an MD simulation, we can sample the displacement of an atom over some time interval, \(t\).
A particle on a 2D random walk, with displacement vector.
Self-diffusion coefficient
- Quantified the diffusion → linked to conductivity.
- Found by fitting a straight line.
\[ \big\langle \Delta \mathbf{r} (t) ^2 \big\rangle = 2dD^*t + c \]
Mean-squared displacement as a function of time interval.
Self-diffusion coefficient
- Quantified the diffusion → linked to conductivity.
- Found by fitting a straight line.
\[ \big\langle \Delta \mathbf{r} (t) ^2 \big\rangle = 2dD^*t + c \]
Mean-squared displacement as a function of time interval, with a line of best fit.
Stochastic simulations
- MD simulations are stochastic.
- There are many possible mean-squared displacements that might be observed.
Examples of different random walks, each with a different line of best fit.
Stochastic simulations
- MD simulations are stochastic.
- There are many possible diffusion coefficients that might be observed.
The distribution of possible \(D^*\) values.
Uncertainty underestimation
- Fitting a straight line underestimates the true uncertainty.
The distribution of possible \(D^*\) values with the estimated uncertainty from a single straight line.
Displacements are not iid
- Independent: the displacement is correlated.
- Identically distributed: the variance increases with \(t\).
Mean-squared displacement, with error bars (left) as a function of time interval and covariance matrix (right).
Account for these aspects
-
We can use:
- Generalised least squares (limited to linear problems),
- Probabilistic sampling with a covariant multivariate normal distribution.
The distribution of possible \(D^*\) values with the GLS estimated uncertainty from a single straight line.
Accurate uncertainty & high efficiency
- When considering the non-iid nature, there is also a significant improvement in statistical efficiency.
Comparison of results from GLS (left) and OLS (right)
Questions?
Tak for at lytteDiscussion
Considering an experimental/simulation technique that you work with regularly:-
Do you know what effect ignoring correlations in your errors
would have on your data/system/decision making? - Are you worried about correlations in your data/system/decision making?
- Should you be?
-
How would you go about determining correlation
between data for your technique?