Kinetic Monte Carlo investigation of tetragonal strain on Onsager matrices

We use three different methods to compute the derivatives of Onsager matrices with respect to strain for vacancy-mediated multicomponent diffusion from kinetic Monte Carlo simulations. We consider a finite difference method, a correlated finite difference method to reduce the relative statistical errors, and a perturbation theory approach to compute the derivatives. We investigate the statistical error behavior of the three methods for uncorrelated single vacancy diffusion in fcc Ni and for correlated vacancy-mediated diffusion of Si in Ni. While perturbation theory performs best for uncorrelated systems, the correlated finite difference method performs best for the vacancy-mediated Si diffusion in Ni, where longer trajectories are required.

Figure 1

Relative statistical errors $\sigma$ and true errors $\varepsilon$ of the strain derivative of diffusivity $D_{xx}^{\prime }$ as a function of the finite difference step size $h$ for the FD and CFD methods. In all the plots, we use different shaped symbols to denote the errors from different methods: squares for FD and triangles for CFD (and diamonds for PT in the other figures). We use open symbols for the relative statistical errors $\sigma$ and solid symbols for the relative true errors $\varepsilon$. We want the statistical errors to be a good estimate of the true errors so that the systematic errors are negligible. In this figure, the relative statistical errors decrease monotonically with increasing $h$. The statistical errors estimate the true errors well for small $h$. However, the good agreement between errors breaks down for $h$ larger than $2\times {10}^{-4}$.

Figure 2

Relative errors of $D_{xx}^{\prime }$ as a function of $N_{\text{step}}$ for a fixed number of trajectories, $N_{\text{traj}}=5\times {10}^5$. The relative statistical errors of the FD and CFD methods do not depend on $N_{\text{step}}$, while for the PT method, ${\sigma }^{\text{PT}}\propto \sqrt{N_{\text{step}}}$. The statistical errors estimate the true errors well for all values of $N_{\text{step}}$.

Figure 3

Relative errors of $D_{xx}^{\prime }$ vs $N_{\text{step}}$ for a fixed total computational effort, $N_{\text{tot}}=N_{\text{step}}{N}_{\text{traj}}={10}^7$. For the FD and CFD methods, ${\sigma }^{\text{FD}}$ and ${\sigma }^{\text{CFD}}\propto \sqrt{N_{\text{step}}}$, whereas for the PT method, ${\sigma }^{\text{PT}}\propto N_{\text{step}}$. The statistical errors estimate the true errors well for all values of $N_{\text{step}}$.

Figure 4

Relative errors of $D_{xx}^{\prime }$ vs the number of trajectories $N_{\text{traj}}$. We choose $h$ and $N_{\text{step}}$ such that the statistical errors are a good estimate of the true errors. As $N_{\text{traj}}$ increases, the statistical errors of all three methods decrease as $\sigma \propto 1/\sqrt{N_{\text{traj}}}$. The CFD method produces lower statistical errors than the FD method using correlated trajectories, and the PT method has the lowest statistical errors.

Figure 5

Relative statistical and true errors of $L_{xx}^{\prime \text{NiSi}}$ as a function of finite difference step size $h$. Similarly to the single vacancy diffusion system, for both finite difference methods the statistical errors decrease monotonically with increasing $h$. The statistical errors are a good estimate of the true errors for small $h$, but the agreement breaks down for $h$ larger than $2\times {10}^{-4}$.

Figure 6

Relative errors of $L_{xx}^{\prime \text{NiSi}}$ vs $N_{\text{step}}$ for a fixed number of trajectories. The statistical errors behave similarly to those of the single vacancy diffusion system. However, the true errors behave differently: there is a threshold value of $N_{\text{step}}=2\times {10}^2$ above which the statistical errors estimate the true errors well, while below this value the true errors become larger than the statistical errors.

Figure 7

Relative errors of $L_{xx}^{\prime \text{NiSi}}$ vs $N_{\text{step}}$ for a fixed total computational effort $N_{\text{step}}{N}_{\text{traj}}=2\times {10}^7$. The statistical errors behave similarly to those of the uncorrelated single vacancy diffusion system: ${\sigma }^{\text{FD}}$ and ${\sigma }^{\text{CFD}}\propto \sqrt{N_{\text{step}}}$, while ${\sigma }^{\text{PT}}\propto N_{\text{step}}$. However, in this correlated case, there is a threshold value of $N_{\text{step}}=2\times {10}^2$ above which the statistical errors are a good estimate of the true errors, while below this value the true errors become larger than the statistical errors.

Figure 8

Relative errors of $L_{xx}^{\prime \text{NiSi}}$ vs $N_{\text{traj}}$. We choose $h$ and $N_{\text{step}}$ such that the statistical errors estimate the true errors well. Similarly to the single vacancy diffusion system, as $N_{\text{traj}}$ increases, the statistical errors of all three methods decay as $\sigma \propto \sqrt{N_{\text{traj}}}$. The CFD and PT methods give similar magnitudes of statistical errors, which are lower than those of the FD method.