Statistical error in simulations of Poisson processes: Example of diffusion in solids

Simulations of diffusion in solids often produce poor statistics of diffusion events. We present an analytical expression for the statistical error in ion conductivity obtained in such simulations. The error expression is not restricted to any computational method in particular, but valid in the context of simulation of Poisson processes in general. This analytical error expression is verified numerically for the case of Gd-doped ceria by running a large number of kinetic Monte Carlo calculations.

Figure 1

The fluorite structure. The $2\times 2\times 2$ (96 sites) cerium oxide supercell structure is shown in thin lines and the cerium oxide unit cell in thick lines. Large green circles are Ce atoms and red small circles are O atoms. From the supercell one O atom is removed to create a vacancy, and two Ce atoms are replaced by Gd atoms at positions (0, 0, 0) and $(a/2,a/2$, 0), marked by small arrows in the figure. $a$ is the lattice constant.

Figure 2

Waiting time distributions originating from: (a) NEMD simulation with field strength $F_{\mathrm{ext}}=0.5$ $\mathrm{eV}/\AA$ (distributions obtained for other color field strengths in the linear regime are similar), and (b) a KMC simulation of $n=10000000$ steps with prefactor $\nu =5\times {10}^{12}$ Hz. The waiting times are binned and presented in a histogram (green bars). Red solid line shows a fitted exponential function. Black dashed vertical line shows the mean waiting time $\overline{\tau }$.

Figure 3

Red solid line is mean waiting time versus simulation length for a KMC simulation with prefactor $\nu =5\times {10}^{12}$ Hz. Dashed black line is the mean waiting over the whole interval.

Figure 4

Mean waiting time $\overline{\tau }$ vs field strength from NEMD (blue dots). The dashed line is a linear fit to the points in the linear response regime, i.e., $F_{\mathrm{ext}}\le 0.5$ eV/Å. Triangles are mean waiting time from KMC (zero field) with prefactors: $3\times {10}^{12},4\times {10}^{12},5\times {10}^{12},6\times {10}^{12}$ Hz, from top to bottom. Error bars are calculated according to Eq. (11) and lie within the plot symbols if not visible.

Figure 5

Histogram of mean waiting time $\overline{\tau }$ for $N=50000$ KMC simulations with prefactor $\nu =5\times {10}^{12}$ Hz (green bars). The mean ${\mu }_{\overline{\tau }}$ and the standard deviation ${\sigma }_{\overline{\tau }}$ of the mean waiting time is shown in each figure. The number in parenthesis is ${\sigma }_{\overline{\tau }}/{\mu }_{\overline{\tau }}$ expressed in percent. Vertical dashed lines mark ${\mu }_{\overline{\tau }}\pm {\sigma }_{\overline{\tau }}$. The red solid line is the normal distribution $Normal({\mu }_{\overline{\tau }},{\sigma }_{\overline{\tau }}^2)$. The subfigures are for different simulation lengths (diffusive jumps) $n$.