Accuracy of lattice Monte Carlo simulation of biased diffusion models

The accuracy of lattice Monte Carlo (LMC) simulation of biased diffusion models is of great importance as far as the simulation credibility is concerned. It is known that the fixed time step LMC algorithm can reproduce the mean and the variance of the particle displacement exactly for all discrete time steps. Thereby, we propose to use the skewness and other quantities to measure the accuracy. For the one-dimensional fixed time step LMC simulation, we obtain an explicit expression for the skewness and find that the algorithm always produces a negative skewness that converges to zero in the long-time limit when the velocity is positive. It is proved that the skewness is inversely proportional to the square root of the number of simulation steps and the first step error only depends on the Péclet number. We further discuss several other measures of the accuracy of the approximation based on appropriately defined mean-square errors, leading to interesting, unexpected results. The accuracy measures can exhibit complicated nonmonotonic behavior and the optimal step size may depend on the measure of accuracy used.

Figure 1

Errors measured by skewness $\left(-{\eta }_t\right)$ as a function of simulation time $\left(s\right)$. Values of $\ell q$ (proportional to the Péclet number) are shown on each curve; individual values of $\ell$ and $q$ are also given.

Figure 2

Errors calculated by ${\Delta }_{\mathrm{II}}^{}$ versus number of simulation steps with different Péclet numbers. Note that the log-log scale is used in the plot.

Figure 3

Errors calculated by ${\Delta }_{\mathrm{II}}^{}$ versus simulation time. Note fitting curves connecting the data points are used in the plot.

Figure 4

Maximum error and first step error calculated by ${\Delta }_{\mathrm{II}}^{}$ versus Péclet number. Note fitting curves connecting the data points are used in the plot.

Figure 5

Maximum error and first step error calculated by ${\Delta }_{\mathrm{III}}^{}$ versus lattice step. Note the curves connecting the data points are fitting curves.