Noise-cancellation algorithm for simulations of Brownian particles

We investigate the usage of a recently introduced noise-cancellation algorithm for Brownian simulations to enhance the precision of measuring transport properties such as the mean-square displacement or the velocity-autocorrelation function. The algorithm is based on explicitly storing the pseudorandom numbers used to create the randomized displacements in computer simulations and subtracting them from the simulated trajectories. The resulting correlation function of the reduced motion is connected to the target correlation function up to a cross-correlation term. Using analytical theory and computer simulations, we demonstrate that the cross-correlation term can be neglected in all three systems studied in this paper. We further expand the algorithm to Monte Carlo simulations and analyze the performance of the algorithm and rationalize that it works particularly well for unbounded, weakly interacting systems in which the precision of the mean-square displacement can be improved by orders of magnitude.

Figure 1

Trajectory of a free Brownian particle $\mathrm{\Delta }{x}^{\text{f}}\left(t\right)$, a Brownian particle in a periodic barrier potential $\mathrm{\Delta }x\left(t\right)$ with identical noise (see Sec. 5), and the difference between the two, the reduced trajectory $\mathrm{\Delta }{x}^{\mathrm{red}}\left(t\right)$.

Figure 2

Analytical results in lines and simulation results in symbols for each term in Eq. (3) for the harmonic relaxator. (a) Reduced MSD $\langle \mathrm{\Delta }{x}^{\mathrm{red}}{\left(t\right)}^2\rangle$ and standard MSD $\langle \mathrm{\Delta }x{\left(t\right)}^2\rangle$ for $\mathrm{\Delta }t/\tau =0.01$. The dashed gray line represents the free diffusion and serves as a guide for the eye. (b) CC term $\langle \mathrm{\Delta }{x}^{\mathrm{red}}\left(t\right)\mathrm{\Delta }x\left(t\right)\rangle$ for different simulation time steps $\mathrm{\Delta }t$.

Figure 3

Comparison of the standard error of the mean $\mathrm{\Sigma }\left(t\right)$ of the reduced MSD computed using the NC algorithm and the standard (STD) MSD for different simulation time steps for a harmonic relaxator.

Figure 4

Simulation results for each term in Eq. (3) for a Brownian particle in a periodic barrier potential for different potential heights $\mathrm{\Delta }U$. (a) Standard MSD $\langle \mathrm{\Delta }x{\left(t\right)}^2\rangle$. (b) Reduced MSD $\langle \mathrm{\Delta }{x}^{\mathrm{red}}{\left(t\right)}^2\rangle$ and the absolute value of the CC term $|\langle \mathrm{\Delta }x\left(t\right)\mathrm{\Delta }{x}^{\mathrm{red}}\left(t\right)\rangle |$.

Figure 5

Comparison of the standard error of the mean $\mathrm{\Sigma }\left(t\right)$ of the reduced MSD computed using the NC algorithm and standard MSD for different barrier heights for a Brownian particle in a periodic barrier potential.

Figure 6

VACF for the periodic barrier potential computed using the two different algorithms, standard and NC on double-logarithmic scales for $U=1.0k_{\text{B}}T$. The dashed line serves as a guide for the eye.

Figure 7

Simulation results for each term in Eq. (3) for a Brownian particle in a cosine potential for different peak-to-peak amplitudes $\mathrm{\Delta }U$. (a) Standard MSD $\langle \mathrm{\Delta }x{\left(t\right)}^2\rangle$. (b) Reduced MSD $\langle \mathrm{\Delta }{x}^{\mathrm{red}}{\left(t\right)}^2\rangle$ and the absolute value of the CC term $|\langle \mathrm{\Delta }{x}^{\mathrm{red}}\left(t\right)\mathrm{\Delta }x\left(t\right)\rangle |$.

Figure 8

Comparison of the standard error of the mean $\mathrm{\Sigma }\left(t\right)$ of the reduced MSD computed using the NC algorithm and standard MSD for different peak-to-peak amplitudes $\mathrm{\Delta }U$ for a Brownian particle in a cosine potential.

Figure 9

VACF $Z\left(t\right)$ for the cosine potential computed using the two different algorithms on double-logarithmic scales for $\mathrm{\Delta }U=0.1k_{\text{B}}T$. $Z\left(t\right)$ from the standard (STD) method is shown in dots and $Z_{\text{NC}}\left(t\right)$ from the NC algorithm is shown in triangles.