One-dimensional diffusion: Discrepancy between exact results and Monte Carlo calculations

The exact expression for the collective diffusion coefficient in one dimension, obtained by Payne and Kreuzer [Phys. Rev. B. 75, 115403 (2007)], is compared with Monte Carlo simulation. Different hopping kinetics are analyzed. For initial- and final-state interaction kinetics no anomalies are observed. However, for the so-called interaction kinetics where both initial- and final-state interactions are involved, it is shown that even when the transition rates satisfy the principle of detail balance, additional constraints are necessary to guarantee the diffusion of particles. These restrictions give rise to a phase diagram that determines the regions where the exact solution of the diffusion coefficient seem to be not physically sound. The Monte Carlo simulation allows us to analyze the mechanism of diffusion in these regions, where in some cases the simulation does not match the exact solution. A possible explanation is presented.

Figure 1

The four relevant hopping processes and their rates for a one-dimensional lattice gas with nearest-neighbor interactions.

Figure 2

Phase diagram for $A_1=-\gamma B_1$ and $C_{11}=0$ showing the different diffusion zones. The white color corresponds to the region $A$. The light gray corresponds to the region $B$, while the dark gray corresponds to the region $C$.

Figure 3

Collective diffusion coefficient vs $\theta$ for $\beta V=-2$ and $\gamma =-2$. Solid line corresponds to exact solution given by Eq. (11), while symbols correspond to MC simulation.

Figure 4

Mean-square displacement of the tracer particles vs time, for $\beta V=2$, $\gamma =-2,$ and $\theta =0.2$. The solid lines correspond to the analytical solution obtained by using Eq. (21).

Figure 5

Collapsed curves corresponding to the mean-square displacement of the tracer particles. In the inset the mean-square displacements of the tracer particles vs time for $\beta V=2$, $\gamma =-2$, are shown (from top to bottom $\theta =0.1,0.2,0.3,0.4,0.5$).

Figure 6

Collapsed curves corresponding to the mean-square displacement of the center of mass of the system. In the inset the mean-square displacements for the center of mass vs time for $\beta V=2$, $\gamma =-2$, are shown (from top to bottom $\theta =0.1,0.2,0.3,0.4,0.5$).