Diffusion on a lattice: Transition rates, interactions, and memory effects

We analyze diffusion of particles on a two-dimensional square lattice. Each lattice site contains an arbitrary number of particles. Interactions affect particles only in the same site, and are macroscopically represented by the excess chemical potential. In a recent work, a general expression for transition rates between neighboring cells as functions of the excess chemical potential was derived. With transition rates, the mean-field tracer diffusivity, $D^{\text{MF}}$, is immediately obtained. The tracer diffusivity, $D=D^{\text{MF}}f$, contains the correlation factor $f$, representing memory effects. An analysis of the joint probability of having given numbers of particles at different sites when a force is applied to a tagged particle allows an approximate expression for $f$ to be derived. The expression is applied to soft core interaction (different values for the maximum number of particles in a site are considered) and extended hard core.

Figure 1

Correlation factor $f$ against density $\rho$ for soft-core interaction and different values of $\mathrm{\Omega }$, the maximum number of particles allowed. Points are numerical results and curves correspond to Eq. (26) with $c=0.156$. Black dotted and dashed curves correspond to analytical results of Nakazato [18] and Chaturvedi [22] respectively, obtained for $\mathrm{\Omega }=1$.

Figure 2

Exclusion regions, shown in blue around the central particle, for different values of $b$. A configuration with maximum concentration is partially represented in each case with crosses at the centers of the particles.

Figure 3

Normalized tracer diffusivity, $D/\nu a^2$, against packing fraction, $\xi$, for different values of $b$. Dots are numerical results and curves represent Eq. (29) with adjusted values of $c_1$ and $c_2$.