One-dimensional diffusion: Validity of various expressions for jump rates

The coverage dependence of the one-dimensional collective diffusion coefficient is analyzed by using the gradient expansion of the local density. The transition probabilities are written as an expansion of the probabilities of the occupation configurations. Since the detail balance principle determines only a part of the diffusion terms in the expansion, different functional relations are proposed for these terms. The diffusion coefficient is obtained for various choices of these relations. However, some of them seem to be not physically sound and the diffusion coefficient does not behave properly. The range of validity of various expressions for the jump rates is determined and phase diagrams are shown. Besides that, it is shown that the transition state theory guarantees physically suitable behavior of the coefficient of one-dimensional diffusion.

Figure 1

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

Figure 2

(a) Diagram $\gamma$ vs $\beta V$ corresponding to case III. The coverage dependence of the diffusivity, $D$, for varying first-neighbor interaction strength and $\gamma$ parameter. (b) $\beta V=-2$ and $\gamma =1,-1$. (c) $\beta V=2$ and $\gamma =1,-1$.

Figure 3

(a) Diagram $\gamma$ vs $\beta V$ corresponding to cases IV and V. The coverage dependence of the diffusivity, $D$, for varying first-neighbor interaction strength and $\gamma$ parameter. (b) Case IV: $\beta V=-2$ and $\gamma =0.5,-0.5$. (c) Case IV: $\beta V=2$ and $\gamma =0.5,-2$. (d) Case V: $\beta V=-2$ and $\gamma =0.5,-0.5$. (e) Case IV: $\beta V=2$ and $\gamma =0.5,-2$.

Figure 4

The coverage dependence of the diffusivity, $D$, for varying $\delta$ parameter using TST. (a) $\beta V=-2$: top to bottom $\delta =2,0.5,-1$. (b) $\beta V=2$: top to bottom $\delta =-1,0.5,2$.