Interacting Brownian particles in a two dimensional periodic potential

We study the dynamic properties of a system of N interacting Brownian particles immersed in a two-dimensional periodic potential in relation to superionic conductors. By varying the density of the mobile ions and the strength of the pair interaction, we investigate commensurability and correlation effects on the conduction process by an analysis of the diffusion coefficient. Both the Frenkel-Kontorova and Yukawa pair potentials between Brownian particles are used for this special investigation. Our numerical results, obtained form Langevin dynamics indicate a large difference between the transport properties of the one-dimensional and the two-dimensional systems, in the range of the weak pair interaction and for certain concentrations around 0.77. The maximum conductivity of two-dimensional conductors is predicted for concentration c = 23, while in the one-dimensional case it is always located at concentration c = 0.77. Furthermore we have evaluated analytically the dc-conductivity, in the two-dimensional case, for any value of the correlation length and for concentration c = 23. The analytical predictions are in good agreement with those of the numerical calculations.