Non-universality of the dynamic exponent in two-dimensional random media

초록

The diffusion of solutes in two-dimensional random media is important in diverse physical situations including the dynamics of proteins in crowded cell membranes and the adsorption on nano-structured substrates. It has generally been thought that the diffusion constant, D, should display universal behavior near the percolation threshold, i.e., D - (phi - phi(c))(mu), where phi is the area fraction of the matrix, phi(c) is the value of phi at the percolation threshold, and mu is the dynamic exponent. The universality of mu is important because it implies that very different processes, such as protein diffusion in membranes and the electrical conductivity in two-dimensional networks, obey similar underlying physical principles. In this work we demonstrate, using computer simulations on a model system, that the exponent mu is not universal, but depends on the microscopic nature of the dynamics. We consider a hard disc that moves via random walk in a matrix of fixed hard discs and show that mu depends on the maximum possible displacement Delta of the mobile hard disc, ranging from 1.31 at Delta <= 0.1 to 2.06 for relatively large values of Delta We also show that this behavior arises from a power-law singularity in the distribution of transition rates due to a failure of the local equilibrium approximation. The non-universal value of mu obeys the prediction of the renormalization group theory. Our simulations do not, however, exclude the possibility that the non-universal values of mu might be a crossover between two different limiting values at very large and small values of Delta The results allow one to rationalize experiments on diffusion in twodimensional systems.

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