CONVOLUTION INTEGRAL FORMULATION OF DISPERSIVE DIFFUSION TRANSPORT - NUMERICAL-SOLUTIONS FOR THE COUPLING MODEL WAITING TIME DISTRIBUTION

Diffusive transport of a species in a correlated system such as entangled polymer chains or Coulombically interacting ions can be formulated as a continuous-time random walk. The physics of the correlations are determined by the waiting-time distribution. Many features of relaxation and transport in correlated systems have been successfully described by the coupling model. Previous work had used solutions in the short-time and asymptotically long-time regimes to show that a waiting-time distribution derived from the coupling model predicts a mean-square displacement which crosses over from a power-law in time to a long-time linear dependence. In the present work, it is demonstrated that convolution integrals can be derived directly for averaged quantities in dispersive diffusion transport which can then be solved for all times by direct numerical methods. Numerical solutions using the coupling model distribution show that the previous analytical solutions for the crossover behavior in the mean-square displacement are valid except for a narrow transition zone. The results also agree with Monte Carlo and molecular dynamics simulations from other workers on self-diffusion of entangled polymer chains and on ion diffusion with Coulomb interactions.