Random walks, trapping and reactions in quasi-one dimensional lattices

Abstract

We performed Monte Carlo simulations of diffusion processes on baguette-like lattices, which have very small width and height, but long length, in effect providing quasi-one dimensional systems. This is done by investigating random walk properties of single particles, and also the well known model bimolecular reactions A + A and A + B. We monitor the number of distinct sites visited, as a function of time, and also the survival probability in the presence of static traps. For the reaction systems we monitor, as usual, the decay of the particle density. The expected one-dimensional behavior is recovered, in the long time limit, for all cases studied. Our interest here is in the crossover time, from 3- (or 2-)dimensionaI behavior (early time) to one-dimensional (long time). We find that this crossover time scales with respect to the baguette's short dimension. However, this scaling deviates significantly from a mean square displacement law, and it is specific to both tube dimensionality and reaction type (e.g. A + A or A + B). Specifically, instead of an expected power of 2, the exponents range between 1 and 4. The densities of the A + B reactions at the dimensional crossover are compared to the densities at the segregation crossover in regular lattices at all three dimesnions. As expected, the time evolution of the A + A reaction parallels the behavior of the average number of distinct sites visited.