Efficiency of trapping processes in regular and disordered networks

Abstract

We use a Markov method to study the efficiency of trapping processes involving both a random walker and a deep trap in regular and disordered networks. The efficiency is gauged by the mean absorption time (average of the mean number of steps performed by the random walker before being absorbed by the trap). We compute this quantity in terms of different control parameters, namely, the length of the walker jumps, the mobility of the trap, and the degree of spatial disorder of the network. For a proper choice of the system size, we find in all cases a nonmonotonic behavior of the efficiency in terms of the corresponding control parameter. We thus arrive at the conclusion that, despite the decrease of the effective system size underlying the increase of the control parameter, the efficiency is reduced as a result of an increase of the escape probability of the walker once it finds itself in the interaction zone of the trap. This somewhat anti-intuitive effect is very robust in the sense that it is observed regardless of the specific choice of the control parameter. For the case of a ring lattice, results for the mean absorption time in systems of arbitrary size are given in terms of a two-parameter scaling function. For the case of a mobile trap, we deal with both trapping via a single channel (walker-trap overlap) and via two channels (walker-trap overlap and walker-trap crossing), thereby generalizing previous work. As for the disordered case, our analysis concerns small world networks, for which we see several crossovers of the absorption time as a function of the control parameter and the system size. The methodology used may be well suited to exploring characteristic time scales of encounter-controlled phenomena in networks with a few interacting elements and the effect of geometric constraints in nanoscale systems with a very small number of particles.