Random walks with imperfect trapping in the decoupled-ring approximation

Abstract

Abstract. We investigate random walks on a lattice with imperfect traps. In one dimension, we perturbatively compute the survival probability by reducing the problem to a particle diffusing on a closed ring containing just one single trap. Numerical simulations reveal this solution, which is exact in the limit of perfect traps, to be remarkably robust with respect to a significant lowering of the trapping probability. We demonstrate that for randomly distributed traps, the long-time asymptotics of our result recovers the known stretched exponential decay. We also study an anisotropic three-dimensional version of our model, where for sufficiently large transverse diffusion the system is described by the mean-field kinetics. We discuss possible applications of some of our findings to the decay of excitons in semiconducting organic polymer materials, and emphasize the crucial influence of the spatial trap distribution on the kinetics.

Random walks with fractally correlated traps: Stretched exponential and power-law survival kinetics

“…In this case, equation (19) predicts stretched exponen-tial relaxation with α = 1/2, and numerical simulation confirms this over the time interval (26). The simulations also show agreement with theoretical predictions for other types of simple (Euclidean) trap configurations embedded in one-and two-dimensional host lattices.…”

“…Due to the presence of arbitrarily large trap-free regions where the walker can survive for a long time, the survival probability decays slower than exponentially, following a stretched exponential function with stretching exponent α = d/(d + 2) for Euclidean host lattices [13][14][15][16][17] and α = d s /(d s + 2) for fractals [11][12][13]. The same asymptotic behavior was also found to hold for randomly distributed imperfect traps [18,19]. While some authors argue that stretched exponential long-time decay due to this mechanism occurs only when f (t) drops to extremely low values and thus is not practically observable, more recent simulations show that the mechanism may manifest itself in an experimentally observable range of values of f (t) [20].…”

“…We stress again that while the interval (21) corresponds only to the initial stage of relaxation, the absolute duration of this stage for weakly absorbing traps (γ ≪ 1) may be significant. For times much larger than t 0 = w −1 γ −1/α , the approximation (16) for P (t) ceases to be valid, and simulation shows that stretched exponential relaxation is replaced by power law decay f (t) ∼ t −α with the same α given by (19). We shall postpone detailed discussion of this regime until Section 7.…”

“…For times much larger than t 0 = w −1 γ −1/α , the approximation (16) for P (t) ceases to be valid, and simulation shows that stretched exponential relaxation is replaced by power law decay f (t) ∼ t −α with the same α given by (19). We shall postpone detailed discussion of this regime until Section 7.…”

Energy Transfer and Emission Decay Kinetics in Mixed Microporous Lanthanide Silicates with Unusual Dimensionality

“…Random walks in 1D systems with annihilation by trapping have been intensively studied in the last 2 decades. 28, [30][31][32][33][34][35][36] The approach we use here is the decoupled-ring approximation. 36 We consider a linear onedimensional chain of infinite regular lattice sites, in which a fraction, f, of the lattice sites are replaced by traps.…”