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

Abstract

We consider the survival probability f (t) of a random walk with a constant hopping rate w on a host lattice of fractal dimension d and spectral dimension ds ≤ 2, with spatially correlated traps. The traps form a sublattice with fractal dimension da < d and are characterized by the absorption rate wa which may be finite (imperfect traps) or infinite (perfect traps). Initial coordinates are chosen randomly at or within a fixed distance of a trap. For weakly absorbing traps (wa ≪ w), we find that f (t) can be closely approximated by a stretched exponential function over the initial stage of relaxation, with stretching exponent α = 1 − (d − da)/dw, where dw is the random walk dimension of the host lattice. At the end of this initial stage there occurs a crossover to power law kinetics f (t) ∼ t −α with the same exponent α as for the stretched exponential regime. For strong absorption wa w, including the limit of perfect traps wa → ∞, the stretched exponential regime is absent and the decay of f (t) follows, after a short transient, the aforementioned power law for all times.

Universal exploration dynamics of random walks

“…The first-return time distribution to this set of traps on the interface is determined by the renewal equation 40 , 49 , 50 that links the probability P trap ( t ) to be at a trap at time t and the distribution of first arrival times F ∞ ( τ ) to a trap at time τ , This equation expresses the partitioning of the total RW path to the interface into a first-passage path to the interface over a time τ and a return path to the interface over the remaining time t − τ ; here we use a continuous-time formulation for simplicity. In this mean-field type equation (detailed in SI Sec.…”

Random walks on uniform and non-uniform combs and brushes

“…This second problem might be avoided if one is interested in such properties as the probability of return to the origin, but becomes inevitable when one wishes to directly evaluate the mean-square displacement, drift in an external field, etc. As shown below (see also [10]), for the Sierpinski brush and similar systems both diculties can https://doi.org/10.1088/1742-5468/aa79b4 J. Stat. Mech.…”

“…Similar to the Sierpinski brush, it is convenient for simulation purposes to design the Cantor comb inductively from smaller to larger scales and to enumerate cells of the base with both decimal and ternary (base-3) integers [10]. As seen from Fig.…”

“…As shown below (see also Ref. [10]), for the Sierpinski brush and similar systems both difficulties can be readily resolved by embedding the fractal in a 2-dimensional Euclidean lattice and writing the Cartesian coordinates of fractal sites as binary numbers.…”

Universal exploration dynamics of random walks

“…The first-return time distribution to this set of traps on the interface is determined by the renewal equation [48][49][50] P trap (t) = δ(t)…”