Weak subordination breaking for the quenched trap model

Abstract

We map the problem of diffusion in the quenched trap model onto a new stochastic process: Brownian motion which is terminated at the coverage "time" Sα = ∞ x=−∞ (nx) α with nx being the number of visits to site x. Here 0 < α = T /Tg < 1 is a measure of the disorder in the original model. This mapping allows us to treat the intricate correlations in the underlying random walk in the random environment. The operational "time" Sα is changed to laboratory time t with a Lévy time transformation. Investigation of Brownian motion stopped at "time" Sα yields the diffusion front of the quenched trap model which is favorably compared with numerical simulations. In the zero temperature limit of α → 0 we recover the renormalization group solution obtained by C. Monthus. Our theory surmounts critical slowing down which is found when α → 1. Above the critical dimension two mapping the problem to a continuous time random walk becomes feasible though still not trivial.

Quenched trap model for Lévy flights

“…In the standard setting, walker in the quenched trap model can move only to the nearest neighbor site (no flights are allowed). This case was extensively studied by both mathematicians [36][37][38][39][40][41][42] and physicists [35,[43][44][45][46]. Surprisingly, in dimensions larger than one, the asymptotic limit of the quenched trap model coincides with the one for annealed CTRW [38].…”

“…The coverage time is an important object in the analysis of systems with quenched disorder. In particular, it was used in [44,45] to construct the corresponding diffusion front and to map the problem of diffusion in the quenched trap model onto an appropriately stopped Brownian motion.…”

“…We now turn to the problem of asymptotic behavior of the so-called coverage time, which is defined as [45] S(n) = i∈Z N(n, i) β , 0 < β < 1.…”

Anomalous diffusion in stochastic systems with nonhomogeneously distributed traps

“…If, on the other hand, the particle exercises a space structure which slowly evolves with time, the trapping time at a given site is the same for each visit of this site and a correlation between trapping times emerges (a quenched disorder) [2]. The quenched trap model [3,4] takes into account that the particle may remember the rest time at a given point. Then the time-dependence of the variance can be derived by averaging over disorder [4] or using a renormalisation group approach [5]; it takes a form ∼ t 2β/(1+β) , where 0 < β < 1 characterises the memory.…”

“…The quenched trap model [3,4] takes into account that the particle may remember the rest time at a given point. Then the time-dependence of the variance can be derived by averaging over disorder [4] or using a renormalisation group approach [5]; it takes a form ∼ t 2β/(1+β) , where 0 < β < 1 characterises the memory.…”