In this minireview we present the main results regarding the transport properties of stochastic movement with relocations to known positions. To do so, we formulate the problem in a general manner to see several cases extensively studied during the last years as particular situations within a framework of random walks with memory. We focus on (i) stochastic motion with resets to its initial position followed by a waiting period, and (ii) diffusive motion with memory-driven relocations to previously visited positions. For both of them we show how the overall transport regime may be actively modified by the details of the relocation mechanism.
Strongly non-Markovian random walks offer a promising modeling framework for understanding animal and human mobility, yet, few analytical results are available for these processes. Here we solve exactly a model with long range memory where a random walker intermittently revisits previously visited sites according to a reinforced rule. The emergence of frequently visited locations generates very slow diffusion, logarithmic in time, whereas the walker probability density tends to a Gaussian. This scaling form does not emerge from the Central Limit Theorem but from an unusual balance between random and long-range memory steps. In single trajectories, occupation patterns are heterogeneous and have a scale-free structure. The model exhibits good agreement with data of free-ranging capuchin monkeys.
Motivated by studies on the recurrent properties of animal and human mobility, we introduce a path-dependent random walk model with long range memory for which not only the mean square displacement (MSD) can be obtained exactly in the asymptotic limit, but also the propagator. The model consists of a random walker on a lattice, which, at a constant rate, stochastically relocates at a site occupied at some earlier time. This time in the past is chosen randomly according to a memory kernel, whose temporal decay can be varied via an exponent parameter. In the weakly non-Markovian regime, memory reduces the diffusion coefficient from the bare value. When the mean backward jump in time diverges, the diffusion coefficient vanishes and a transition to an anomalous subdiffusive regime occurs. Paradoxically, at the transition, the process is an anti-correlated Levy flight. Although in the subdiffusive regime the model exhibits some features of the continuous time random walk with infinite mean waiting time, it belongs to another universality class. If memory is very long-ranged, a second transition takes place to a regime characterized by a logarithmic growth of the MSD with time. In this case the process is asymptotically Gaussian and effectively described as a scaled Brownian motion with a diffusion coefficient decaying as 1/t.
We study how the Hurst exponent $alpha$ depends on the fraction $f$ of the total time $t$ remembered by non-Markovian random walkers that recall only the distant past. We find that otherwise nonpersistent random walkers switch to persistent behavior when inflicted with significant memory loss. Such memory losses induce the probability density function of the walkers position to undergo a transition from Gaussian to non-Gaussian. We interpret these findings of persistence in terms of a breakdown of self-regulation mechanisms and discuss their possible relevance to some of the burdensome behavioral and psychological symptoms of Alzheimers disease and other dementias.
We formulate the generalized master equation for a class of continuous time random walks in the presence of a prescribed deterministic evolution between successive transitions. This formulation is exemplified by means of an advection-diffusion and a jump-diffusion scheme. Based on this master equation, we also derive reaction-diffusion equations for subdiffusive chemical species, using a mean field approximation.
We investigate diffusion-limited aggregation (DLA) in a wedge geometry. Arneodo and collaborators have suggested that the ensemble average of DLA cluster density should be close to the noise-free selected Saffman-Taylor finger. We show that a different, but related, ensemble average, that of the conformal maps associated with random clusters, yields a non-trivial shape which is also not far from the Saffman-Taylor finger. However, we have previously demonstrated that the same average of DLA in a channel geometry is not the Saffman-Taylor finger. This casts doubt on the idea that the average of noisy diffusion-limited growth is governed by a simple transcription of noise-free results.
Axel Maso-Puigdellosas
,Daniel Campos
,Vicenc{c} Mendez
.
(2019)
.
"Anomalous diffusion in random-walks with memory-induced relocations"
.
Axel Mas\\'o-Puigdellosas
هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا