Amnestically Induced Persistence in Random Walks

Cressoni, J. C.; Silva, Marco Antonio Alves da; Viswanathan, G. M.

doi:10.1103/physrevlett.98.070603

Cited by 82 publications

(116 citation statements)

References 19 publications

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“…This results follow straightforwardly from Eqs. (42) and (30). On the other hand, in the limit ∆ → ∞ (with finite λ), the parameter a goes to 1 + λ, while b vanishes.…”

Section: Probability Densitiesmentioning

confidence: 98%

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Budini

2016

Phys. Rev. E

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We introduce a class of discrete random walk model driven by global memory effects. At any time the right-left transitions depend on the whole previous history of the walker, being defined by an urn-like memory mechanism. The characteristic function is calculated in an exact way, which allows us to demonstrate that the ensemble of realizations is ballistic. Asymptotically each realization is equivalent to that of a biased Markovian diffusion process with transition rates that strongly differs from one trajectory to another. Using this "inhomogeneous diffusion" feature the ergodic properties of the dynamics are analytically studied through the time-averaged moments. Even in the long time regime they remain random objects. While their average over realizations recover the corresponding ensemble averages, departure between time and ensemble averages is explicitly shown through their probability densities. For the density of the second time-averaged moment an ergodic limit and the limit of infinite lag times do not commutate. All these effects are induced by the memory effects. A generalized Einstein fluctuation-dissipation relation is also obtained for the time-averaged moments.

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“…This results follow straightforwardly from Eqs. (42) and (30). On the other hand, in the limit ∆ → ∞ (with finite λ), the parameter a goes to 1 + λ, while b vanishes.…”

Section: Probability Densitiesmentioning

confidence: 98%

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Budini

2016

Phys. Rev. E

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“…For p sufficiently larger than p = 1/2, the behavior becomes persistent (i.e., H > 1/2). But the finding of persistence for p < 1/2 and small f overturned commonly held beliefs concerning repetitive behavior and memory loss [2]. Very recently, Kenkre [3] has found an exact solution to this problem for the behavior of the first moment, for all f , and generalized it in important ways, with excellent agreement with the numerical results over 6 orders of magnitude in time.…”

mentioning

confidence: 61%

“…Most random walks with and without memory display continuous scale invariance symmetry, i.e., continuous scale transformations by a "zoom" factor λ leave the Hurst exponent unchanged: t 2H → λ 2H t 2H as t → λt. Schütz and Trimper [1] pioneered a novel approach for studying walks with long-range memory [6,7,8,9], which we have adapted [2] for studying memory loss. Consider a random walker that starts at the origin at time t 0 = 0, with memory of the initial f t time steps of its complete history (0 ≤ f ≤ 1).…”

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confidence: 99%

Spontaneous symmetry breaking in amnestically induced persistence

et al. 2008

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We investigate a recently proposed non-Markovian random walk model characterized by loss of memories of the recent past and amnestically induced persistence. We report numerical and analytical results showing the complete phase diagram, consisting of 4 phases, for this system: (i) classical nonpersistence, (ii) classical persistence (iii) log-periodic nonpersistence and (iv) logperiodic persistence driven by negative feedback. The first two phases possess continuous scale invariance symmetry, however log-periodicity breaks this symmetry. Instead, log-periodic motion satisfies discrete scale invariance symmetry, with complex rather than real fractal dimensions. We find for log-periodic persistence evidence not only of statistical but also of geometric self-similarity.Nonpersistent random walkers with negative feedback tend not to repeat past behavior [1], but what happens when they forget their recent past [2]? Remarkably, they become persistent for sufficiently large memory losses. This recently reported phenomenon of amnestically induced persistence [2, 3] allows log-periodic [4] superdiffusion [5,6,7] driven by negative feedback. Its practical importance stems from the conceptual advance of quantitatively relating, on a causal level, two otherwise apparently unconnected phenomena: repetitive or persistent behavior on the one hand, and recent memory loss on the other, e.g., in Alzheimer's disease [2]. Precisely how does persistence depend on recent memory loss? Here, we answer this question and report numerical and analytical results showing the complete phase diagram for the problem, comprising 4 phases: (i) classical nonpersistence (ii) classical persistence, (iii) log-periodic nonpersistence and (iv) log-periodic persistence driven by negative feedback. The former two phases possess continuous scale invariance symmetry, which breaks down in the other two.Random walkers without memory have a mean square displacement x 2 that scales with time t according to x 2 ∼ t 2H , with Hurst exponent H = 1/2 as demanded by the Central Limit Theorem, assuming finite moments. Hurst exponents H > 1/2 indicate persistence and can arise due to long-range memory. Most random walks with and without memory display continuous scale invariance symmetry, i.e., continuous scale transformations by a "zoom" factor λ leave the Hurst exponent unchanged: t 2H → λ 2H t 2H as t → λt. Schütz and Trimper [1] pioneered a novel approach for studying walks with long-range memory [6,7,8,9], which we have adapted [2] for studying memory loss. Consider a random walker that starts at the origin at time t 0 = 0, with memory of the initial f t time steps of its complete history (0 ≤ f ≤ 1). At each time step the random walker moves either one step to the right or left. Let v t = ±1 represent the "velocity" at time t, such that the position followsAt time t, we randomly choose a previous time 1 ≤ t ′ < f t with equal a priori probabilities. The walker then chooses the current step direction v t based on the value of v t ′ , using the following rule. W...

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“…The properties of the phases include log-periodic oscillations that appear for small sizes of the long-range memory. Log-periodic oscillations in RW have been reported to appear elsewhere (see e.g., [21][22][23]). We also show that the size of the region of the phase diagram with superdiffusion is controlled by the memoryless noise.…”

mentioning

confidence: 80%

“…The onset of superdiffusion in the elephant and Alzheimer walk models has been shown to be related to a tendency to repeat [20] or negate [21] previous actions of the walker. The former leads to classical superdiffusion while the latter produces log-periodic superdiffusion.…”

mentioning

confidence: 99%

Robustness of the non-Markovian Alzheimer walk under stochastic perturbation

Cressoni¹,

Silva²,

Viswanathan³

et al. 2012

EPL

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-The elephant walk model originally proposed by Schütz and Trimper to investigate non-Markovian processes led to the investigation of a series of other random-walk models. Of these, the best known is the Alzheimer walk model, because it was the first model shown to have amnestically induced persistence -i.e. superdiffusion caused by loss of memory. Here we study the robustness of the Alzheimer walk by adding a memoryless stochastic perturbation. Surprisingly, the solution of the perturbed model can be formally reduced to the solutions of the unperturbed model. Specifically, we give an exact solution of the perturbed model by finding a surjective mapping to the unperturbed model. Copyright c EPLA, 2012Introduction. -The theoretical foundations of the Brownian motion are well known and were laid more than one hundred years ago (see [1][2][3] and references therein). Since then, the random walk (RW), the simplest realization of the Brownian motion, has been used in the scientific literature as a prototype for modelling applications in various fields. The diffusion of the traditional RW is known as normal diffusion.Superdiffusion [4][5][6][7] is an accelerated diffusion, for which the mean squared displacement grows faster than linearly in time. Theoretical studies of anomalous diffusion are based on generalized Langevin equations [8,9], the fractional Fokker-Planck equation [10,11] and the continuous-time random-walk [12-14] approaches. Superdiffusion is possible only if the necessary and sufficient conditions of the central limit theorem for sums of random variables are not met, otherwise the behavior is always normal diffusion (e.g., Brownian motion). There are two basic mechanisms (not mutually exclusive) by which a random walker can undergo superdiffusion. The first is an infinite second moment for the random-walk step size distribution, as seen in Lévy processes [15][16][17]. The second mechanism is long-range power law correlations, i.e. long-range memory [18,19].

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Amnestically Induced Persistence in Random Walks

Cited by 82 publications

References 19 publications

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Spontaneous symmetry breaking in amnestically induced persistence

Robustness of the non-Markovian Alzheimer walk under stochastic perturbation

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