Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk

Schütz, Gunter M.; Trimper, Steffen

doi:10.1103/physreve.70.045101

Cited by 257 publications

(402 citation statements)

References 19 publications

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“…The equations for the dot-dashed line are given by p max + s = 1 (p + q + s = 1 for q = 0). We see that for f = 1 only three phases exist, in accordance with [20]. Anomalous diffusion do not exist for s > s .…”

Section: Regions and Parameters Regionsupporting

confidence: 81%

“…Note that f = 1 gives β = 1/2 (or p = 3/4) in agreement with ref. [20]. There are other regimes for δ < 1/2, leaving a total of six different phases.…”

Section: (A)-(d)mentioning

confidence: 99%

“…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%

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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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Section: Regions and Parameters Regionsupporting

confidence: 81%

“…Note that f = 1 gives β = 1/2 (or p = 3/4) in agreement with ref. [20]. There are other regimes for δ < 1/2, leaving a total of six different phases.…”

Section: (A)-(d)mentioning

confidence: 99%

See 1 more Smart Citation

Robustness of the non-Markovian Alzheimer walk under stochastic perturbation

Cressoni¹,

Silva²,

Viswanathan³

et al. 2012

EPL

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show abstract

“…The dynamics and limit distributions of constrained LFs are not well understood, except for processes subjected to long waiting times or in external potentials, mainly [8,31]. Several limit theorems also exist for specific problems of sums of correlated random variables [32], and a few random walks with infinite memory of their previous displacements have exactly solvable first moments [33][34][35]. Yet, very little is known on LFs composed of non-independent steps, in particular processes with self-attraction.…”

Section: Introductionmentioning

confidence: 99%

Slow Lévy flights

Boyer

Pineda

2016

Phys. Rev. E

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Among Markovian processes, the hallmark of Lévy flights is superdiffusion, or faster-thanBrownian dynamics. Here we show that Lévy laws, as well as Gaussians, can also be the limit distributions of processes with long range memory that exhibit very slow diffusion, logarithmic in time. These processes are path-dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the Central Limit Theorem is modified in this context, keeping the usual distinction between analytic and non-analytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.

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“…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.…”

mentioning

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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Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk

Cited by 257 publications

References 19 publications

Robustness of the non-Markovian Alzheimer walk under stochastic perturbation

Robustness of the non-Markovian Alzheimer walk under stochastic perturbation

Slow Lévy flights

Spontaneous symmetry breaking in amnestically induced persistence

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