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 walker's 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 Alzheimer's disease and other dementias.

We report numerically and analytically estimated values for the Hurst exponent for a recently proposed non-Markovian walk characterized by amnestically induced persistence. These results are consistent with earlier studies showing that log-periodic oscillations arise only for large memory losses of the recent past. We also report numerical estimates of the Hurst exponent for non-Markovian walks with diluted memory. Finally, we study walks with a fractal memory of the past for a Thue-Morse and Fibonacci memory patterns. These results are interpreted and discussed in the context of the necessary and sufficient conditions for the central limit theorem to hold.

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...

A reduced (stereo-chemical) model is employed to study kinetic aspects of globular protein folding process, by Monte Carlo simulation. Nonextensive statistical approach is used: transition probability p i j between configurations i → j is given by p i j = [1 + (1 − q)∆G i j /k B T ] 1/(1−q) , where q is the nonextensive (Tsallis) parameter. The system model consists of a chain of 27 beads immerse in its solvent; the beads represent the sequence of amino acids along the chain by means of a 10-letter stereo-chemical alphabet; a syntax (rule) to design the amino acid sequence for any given 3D structure is embedded in the model. The study focuses mainly kinetic aspects of the folding problem related with the protein folding time, represented in this work by the concept of first passage time (FPT). Many distinct proteins, whose native structures are represented here by compact self avoiding (CSA) configurations, were employed in our analysis, although our results are presented exclusively for one representative protein, for which a rich statistics was achieved. Our results reveal that there is a specific combinations of value for the nonextensive parameter q and temperature T, which gives the smallest estimated folding characteristic time t . Additionally, for q = 1.1, t stays almost invariable in the range 0.9 ≤ T ≤ 1.3, slightly oscillating about its average value t = 27 ±σ, where σ = 2 is the standard deviation. This behavior is explained by comparing the distribution of the folding times for the Boltzmann statistics (q → 1), with respect to the nonextensive statistics for q = 1.1, which shows that the effect of the nonextensive parameter q is to cut off the larger folding times present in the original (q → 1) distribution. The distribution of natural logarithm of the folding times for Boltzmann statistics is a triple peaked Gaussian, while, for q = 1.1 (Tsallis), it is a double peaked Gaussian, suggesting that a log-normal process with two characteristic times replaced the original process with three characteristic times. Finally we comment on the physical meaning of the present results, as well its significance in the near future works.