Lévy walkers inside spherical shells with absorbing boundaries: Towards settling the optimal Lévy walk strategy for random searches.

The Lévy flight foraging hypothesis states that organisms must have evolved adaptations to exploit Lévy walk search strategies. Indeed, it is widely accepted that inverse square Lévy walks optimize the search efficiency in foraging with unrestricted revisits (also known as nondestructive foraging). However, a mathematically rigorous demonstration of this for dimensions D≥2 is still lacking. Here we study the very closely related problem of a Lévy walker inside annuli or spherical shells with absorbing boundaries. In the limit that corresponds to the foraging with unrestricted revisits, we show that inverse square Lévy walks optimize the search. This constitutes the strongest formal result to date supporting the optimality of inverse square Lévy walks search strategies.

Effect of the search space dimensionality for finding close and faraway targets in random searches.

We investigate the dependence on the search space dimension of statistical properties of random searches with Lévy α-stable and power-law distributions of step lengths. We find that the probabilities to return to the last target found (P_{0}) and to encounter faraway targets (P_{L}), as well as the associated Shannon entropy S, behave as a function of α quite differently in one (1D) and two (2D) dimensions, a somewhat surprising result not reported until now. While in 1D one always has P_{0}≥P_{L}, an interesting crossover takes place in 2D that separates the search regimes with P_{0}>P_{L} for higher α and P_{0}

Spectrum of the tight-binding model on Cayley trees and comparison with Bethe lattices.

There are few exactly solvable lattice models and even fewer solvable quantum lattice models. Here we address the problem of finding the spectrum of the tight-binding model (equivalently, the spectrum of the adjacency matrix) on Cayley trees. Recent approaches to the problem have relied on the similarity between the Cayley tree and the Bethe lattice. Here, we avoid to make any ansatz related to the Bethe lattice due to fundamental differences between the two lattices that persist even when taking the thermodynamic limit. Instead, we show that one can use a recursive procedure that starts from the boundary and then use the canonical basis to derive the complete spectrum of the tight-binding model on Cayley trees. Our resulting algorithm is extremely efficient, as witnessed with remarkable large trees having hundreds of shells. We also show that, in the thermodynamic limit, the density of states is dramatically different from that of the Bethe lattice.

Landscape-scaled strategies can outperform Lévy random searches.

Information on the relevant global scales of the search space, even if partial, should conceivably enhance the performance of random searches. Here we show numerically and analytically that the paradigmatic uninformed optimal Lévy searches can be outperformed by informed multiple-scale random searches in one (1D) and two (2D) dimensions, even when the knowledge about the relevant landscape scales is incomplete. We show in the low-density nondestructive regime that the optimal efficiency of biexponential searches that incorporate all key scales of the 1D landscape of size L decays asymptotically as η_{opt}∼1/sqrt[L], overcoming the result η_{opt}∼1/(sqrt[L]lnL) of optimal Lévy searches. We further characterize the level of limited information the searcher can have on these scales. We obtain the phase diagram of bi- and triexponential searches in 1D and 2D. Remarkably, even for a certain degree of lack of information, partially informed searches can still outperform optimal Lévy searches. We discuss our results in connection with the foraging problem.

Transient dynamics in a nonequilibrium superdiffusive reaction-diffusion process: Nonequilibrium random search as a case study.

Transient regimes, often difficult to characterize, can be fundamental in establishing final steady states features of reaction-diffusion phenomena. This is particularly true in ecological problems. Here, through both numerical simulations and an analytic approximation, we analyze the transient of a nonequilibrium superdiffusive random search when the targets are created at a certain rate and annihilated upon encounters (a key dynamics, e.g., in biological foraging). The steady state is achieved when the number of targets stabilizes to a constant value. Our results unveil how key features of the steady state are closely associated to the particularities of the initial evolution. The searching efficiency variation in time is also obtained. It presents a rather surprising universal behavior at the asymptotic limit. These analyses shed some light into the general relevance of transients in reaction-diffusion systems.

Why Lévy α-stable distributions lack general closed-form expressions for arbitrary α.

The ubiquitous Lévy α-stable distributions lack general closed-form expressions in terms of elementary functions-Gaussian and Cauchy cases being notable exceptions. To better understand this 80-year-old conundrum, we study the complex analytic continuation p_{α}(z), z∈C, of the symmetric Lévy α-stable distribution family p_{α}(x), x∈R, parametrized by 0<α≤2. We first extend known but obscure results, and give a new proof that p_{α}(z) is holomorphic on the entire complex plane for 1<α≤2, whereas p_{α}(z) is not even meromorphic on C for 0<α<1. Next, we unveil the complete complex analytic structure of p_{α}(z) using domain coloring. Finally, motivated by these insights, we argue that there cannot be closed-form expressions in terms of elementary functions for p_{α}(x) for general α.

Shannon entropy of brain functional complex networks under the influence of the psychedelic Ayahuasca.

The entropic brain hypothesis holds that the key facts concerning psychedelics are partially explained in terms of increased entropy of the brain's functional connectivity. Ayahuasca is a psychedelic beverage of Amazonian indigenous origin with legal status in Brazil in religious and scientific settings. In this context, we use tools and concepts from the theory of complex networks to analyze resting state fMRI data of the brains of human subjects under two distinct conditions: (i) under ordinary waking state and (ii) in an altered state of consciousness induced by ingestion of Ayahuasca. We report an increase in the Shannon entropy of the degree distribution of the networks subsequent to Ayahuasca ingestion. We also find increased local and decreased global network integration. Our results are broadly consistent with the entropic brain hypothesis. Finally, we discuss our findings in the context of descriptions of "mind-expansion" frequently seen in self-reports of users of psychedelic drugs.

Correspondence between spanning trees and the Ising model on a square lattice.

An important problem in statistical physics concerns the fascinating connections between partition functions of lattice models studied in equilibrium statistical mechanics on the one hand and graph theoretical enumeration problems on the other hand. We investigate the nature of the relationship between the number of spanning trees and the partition function of the Ising model on the square lattice. The spanning tree generating function T(z) gives the spanning tree constant when evaluated at z=1, while giving the lattice green function when differentiated. It is known that for the infinite square lattice the partition function Z(K) of the Ising model evaluated at the critical temperature K=K_{c} is related to T(1). Here we show that this idea in fact generalizes to all real temperatures. We prove that [Z(K)sech2K]^{2}=kexp[T(k)] , where k=2tanh(2K)sech(2K). The identical Mahler measure connects the two seemingly disparate quantities T(z) and Z(K). In turn, the Mahler measure is determined by the random walk structure function. Finally, we show that the the above correspondence does not generalize in a straightforward manner to nonplanar lattices.

Inferring Lévy walks from curved trajectories: A rescaling method.

An important problem in the study of anomalous diffusion and transport concerns the proper analysis of trajectory data. The analysis and inference of Lévy walk patterns from empirical or simulated trajectories of particles in two and three-dimensional spaces (2D and 3D) is much more difficult than in 1D because path curvature is nonexistent in 1D but quite common in higher dimensions. Recently, a new method for detecting Lévy walks, which considers 1D projections of 2D or 3D trajectory data, has been proposed by Humphries et al. The key new idea is to exploit the fact that the 1D projection of a high-dimensional Lévy walk is itself a Lévy walk. Here, we ask whether or not this projection method is powerful enough to cleanly distinguish 2D Lévy walk with added curvature from a simple Markovian correlated random walk. We study the especially challenging case in which both 2D walks have exactly identical probability density functions (pdf) of step sizes as well as of turning angles between successive steps. Our approach extends the original projection method by introducing a rescaling of the projected data. Upon projection and coarse-graining, the renormalized pdf for the travel distances between successive turnings is seen to possess a fat tail when there is an underlying Lévy process. We exploit this effect to infer a Lévy walk process in the original high-dimensional curved trajectory. In contrast, no fat tail appears when a (Markovian) correlated random walk is analyzed in this way. We show that this procedure works extremely well in clearly identifying a Lévy walk even when there is noise from curvature. The present protocol may be useful in realistic contexts involving ongoing debates on the presence (or not) of Lévy walks related to animal movement on land (2D) and in air and oceans (3D).

Reply to "Comment on 'Third law of thermodynamics as a key test of generalized entropies' ".

In Bento et al. [Phys. Rev. E 91, 039901 (2015)] we develop a method to verify if an arbitrary generalized statistics does or does not obey the third law of thermodynamics. As examples, we address two important formulations, Kaniadakis and Tsallis. In their Comment on the paper, Bagci and Oikonomou suggest that our examination of the Tsallis statistics is valid only for q≥1, using arguments like there is no distribution maximizing the Tsallis entropy for the interval q<0 (in which the third law is not verified) compatible with the problem energy expression. In this Reply, we first (and most importantly) show that the Comment misses the point. In our original work we have considered the now already standard construction of the Tsallis statistics. So, if indeed such statistics lacks a maximization principle (a fact irrelevant in our protocol), this is an inherent feature of the statistics itself and not a problem with our analysis. Second, some arguments used by Bagci and Oikonomou (for 0

Robustness of optimal random searches in fragmented environments.

The random search problem is a challenging and interdisciplinary topic of research in statistical physics. Realistic searches usually take place in nonuniform heterogeneous distributions of targets, e.g., patchy environments and fragmented habitats in ecological systems. Here we present a comprehensive numerical study of search efficiency in arbitrarily fragmented landscapes with unlimited visits to targets that can only be found within patches. We assume a random walker selecting uniformly distributed turning angles and step lengths from an inverse power-law tailed distribution with exponent µ. Our main finding is that for a large class of fragmented environments the optimal strategy corresponds approximately to the same value µ(opt)≈2. Moreover, this exponent is indistinguishable from the well-known exact optimal value µ(opt)=2 for the low-density limit of homogeneously distributed revisitable targets. Surprisingly, the best search strategies do not depend (or depend only weakly) on the specific details of the fragmentation. Finally, we discuss the mechanisms behind this observed robustness and comment on the relevance of our results to both the random search theory in general, as well as specifically to the foraging problem in the biological context.

Third law of thermodynamics as a key test of generalized entropies.

The laws of thermodynamics constrain the formulation of statistical mechanics at the microscopic level. The third law of thermodynamics states that the entropy must vanish at absolute zero temperature for systems with nondegenerate ground states in equilibrium. Conversely, the entropy can vanish only at absolute zero temperature. Here we ask whether or not generalized entropies satisfy this fundamental property. We propose a direct analytical procedure to test if a generalized entropy satisfies the third law, assuming only very general assumptions for the entropy S and energy U of an arbitrary N-level classical system. Mathematically, the method relies on exact calculation of ß=dS/dU in terms of the microstate probabilities p(i). To illustrate this approach, we present exact results for the two best known generalizations of statistical mechanics. Specifically, we study the Kaniadakis entropy S(κ), which is additive, and the Tsallis entropy S(q), which is nonadditive. We show that the Kaniadakis entropy correctly satisfies the third law only for -1<κ<+1, thereby shedding light on why κ is conventionally restricted to this interval. Surprisingly, however, the Tsallis entropy violates the third law for q<1. Finally, we give a concrete example of the power of our proposed method by applying it to a paradigmatic system: the one-dimensional ferromagnetic Ising model with nearest-neighbor interactions.

Efficient search of multiple types of targets.

Random searches often take place in fragmented landscapes. Also, in many instances like animal foraging, significant benefits to the searcher arise from visits to a large diversity of patches with a well-balanced distribution of targets found. Up to date, such aspects have been widely ignored in the usual single-objective analysis of search efficiency, in which one seeks to maximize just the number of targets found per distance traversed. Here we address the problem of determining the best strategies for the random search when these multiple-objective factors play a key role in the process. We consider a figure of merit (efficiency function), which properly "scores" the mentioned tasks. By considering random walk searchers with a power-law asymptotic Lévy distribution of step lengths, p(ℓ)∼ℓ(-µ), with 1<µ≤3, we show that the standard optimal strategy with µ(opt)≈2 no longer holds universally. Instead, optimal searches with enhanced superdiffusivity emerge, including values as low as µ(opt)≈1.3 (i.e., tending to the ballistic limit). For the general theory of random search optimization, our findings emphasize the necessity to correctly characterize the multitude of aims in any concrete metric to compare among possible candidates to efficient strategies. In the context of animal foraging, our results might explain some empirical data pointing to stronger superdiffusion (µ<2) in the search behavior of different animal species, conceivably associated to multiple goals to be achieved in fragmented landscapes.

Ultraslow diffusion in an exactly solvable non-Markovian random walk.

We study a one-dimensional discrete-time non-Markovian random walk with strong memory correlations subjected to pauses. Unlike the Scher-Montroll continuous-time random walk, which can be made Markovian by defining an operational time equal to the random-walk step number, the model we study keeps a record of the entire history of the walk. This new model is closely related to the one proposed recently by Kumar, Harbola, and Lindenberg [Phys. Rev. E 82, 021101 (2010)], with the difference that in our model the stochastic dynamics does not stop even in the extreme limit of subdiffusion. Surprisingly, this small difference leads to large consequences. The main results we report here are exact results showing ultraslow diffusion and a stationary diffusion regime (i.e., localization). Specifically, the equations of motion are solved analytically for the first two moments, allowing the determination of the Hurst exponent. Several anomalous diffusion regimes are apparent, ranging from superdiffusion to subdiffusion, as well as ultraslow and stationary regimes. We present the complete phase diffusion diagram, along with a study of the persistence and the statistics in the regions of interest.

Unveiling a mechanism for species decline in fragmented habitats: fragmentation induced reduction in encounter rates.

Several studies have reported that fragmentation (e.g. of anthropogenic origin) of habitats often leads to a decrease in the number of species in the region. An important mechanism causing this adverse ecological impact is the change in the encounter rates (i.e. the rates at which individuals meet other organisms of the same or different species). Yet, how fragmentation can change encounter rates is poorly understood. To gain insight into the problem, here we ask how landscape fragmentation affects encounter rates when all other relevant variables remain fixed. We present strong numerical evidence that fragmentation decreases search efficiencies thus encounter rates. What is surprising is that it falls even when the global average densities of interacting organisms are held constant. In other words, fragmentation per se can reduce encounter rates. As encounter rates are fundamental for biological interactions, it can explain part of the observed diminishing in animal biodiversity. Neglecting this effect may underestimate the negative outcomes of fragmentation. Partial deforestation and roads that cut through forests, for instance, might be responsible for far greater damage than thought. Preservation policies should take into account this previously overlooked scientific fact.

Non-Gaussian propagator for elephant random walks.

For almost a decade the consensus has held that the random walk propagator for the elephant random walk (ERW) model is a Gaussian. Here we present strong numerical evidence that the propagator is, in general, non-Gaussian and, in fact, non-Lévy. Motivated by this surprising finding, we seek a second, non-Gaussian solution to the associated Fokker-Planck equation. We prove mathematically, by calculating the skewness, that the ERW Fokker-Planck equation has a non-Gaussian propagator for the superdiffusive regime. Finally, we discuss some unusual aspects of the propagator in the context of higher order terms needed in the Fokker-Planck equation.

Alzheimer random walk model: two previously overlooked diffusion regimes.

A non-Markovian one-dimensional random walk model is studied with emphasis on the phase-diagram, showing all the diffusion regimes, along with the exactly determined critical lines. The model, known as the Alzheimer walk, is endowed with memory-controlled diffusion, responsible for the model's long-range correlations, and is characterized by a rich variety of diffusive regimes. The importance of this model is that superdiffusion arises due not to memory per se, but rather also due to loss of memory. The recently reported numerically and analytically estimated values for the Hurst exponent are hereby reviewed. We report the finding of two, previously overlooked, phases, namely, evanescent log-periodic diffusion and log-periodic diffusion with escape, both with Hurst exponent H=1/2. In the former, the log-periodicity gets damped, whereas in the latter the first moment diverges. These phases further enrich the already intricate phase diagram. The results are discussed in the context of phase transitions, aging phenomena, and symmetry breaking.

Conditions under which a superdiffusive random-search strategy is necessary.

Intuitively, lower target densities and lower detection capabilities should demand more sophisticated search strategies for a random search reasonable outcome. In contrast, when targets are easily found, a simple Brownian random walk strategy is enough. But where is the threshold between these two scenarios and when is optimization really necessary? We address this considering the interplay between two essential scales in random search, the average distance between neighbor targets l(0) and the detection capability r(v). In the limit cases the ratio ß=r(v)/l(0) suffices to characterize the problem. For low (high) ß a superdiffusive behavior is (is not) crucial for the process optimization. However, there is a crossover range, which is a nontrivial function of r(v) and l(0), separating the two regimes. We analyze this intermediate region, common in nature, and discuss the often overlooked important trade between resources availability and the searcher location power. Our results highlight contexts where efficient random search is a key factor for survival, such as in animal foraging.