How landscape heterogeneity frames optimal diffusivity in searching processes.

Theoretical and empirical investigations of search strategies typically have failed to distinguish the distinct roles played by density versus patchiness of resources. It is well known that motility and diffusivity of organisms often increase in environments with low density of resources, but thus far there has been little progress in understanding the specific role of landscape heterogeneity and disorder on random, non-oriented motility. Here we address the general question of how the landscape heterogeneity affects the efficiency of encounter interactions under global constant density of scarce resources. We unveil the key mechanism coupling the landscape structure with optimal search diffusivity. In particular, our main result leads to an empirically testable prediction: enhanced diffusivity (including superdiffusive searches), with shift in the diffusion exponent, favors the success of target encounters in heterogeneous landscapes.

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.

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}

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.

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.

The influence of turning angles on the success of non-oriented animal searches.

Animal searches cover a full range of possibilities from highly deterministic to apparently completely random behaviors. However, even those stochastic components of animal movement can be adaptive, since not all random distributions lead to similar success in finding targets. Here we address the general problem of optimizing encounter rates in non-deterministic, non-oriented searches, both in homogeneous and patchy target landscapes. Specifically, we investigate how two different features related to turning angle distributions influence encounter success: (i) the shape (relative kurtosis) of the angular distribution and (ii) the correlations between successive relative orientations (directional memory). Such influence is analyzed in correlated random walk models using a proper choice of representative turning angle distributions of the recently proposed Jones and Pewsey class. We consider the cases of distributions with nearly the same shape but considerably distinct correlation lengths, and distributions with same correlation but with contrasting relative kurtosis. In homogeneous landscapes, we find that the correlation length has a large influence in the search efficiency. Moreover, similar search efficiencies can be reached by means of distinctly shaped turning angle distributions, provided that the resulting correlation length is the same. In contrast, in patchy landscapes the particular shape of the distribution also becomes relevant for the search efficiency, specially at high target densities. Excessively sharp distributions generate very inefficient searches in landscapes where local target density fluctuations are large. These results are of evolutionary interest. On the one hand, it is shown that equally successful directional memory can arise from contrasting turning behaviors, therefore increasing the likelihood of robust adaptive stochastic behavior. On the other hand, when target landscape is patchy, adequate tumbling may help to explore better local scale heterogeneities, being some details of the shape of the distribution also potentially adaptive.

Multifractality of random walks in the theory of vehicular traffic.

We investigate the origin of the experimentally observed multifractal scaling of vehicular traffic flows by studying a hydrodynamic model of traffic. We first extend and apply the formalism of generalized Hurst exponents H(q) to the case of random walkers that not only diffuse but rather also undergo nonlinear convection due to interactions with other walkers. We recover analytically, as expected, that H(q) equals 12 for a single random walker starting at the origin whose probability density function satisfies Burger's equation. Despite this result for a single walker, we find that for a collection of nonlinearly convecting diffusive particles, transient effects can give rise to multiscaling at given time scales for many initial conditions. In the Lighthill-Whitham-Richards hydrodynamic model of traffic, this multiscaling effect becomes more prominent for smaller diffusion constants and larger speed limits. We discuss the relevance of these findings for the realistic scenario of traffic that flows from small roads to large highways and vice versa, where transient effects can be expected to play a significant role.

Spontaneous symmetry breaking in amnestically induced persistence.

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 four phases, for this system: (i) classical nonpersistence, (ii) classical persistence, (iii) log-periodic nonpersistence, and (iv) log-periodic 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.

Optimization of random searches on defective lattice networks.

We study the general problem of how to search efficiently for targets randomly located on defective lattice networks--i.e., regular lattices which have some fraction of its nodes randomly removed. We consider large but finite triangular lattices and assume for the search dynamics that the walker chooses steps lengths lj from the power-law distribution P(lj) approximately lj(-mu) , with the exponent mu regulating the strategy of the search process. At each step lj, the searcher moves in straight lines and constantly looks within a detection radius of vision rv for the targets along the way. If there is contact with a defect, the movement stops and a new step length is chosen. Hence, the presence of defects decreases the efficiency of the overall process. We study numerically how three different aspects of the lattice influence the optimization of the search efficiency: (i) the type of boundary conditions, (ii) the concentration of targets and defects, and (iii) the category or class of search--destructive, nondestructive, or regenerative. Motivated by the results, we develop a type of mean-field model for the problem and obtain an analytical approximation for the search efficiency function. Finally we discuss, in the context of searches, how defective lattices compare with perfect lattices and with continuous environments.

Origin of power-law distributions in deterministic walks: the influence of landscape geometry.

We investigate the properties of a deterministic walk, whose locomotion rule is always to travel to the nearest site. Initially the sites are randomly distributed in a closed rectangular (ALxL) landscape and, once reached, they become unavailable for future visits. As expected, the walker step lengths present characteristic scales in one (L-->0) and two (AL approximately L) dimensions. However, we find scale invariance for an intermediate geometry, when the landscape is a thin striplike region. This result is induced geometrically by a dynamical trapping mechanism, leading to a power-law distribution for the step lengths. The relevance of our findings in broader contexts--of both deterministic and random walks--is also briefly discussed.

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.

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.

Necessary criterion for distinguishing true superdiffusion from correlated random walk processes.

A difficulty in interpreting phenomena related to anomalous diffusion concerns how to identify scale invariant superdiffusive from Markovian correlated random walk processes. Here we propose a criterion that can distinguish between these two kinds of random walks and describe its usefulness in interpreting real data. To do so, we estimate the correlation time tau of the orientation persistence of a general correlated random walk. If the experimentally observed random walk appears diffusive on scales larger than tau, then the data cannot support the possibility of superdiffusion. We argue that the criterion is a necessary but not sufficient condition for establishing true superdiffusive behavior.

Optimization of random searches on regular lattices.

We investigate random searches on isotropic and topologically regular square and triangular lattices with periodic boundary conditions and study the efficiency of search strategies based on a power-law distribution P() approximately (-mu) of step lengths . We consider both destructive searches, in which a target can be visited only once, and nondestructive searches, when a target site is always available for future visits. We discuss (i) the dependence of the search efficiency on the choice of the lattice topology, (ii) the relevance of the periodic boundary conditions, (iii) the behavior of the optimal power-law exponent mu(opt) as a function of target site density, (iv) the differences between destructive and nondestructive environments, and finally (v) how the results for the discrete searches differ from the continuous cases previously studied.

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.

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

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.