Analyzing Lévy walks in 2-D and 3-D

Lévy flights and walks form a class of random walks. Whereas in Brownian motion the probability density function (PDF) of the random walk steps has finite statistical moments, in contrast the PDF of the step sizes for a Lévy flight are not all finite. Higher moments are infinite, because the PDF decays asymptotically as a power law:

\displaystyle P(\ell) \sim \ell^{-\mu} ~. \ \ \ \ \ (1)

If the random walk steps are uncorrelated, it is fairly easy to show that the central limit theorem (CLT) is applicable so long as {\mu} is large enough for the variance of the PDF to be finite. In 1 dimension (1-D), the critical value of {\mu} is {\mu=3}. For {\mu} larger than this critical value, the result is Brownian motion due to the CLT.

However, for {\mu<3} in 1-D, the variance is infinite and the CLT is not applicable. There is a generalization of the CLT due to Paul Lévy and others for sums of random variables — typically independent and identically distributed (IID) — whose variance is not finite. The equivalent of the Gaussian distribution in this generalization is known as the Lévy stable distribution. The PDF for the Lévy stable distribution is usually expressed as a Fourier transform and for our purposes here we do not need to know it explicitly.

The only important property of the Lévy stable distribution that is relevant for our purposes here is the exponent {\mu}. For IID random variables described by a power law tailed PDF with exponent {\mu}, the PDF of the sums of these variables converges to a Lévy stable distribution which has the same power law decay exponent {\mu}. This is the crucial point. For {\mu>3}, the relevant Lévy stable distribution is the Gaussian, which is a special case of the Lévy stable distribution. The critical value {\mu=3} leads to a logarithmic blow-up of the variance, thus separating Brownian motion from Lévy flights. For {\mu=1}, the PDF is non-normalizable. So the relevant range of exponents for a 1-D Lévy flight is {1< \mu <3}. The same basic ideas generalize to higher dimensions, although the value of the exponent must be handled with some care.

What about Lévy walks versus Lévy flights? The difference is minimal, in fact, and can usually be ignored in a first reading. In a Lévy flight, arbitrarily large jumps can be taken in one time step. For this reason, the instantaneous velocity can be arbitrarily large — even larger than the speed of light, for instance. For this reason, Lévy flights of matter or energy in physical space are not realistic. They are, however, plausible in other kinds of “spaces,” such as along a crumpled polymer with the metric given by chemical distance, i.e. distance measured along the polymer chain.

In contrast, a Lévy walk traverses the same path as the corresponding Lévy flight, but always at a finite (typically constant) velocity. Whereas it is impossible to imagine an animal performing a Lévy flight, it is not so far-fetched to imagine a Lévy walk.

Finally, I should mention that power laws always appear truncated in the physical universe. There are no magnitude 100 earthquakes, for example. Similarly, the inverse square decay of gravity cannot extend beyond the size of the universe. The power law behavior seen in second-order phase transitions is, in fact, a mathematical idealization for an infinite system. Finite sized systems have inflection points (or similar behavior) rather than singularities at the critical point of the phase transition. Even the singularity inside a black hole is hypothetical, since these have never been directly observed. (How do you even physically measure a singularity?) So when we say that a system is “fractal” or that this or that path is a Lévy walk or flight, upper and lower cutoffs to the power law behavior are always to be understood as implicit.

Having reviewed the basics about Lévy flights, let us turn to the topic of this post. My collaborators and I have written a book about how animals move. In particular, animals seem to perform Lévy walks. There is still some debate on this point, but I would venture to say that a majority of researchers in the field are convinced that Lévy walks correctly describe important aspects of actual animal motion.

There are several reasons why the debate continues. Some of these reasons (but not all) are technical. One obstacle has to do with statistical data analysis and statistical inference. It is not entirely straightforward to analyze data that are described by power laws. An example: when studying earthquakes, what size earthquake should be considered an “outlier”? The answer, of course, is that earthquakes are described by the Gutenberg-Richter law, which is a power law. So there is no conventional meaning for the term “outlier”. This property is not the topic of this post, although I will perhaps return to it in the future. Instead, here I wish to discuss how to detect Lévy walks and flights.

A basic strategy to infer a Lévy walk or flight is to study the histogram of the random walk step sizes. A pure Lévy walk will give rise to a power law tailed histogram. There are other equivalent methods. So far, so good.

The problem, of course, is that empirical data are never perfectly clean. There is always some noise. Superimposed on the Lévy flight or walk might be some Brownian motion. The ultra-large steps of the Lévy flight might become “chopped up” due to this noise. In 1-D, this problem is easy to solve, because one can simply coarse-grain (i.e., smooth) the random walk and make a new histogram. Since the noise has a well described characteristic scale, it will attenuate under coarse-graining. In contrast, the underlying Lévy flight signal has scale-free properties, because power laws are self-affine. So the Lévy walk “backbone” or skeleton will survive a coarse-graining. Speaking somewhat loosely, we can say that coarse-graining can give us a better signal-to-noise ratio. Put in slightly different terms, the basic idea is to look at the large scale structure, not the fine scale structure.

This approach cannot work in 2-D or 3-D without modification. The reason for this difficulty is that in 2-D and 3-D the paths can be curved. The image below shows curved trajectories.


The left panel (a) shows 2 random walk models seen at small scales and (b) shows the same walks at much larger scales. One model has an underlying Lévy walk backbone, with added curvature. Both models have the same random walk step sizes and the same turning angle distribution. The difference is that one model is Markovian in time and the other (non-Markovian) model has long-range power law correlations induced by the underlying Lévy walk. The image is adapted from this paper, which contains the somewhat technical explanation of how the 2 random walks are generated.

How do you even make histograms with such “data”? Where does a random walk step “begin” and where does it “end”? The presence of curvature introduces a set of challenges which are not present in the 1-D counterpart. The paper cited above was, to a large extent, motivated by such questions. The basic problem is the so-called “discretization” issue. How do you take a curved path and segment it into discrete random walk steps? Is there an objective criterion that we can use to know where one step ends and the other begins?

In this context, there is an interesting article which was published in the journal Methods in Ecology and Evolution, that makes significant progress on such questions. Nicolas E. Humphries, Henri Weimerskirch and David W. Sims, have written a paper titled “A new approach for objective identification of turns and steps in organism movement data relevant to random walk modelling” in which they present a new approach to the problem of how to analyze 2-D and 3-D trajectories. (Clicking on the link will take you to their paper.)

It is well known that the projection onto 1-D of a 2-D or 3-D Brownian motion is itself Brownian motion. It is less well known that the projection of a 2-D or 3-D Lévy flight is a 1-D Lévy flight. In other words, the projection of a Lévy flight is itself a Lévy flight. Humphries, Weimerskirch and Sims exploit this key idea to improve our ability to detect Lévy walks.


The image above is the 1-D projection (with time on the horizontal axis now) of the 2-D walks shown above. The ultra-long jumps are evident. Coarse graining makes them even more visible:


The essence of their method is as follows. Given a 2-D or 3-D Lévy walk, they look at its 1-D projection. This projection, being 1-D, cannot contain any curvature. At this point, the usual methods for studying 1-D Lévy walks can be successfully applied. There is an elegance to the simplicity of the idea. More importantly, the method seems to work well.

Book: Perspectives and Challenges in Statistical Physics and Complex Systems for the Next Decade


Edited by:
Gandhimohan M Viswanathan (Universidade Federal do Rio Grande do Norte),
Ernesto P Raposo (Universidade Federal de Pernambuco),
Marcos Gomes Eleutério da Luz (Universidade Federal do Paraná)


Statistical Physics (SP) has followed an unusual evolutionary path in science. Originally aiming to provide a fundamental basis for another important branch of Physics, namely Thermodynamics, SP gradually became an independent field of research in its own right. But despite more than a century of steady progress, there are still plenty of challenges and open questions in the SP realm.

In fact, the area is still rapidly evolving, in contrast to other branches of science, which already have well defined scopes and borderlines of applicability. This difference is due to the steadily expanding number of applications, as well as ongoing improvements and revisions of concepts and methods in SP. Such particular aspects of SP lend further significance and timeliness to this book about perspectives and trends within the field.

Here, the aim is to present the state-of-the-art vision of expert researchers who study SP and Complex Systems. Although a comprehensive treatment is well beyond what can be treated in a single volume, the book provides a snapshot of the field today, as well as a glimpse of where the field may be heading during the next decade.

The book is aimed at graduate and advanced undergraduate physics students, as well as researchers who work with SP, Complex Systems, Computational Physics, Biological Physics and related topics. It addresses questions such as: What insights can be gained from recent advances in the study of traditional problems in SP? How can SP help us understand problems that arise in the biological sciences and in the study of complex systems? How can new problems be formulated using the ‘language’ of SP? In this way, it attempts to document partial progress in answering these and related questions.

The book also commemorates the occasion of the 70th anniversary in 2011 of two important physicists and friends who dedicated their lives to the understanding of nature in general and to the development of Statistical Physics and the science of Complexity in particular: Liacir Lucena and H Eugene Stanley.

Sample Chapter(s)
Comparing methods and Monte Carlo algorithms at phase transition regimes: A general overview (290 KB)

  • Further Advances in the Analysis of Traditional Problems in Statistical Physics:
    • Comparing Methods and Monte Carlo Algorithms at Phase Transition Regimes: A General Overview (C E Fiore)
    • Density of States of the Ising Model in the Field (B D Stošić)
    • A Renormalization Group Study of the Three-Color Ashkin-Teller Model on a Wheatstone Hierarchical Lattice (R Teodoro, C G Bezerra, A M Mariz, F A da Costa and J M de Araújo)
    • Applying Virial Theorem in Continuous Potential of Two Scales (N M Barraz Jr and M C Barbosa)
    • Elementary Statistical Models for Nematic Transitions in Liquid-Crystalline Systems (D B Liarte & S R Salinas)
    • Phase Diagram and Layer-Thinning Transitions in Free-Standing Liquid Crystal Films (M S S Pereira, I N de Oliveira and M L Lyra)
    • On Some Experimental Reasons for an Inhomogeneous Structure of Ambient Water on the Nanometer Length Scale (F Mallamace, C Corsaro and C Vasi)
    • Polyamorphism and Polymorphism of a Confined Water Monolayer: Liquid-Liquid Critical Point, Liquid-Crystal and Crystal-Crystal Phase Transitions (V Bianco, O Vilanova and G Franzese)
  • New Insights into Traditional Problems in Statistical Physics:
    • Topological and Geometrical Aspects of Phase Transitions (F A N Santos, J A Rehn and M D Coutinho-Filho)
    • Pacman Percolation and the Glass Transition (R Pastore, M P Ciamarra and A Coniglio)
    • Exact Solution for a Diffusive Process on a Backbone Structure: Green Function Approach and External Force (E K Lenzi, L R da Silva, A A Tateishi, M K Lenzi and H V Ribeiro)
    • Multifractal Surfaces: Lucena and Stanley Approaches (G Corso and D A Moreira)
  • Applications to Biological Problems:
    • Nanoelectronics of a DNA Molecule (E L Albuquerque, U L Fulco, E W S Caetano, V N Freire, M L Lyra and F A B F Moura)
    • Magic Trees in Mammalians Respiration or When Evolution Selected Clever Physical Systems (B Sapoval and M Filoche)
    • Social Distancing Strategies Against Disease Spreading (L D Valdez, C Buono, P A Macri and L A Braunstein)
  • Non-Traditional Problems Related to Complex Systems:
    • Thermodynamics and Kinetic Theory of Granular Materials (G M Kremer)
    • Continuous and First-Order Jamming Transition in Crossing Pedestrian Traffic Flows (H J Hilhorst, J Cividini and C Appert-Rolland)
    • Multiplicative Processes in Visual Cognition (H F Credidio, E N Teixeira, S D S Reis, A A Moreira and J S Andrade Jr)
    • Search Strategy: Hedging Your Bet (M F Shlesinger)

Readership: Advanced undergraduates, graduate students, researchers, academics and professionals in statistical physics, complex systems and nonlinear science.

TelexFREE é uma pirâmide?

Recentemente muitas pessoas sofreram prejuízos desnecessariamente, devido a investimentos em esquemas financeiros.  Um exemplo é o TelexFREE.   Em julho deste ano esta empresa foi impedida judicialmente de exercer suas funções.  Muitas pessoas estão confusas sobre o que está acontecendo. Aparentemente, algumas dessas pessoas não sabem o que é uma pirâmide financeira.   Neste artigo, venho tentar esclarecer as questões fundamentais sobre pirâmides financeiras.   Notem, por favor, que embora o título contenha a palavra TelexFREE, de fato este artigo não é sobre o TelexFREE, mas sim sobre a natureza de pirâmides financeiras.

Mas, antes, venho lamentar o estado da educação no Brasil.  Do meu ponto de vista, essa confusão sobre pirâmides é consequência natural do péssimo estado de educação no país. Já fui professor de ensino fundamental e médio e conheço a primeira mão os vários problemas. Em 2013 o Brasil ficou em último lugar na América Latina no ranking da PNUD da ONU.  Enquanto que a escolaridade média do brasileiro é inferior a 8 anos, por outro lado a dos Norte-Americanos é superior a 13 anos.   A falta de escolaridade e de educação cria muitos problemas, sendo um deles o analfabetismo financeiro.   Por exemplo, os defensores das empresas, reagindo às acusações sobre serem pirâmides, reagem fazendo afirmações absurdas.  Alguns andam dizendo que a pirâmide é o Congresso Nacional!  Só falta então dizer que as pirâmides do Egito foram construídas por aliens!   O analfabetismo científico e o analfabetismo financeiro tem origem, em parte, na falta de educação de qualidade.  Levam a consequências negativas tanto para o indivíduo como para a sociedade.

Uma pirâmide financeira é uma maneira de ganhar dinheiro que depende da entrada de um número cada vez maior de pessoas participando para o esquema não quebrar. Como a população do planeta é finita, uma pirâmide financeira não pode operar por muito tempo. Ou ela quebra, devido a falta de entrada de novas pessoas, ou então, muito raramente, o esquema se transforma, via calote parcial, para se tornar sustentável, não sendo mais, então, uma pirâmide.

Um exemplo para ilustrar a ideia: Digamos que uma empresa PIRAMIDE S/A ofereça um retorno financeiro mensal de 50%, muito acima do que é encontrado no mercado financeiro.  Obviamente, muitas pessoas sem conhecimento sobre finanças irão querer entrar nesse esquema.  Vamos supor que os gerentes da empresa usem todo o dinheiro que entra na empresa para pagar o retorno financeiro aos clientes.  Digamos que no primeiro mês a empresa captou 1000 reais de 10 clientes que entraram com 100 reais cada, tal que 30 dias depois a empresa precisa ‘devolver’ 1500 reais. Cada uma das 10 pessoas precisa receber de volta 150 reais. Como eles entraram com 100 reais, o retorno financeiro é 50%, altíssimo.  De onde vem esses 500 reais adicionais?   Ela precisa sair de algum lugar. De onde ela vem?  A resposta é simples: Os 500 adicionais vem do bolso de novos clientes!    Então, supondo que cada cliente possa comprar uma participação no esquema em parcelas de 100 reais, são necessários 5 novos clientes ou parcelas.

É importante notar que os 500 reais que entram vindo dessas 5 novas pessoas servem para pagar apenas os ‘juros’ das primeiras 10 pessoas! Se as 10 primeiras pessoas continuarem com os 1000 reais investidos, então no segundo mês será necessário a entrada de mais 5 pessoas, além dos primeiros 5 que já entraram!   Veja bem que ainda nem contabilizamos o retorno financeiro dos 5 novatos inicais.  Como o retorno financeiro é alto, existe um forte incentivo para os clientes não pedirem para sacar o valor investido. Os gerentes podem, assim, manipular seus clientes (na verdade, suas vítimas) por um tempo prolongado.

O nome ‘pirâmide’ vem do fato de que os clientes iniciais precisam recrutar  novatos, que formam então a segunda camada. Esses então precisam recrutar a terceira camada, etc.   Em princípio, camadas inferiores precisam ser maior do que as camadas superiores, para permitir que as pessoas nas camadas superiores possam começar a sacar.  O recrutamento de mais e mais novos clientes, que então se tornam novos recrutadores,  é um dos requisitos de pirâmides financeiras.

Note também que é impossível devolver o dinheiro investido para todo mundo simultaneamente.  Em pirâmides financeiros, é impossível todo mundo sacar o dinheiro simultaneamente, mesmo em teoria.  Já em fundos de investimento, por exemplo, é possível liquidar todos os ativos para devolver integralmente todo o saldo dos clientes, pelo menos em teoria se não em prática.

Existem variações de pirâmides, como por exemplo o esquema de Ponzi, que é um pouco mais centralizado relativo às pirâmides financeiras.   Mas a ideia básica é a mesma:  O aspecto mais importante é a insustentabilidade desses esquemas.

Como eu trabalhei por vários anos na área de Econofísica, conheço um pouco sobre finanças.   Posso afirmar, com uma certa segurança, que pirâmides financeiras representam, quase sempre, fraudes. Os clientes são as vítimas.  É sempre difícil dizer isto para quem acredita piamente nos esquemas. Tipicamente, as vítimas dessas fraudes se tornam grandes defensores dos esquemas. É irônico e triste ver a vítima defendendo o agressor.

Um pouco mais de 1 ano atrás, quando conhecidos me falaram sobre o assunto e as maravilhas que eram esses negócios, eu imediatamente pensei na questão da sustentabilidade do esquema.   Expliquei que eu jamais iria querer entrar num esquema desses, pois rentabilidade e risco caminham juntos. Um retorno financeiro astronômico é possível sinal de risco também astronômico.  Essas informações estão na Internet e qualquer pessoa pode usufruir delas. Mesmo assim, as pessoas viram vítimas e acabam sofrendo desnecessariamente. O que fazer?

Na verdade, é sim possível ganhar dinheiro numa pirâmide, se você se cuidar de sair da camada mais baixa o mais rápido possível e então sacar todo seu saldo antes da quebra. Isso é possível, mas é ariscado.  Em jogos de loteria, muitos perdem para que poucos possam ganhar muito.  Da mesma forma, é possível ganhar dinheiro numa pirâmide.  Certamente, os criadores dessas fraudes tipicamente ganham muito dinheiro.  Obviamente, existem problemas éticos e essas práticas frequentemente representam crimes.  Um caso famoso recente é Bernard Madoff. Ele era considerado um grande investidor, mas hoje está na prisão.

Voltando ao assunto:  A empresa é ou não é pirâmide?

Basta decidir se a empresa poderia sobreviver mesmo sem a entrada de novos clientes (‘divulgadores’).   Se a resposta ainda não for óbvia por algum motivo, então a seguinte indagação talvez ajude a esclarecer um pouco mais o fato crucial:

O que aconteceria se todo mundo no planeta já fosse cliente?

Simulações de Física

Venho compartilhar com todos esse link de simulações interativas de experiências de Física.

Um aluno de Iniciação Científica, Gustavo Lamenha, e eu traduzimos para Português os textos, originalmente em Inglês.  O site original foi criado pelo Professor Fu-Kwun Hwang, da National Taiwan Normal University. Com o apoio da Reitoria da Universidade Federal de Alagoas (UFAL) em Maceió-AL, onde eu era professor na época, conseguimos financiamento no valor de R$7000,00 para reformar uma sala do Departamento de Física da UFAL, com a finalidade de montar o primeiro Laboratório de Multimídia da UFAL.   Mesmo com essa pequena quantia, conseguimos terminar as obras e o projeto foi um grande sucesso.

Durante sua existência física, o laboratório era bastante popular entre os alunos de graduação da UFAL. Os novos métodos de ensino repercutiram positivamente, veja aqui um artigo que saiu publicado no jornal Tribuna de Alagoas, na edição de Domingo do dia 06/01/2002. No entanto, depois de 2 ou 3 anos, o laboratório foi demolido para construir salas para professores.

Quando Professor Fu-Kwun Hwang soube que nosso laboratório seria desativado, ele muito gentilmente se ofereceu para hospedar em Taiwan um clone do nosso site original, em Português, para permitir a continuidade de sua existência na Internet.  Agradeço muito ao Professor Fu-Kwun Hwang por tudo que tem feito para melhorar o ensino de Física para as gerações futuras.

Movement Ecology journal now accepting manuscript submissions

Ran Nathan, of The Hebrew University of Jerusalem, and Luca Giuggioli, of the University of Bristol, have co-founded the new peer-reviewed interdisciplinary journal Movement Ecology I support open-access publication, so I am particularly pleased to be on the Editorial Board of this new open-access journal. Its scope spans empirical and theoretical approaches to the ecology of the movement of animals, plants and microorganisms.  Movement Ecology is now accepting submissions through their web site.

List of publications

Click here to see the list of my publications. In most cases the full PDF file is available, along with other bibliographic information. I aim to keep the page fairly up-to-date.

Lévy flights of wandering albatrosses: fact or fiction?

A very recent and remarkable study appears to overturn the conventional wisdom about how wandering albatrosses move when they search for food. A new paper published in the Proceedings of the National Academy of Sciences USA (PNAS) by Nicolas E. Humphries et al. looks at whether or not the idea of Lévy flight foraging is supported by empirical data for two species of albatrosses: the wandering albatross and the black-browed albatross. A Lévy flight is a type of random walk in which the zig-zag patterns contain rare but extra long jumps.  In mathematical terms, Lévy flights contain a power law tailed probability density function of jump lengths. The latest results add a new twist to the controversy surrounding wandering albatrosses.

A little background: More than a decade ago in 1994, as a graduate student of Professor Gene Stanley at Boston University, I analyzed flight data of wandering albatrosses.  Our motivation was to try to gain insight into nebulous concepts such as the “free will” of animals.  If  free will really exists and confers real advantages, then it ought to operate in complex and sophisticated behaviors such as foraging.  At least, that was our thinking.  The research progressed rapidly and, in 1996, we published in Nature a paper titled Lévy flight search patterns of wandering albatrosses in which we reported what we thought was evidence of Lévy flights.  A year later, in 1997, I returned to Brazil.  In Natal and later in Maceió, I continued to investigate this phenomenon.  In 1998, my collaborators and I made some unexpected discoveries about previously unknown properties of Lévy flights.  In 1999, Sergey Buldyrev, Shlomo Havlin, Marcos da Luz, Ernesto Raposo, Gene Stanley and I published these results in another paper in Nature.  We showed analytically  that Lévy flights can optimize random searches under conditions of scarcity when the targets are revisitable and randomly located.  (Actually we also assumed some other simplifying conditions, such as negligible learning and memory, but it turns out that these issues do not significantly alter the main findings of the 1999 paper.)

The main reason why Lévy flights increase efficiency is that they can
reduce the expensive habit (known as “oversampling”) of Brownian random walkers to revisit previously visited sites.  They also increase, relative to ballistic motion, the chance of reaching nearby targets at the cost of reaching far away targets, thereby reducing overall traveled distances.  These and many other issues are discussed in our new book, The Physics of Foraging.

Over the next 5 years or so, a growing number of studies added weight to the Lévy flight foraging hypothesis.  Naturally, so did attempts to overturn or falsify the hypothesis.  As Karl Popper explained, scientists do not just verify hypotheses, rather they work to falsify them. This is in large part how science advances.   Supplanting old theories with new and better theories is how our understanding grows and improves.

The Lévy flight foraging hypothesis exploded into controversy in 2007, when another  paper published in Nature by A. M. Edwards et al.  questioned the validity of the hypothesis.  The hypothesis holds that animals should have evolved to move in a superdiffusive Lévy flight type of behavior under conditions of scarcity, because such behavior improves search efficiencies and encounter rates according to mathematical predictions.  Early papers from the 1980s and early 1990s looked at Lévy walks and similar behavior of microorganisms. But it was only in 1996 that the idea of Lévy flight foraging became a hot topic, when we reported evidence  (apparently) showing that Diomedea exulans, the wandering albatross, performs Lévy flight search patterns.   But there was a problem with the data. In 2007, in a paper authored by A. M. Edwards and many others,  we reported evidence which corrected the 1996 results.

Journalists had a field day.  For example, it was claimed by a journalist (Alexandra Witze of Science News) that the 1996 albatross results had been ‘debunked.’   After 2007  few people thought that albatrosses used Lévy flight patterns to forage.

It is in this context that the results reported in this new PNAS paper are simply extraordinary.  The crucial point is that this study directly
contradicts the 2007 paper of Edwards et al.  How could this be?  The authors provide an explanation for the inconsistency: Whereas
the 2007 study only looked at pooled data sets, the present paper
looks at individual birds.  If I understood the paper correctly, the
authors have re-tested the 2007 data and found that the behavior of
individual birds can be strikingly different from the average
behavior.  What I understood was that individual birds do not always
use the Lévy behavior, so when you average the data for many birds,
the signal-to-noise ratio degrades and the power law tail is less
clear.  The authors also reconfirm what was suspected all along, viz.,
that the 2007 paper did not explicitly take into account power law
truncation. But the bigger effect seems to be the pooling.

Maybe albatrosses do in fact use Lévy flight foraging patterns after all.