47 research outputs found
L\'evy walks
Random walk is a fundamental concept with applications ranging from quantum
physics to econometrics. Remarkably, one specific model of random walks appears
to be ubiquitous across many fields as a tool to analyze transport phenomena in
which the dispersal process is faster than dictated by Brownian diffusion. The
L\'{e}vy walk model combines two key features, the ability to generate
anomalously fast diffusion and a finite velocity of a random walker. Recent
results in optics, Hamiltonian chaos, cold atom dynamics, bio-physics, and
behavioral science demonstrate that this particular type of random walks
provides significant insight into complex transport phenomena. This review
provides a self-consistent introduction to L\'{e}vy walks, surveys their
existing applications, including latest advances, and outlines further
perspectives.Comment: 50 page
採餌問題のための確率的探索戦略の設計と最適化
Autonomous robot’s search strategy is the set of rules that it employs while looking for targets in its environment. In this study, the stochastic movement of robots in unknown environments is statistically studied, using a Levy walk method. Biological systems (e.g., foraging animals) provide useful models for designing optimal stochastic search algorithms. Observations of biological systems, ranging from large animals to immune cells, have inspired the design of efficient search strategies that incorporate stochastic movement. In this study, we seek to identify the optimal stochastic strategies for autonomous robots. Given the complexity of interaction between the robot and its environment, optimization must be performed in high-dimensional parameter space. The effect of the explanatory variable on the forger robot movement with the minimum required energy was also studied using experiments done by the response surface methodology (RSM). We analyzed the extent to which search efficiency requires these characteristics, using RSM. Correlation between the involved parameters via a Lévy walk process was examined through designing a setup for the experiments to determine the interaction of the involved variables and the robot movement. The extracted statistical model represents the priority influence of those variables on the robot by developing the statistical model of the mentioned unknown area. The efficiency of a simple strategy was investigated based on Lévy walk search in two-dimensional landscapes with clumped resource distributions. We show how RSM techniques can be used to identify optimal parameter values as well as to describe how sensitive efficiency reacts to the changes in these values. Here, we identified optimal parameter for designing robot by using stochastic search pattern and applying mood-switching criteria on a mixture of speed and sensor and μ to determine how many robots are needed for a solution. Fractal criterion-based robot strategies were more efficient than those based on the resource encounter criterion, and the former was found to be more robust to changes in resource distribution as well.九州工業大学博士学位論文 学位記番号:生工博甲第358号 学位授与年月日:令和元年9月20日1 Introduction|2 Levy Walk|3 Design of Experiment (DOE)|4 Response Surface Methodology|5 Results and Discussions|6 Conclusion九州工業大学令和元年
Fractional Calculus and the Future of Science
Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding
Towards a Comprehensive Framework for the Analysis of Anomalous Diffusive Systems
PhDThe modelling of transport processes in biological systems is one of the main theoretical
challenges in physics, chemistry and biology. This is motivated by their essential role in
the emergence of diseases, like tumour metastases, which originate from the spontaneous
migration of cancer cells. Thus, improvements in their understanding could potentially
pave the way for an outstanding innovation of present-day techniques in medicine. These
processes often exhibit anomalous properties, which are qualitatively described by the
power-law scaling of their mean square displacement, compared to the linear one of normal
diffusion. Such behaviour has been often successfully explained by the celebrated
continuous-time random walk model. However, recent experimental studies revealed the
existence of both more complicated mean square displacement behaviour and anomalous
features in other characteristic observables, e.g. the position-velocity statistics or the two
point correlation functions of either the velocity or the position. Thus, in order to understand
the anomalous diffusion recorded in these experiments and assess the microscopic
processes underlying the observed macroscopic dynamics, one needs to have a complete
tool-kit of techniques and models that can be readily compared with the experimental
datasets. In this Thesis, we contribute to the construction of such a complete framework
by fully characterising anomalous processes, which are described by means of a continuoustime
random walk with general waiting time distributions and/or external forces that are
exerted both during the jumps (as in the original model) and the waiting times. In the first
case we derive both the joint statistics of these processes and their observables, specifically
by obtaining a generalised fractional Feynman-Kac formula, and their multipoint correlation
functions and employ them to fit the mean square displacement data of diffusing
mitochondria. This result supports the experimental relevance of our formalism, which
comprises general formulas for several quantities that can provide readily predictable tests
to be checked in experiments. In the second case, we characterise the new anomalous
processes by means of Langevin equations driven by a novel type of non Gaussian noise,
which reproduces the typical fluctuations of a free diffusive continuous-time random walk.
For a constant external force, we also obtain the fractional evolution equations of their
position probability density function and show that, contrarily to continuous-time random
walks, they are weak Galilean invariant, i.e., their position distribution in different Galilean
frames is obtained by shifting the sample variable according to the relative motion of the
frames. Thus, these processes provide a suitable frame-invariant framework, that could be
employed to investigate the stochastic thermodynamics of anomalous diffusive processes
Behavioural analysis of marine predator movements in relation to heterogeneous environments
An understanding of the spatio-temporal dynamics of marine predator populations is essential for the sustainable management of marine resources. Tagging studies are providing ever more information about the movements and migrations of marine predators and much has been learned about where these predators spend their time. However little is known about their underlying motivations, making it difficult to make predictions about how apex predators will respond to changing environments. While much progress has been made in behavioural ecology through the use of optimality models, in the marine environment the necessary costs and benefits are difficult to quantify making this approach less successful than with terrestrial studies. One aspect of foraging behaviour that has proved tractable however is the optimisation of random searches. Work by statistical physicists has shown that a specialised movement, known as Lévy flight, can optimise the rate of new prey patch encounters when new prey patches are beyond sensory range. The resulting Lévy flight foraging (LFF) hypothesis makes testable predictions about marine predator search behaviour that can be addressed with the theoretical and empirical studies that form the basis of this thesis. Results presented here resolve the controversy surrounding the hypothesis, demonstrating the optimality of Lévy searches under a broader set of conditions than previously considered, including whether observed Lévy patterns are innate or emergent. Empirical studies provide robust evidence for the prevalence of Lévy search patterns in the movements of diverse marine pelagic predators such as sharks, tunas and billfish as well as in the foraging patterns of albatrosses, overturning a previous study. Predictions from the LFF hypothesis concerning fast moving prey are confirmed leading to simulation studies of ambush predator’s activity patterns. Movement analysis is then applied to the assessment of by-catch mitigation efforts involving VMS data from long-liners and simulated sharks
Dynamical Systems
Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...
Intermittency and Self-Organisation in Turbulence and Statistical Mechanics
There is overwhelming evidence, from laboratory experiments, observations, and computational studies, that coherent structures can cause intermittent transport, dramatically enhancing transport. A proper description of this intermittent phenomenon, however, is extremely difficult, requiring a new non-perturbative theory, such as statistical description. Furthermore, multi-scale interactions are responsible for inevitably complex dynamics in strongly non-equilibrium systems, a proper understanding of which remains a main challenge in classical physics. As a remarkable consequence of multi-scale interaction, a quasi-equilibrium state (the so-called self-organisation) can however be maintained. This special issue aims to present different theories of statistical mechanics to understand this challenging multiscale problem in turbulence. The 14 contributions to this Special issue focus on the various aspects of intermittency, coherent structures, self-organisation, bifurcation and nonlocality. Given the ubiquity of turbulence, the contributions cover a broad range of systems covering laboratory fluids (channel flow, the Von Kármán flow), plasmas (magnetic fusion), laser cavity, wind turbine, air flow around a high-speed train, solar wind and industrial application
Modeling and predicting time series of social activities with fat-tailed distributions
Fat-tailed distributions, characterized by the relation P(x) ∝ x^{−α−1}, are an emergent statistical signature of many complex systems, and in particular of social activities. These fat-tailed distributions are the outcome of dynamical processes that, contrary to the shape of the distributions, is in most cases are unknown. Knowledge of these processes’ properties sheds light on how the events in these fat tails, i.e. extreme events, appear and if it is possible to anticipate them. In this Thesis, we study how to model the dynamics that lead to fat-tailed distributions and the possibility of an accurate prediction in this context. To approach these problems, we focus on the study of attention to items (such as videos, forum posts or papers) in the Internet, since human interactions through the online media leave digital traces that can be analysed quantitatively. We collected four sets of time series of online activity that show fat tails and we characterize them.
Of the many features that items in the datasets have, we need to know which ones are the most relevant to describe the dynamics, in order to include them in a model; we select the features that show high predictability, i.e. the capacity of realizing an accurate prediction based on that information. To quantify predictability we propose to measure the quality of the optimal forecasting method for extreme events, and we construct this measure. Applying these methods to data, we find that more extreme events (i.e. higher value of activity) are systematically more predictable, indicating that the possibility of discriminate successful items is enhanced. The simplest model that describes the dynamics of activity is to relate linearly the increment of activity with the last value of activity recorded. This starting point is known as proportional effect, a celebrated and widely used class of growth models in complex systems, which leads to a distribution of activity that is fat-tailed. On the one hand, we show that this process can be described and generalized in the framework of Stochastic Differential Equations (SDE) with Normal noise; moreover, we formalize the methods to estimate the parameters of such SDE. On the other hand, we show that the fluctuations of activity resulting from these models are not compatible with the data. We propose a model with proportional effect and Lévy-distributed noise, that proves to be superior describing the fluctuations around the average of the data and predicting the possibility of an item to become an extreme event.
However, it is possible to model the dynamics using more than just the last value of activity; we generalize the growth models used previously, and perform an analysis that indicates that the most relevant variable for a model is the last increment in activity. We propose a new model using only this variable and the fat-tailed noise, and we find that, in our data, this model is superior to the previous models, including the one we proposed. These results indicate that, even if present, the relevance of proportional effect as a generative mechanism for fat-tailed distributions is greatly reduced, since the dynamical equations of our models contain this feature in the noise. The implications of this new interpretation of growth models to the quantification of predictability are discussed along with applications to other complex systems