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This book focuses on a central question in the field of complex systems: Given a fluctuating (in time or space), uni- or multi-variant sequentially measured set of experimental data (even noisy data), how should one analyse non-parametrically the data, assess underlying trends, uncover characteristics of the fluctuations (including diffusion and jump contributions), and construct a stochastic evolution equation?
Here, the term "non-parametrically" exemplifies that all the functions and parameters of the constructed stochastic evolution equation can be determined directly from the measured data.
The book provides an overview of methods that have been developed for the analysis of... 

Packing of cubic particles arises in a variety of problems, ranging from biological materials to colloids and the fabrication of new types of porous materials with controlled morphology. The properties of such packings may also be relevant to problems involving suspensions of cubic zeolites, precipitation of salt crystals during CO2 sequestration in rock, and intrusion of fresh water in aquifers by saline water. Not much is known, however, about the structure and statistical descriptors of such packings. We present a detailed simulation and microstructural characterization of packings of nonoverlapping monodisperse cubic particles, following up on our preliminary results [H. Malmir, Sci.... 

Understanding the properties of random packings of solid objects is of critical importance to a wide variety of fundamental scientific and practical problems. The great majority of the previous works focused, however, on packings of spherical and sphere-like particles. We report the first detailed simulation and characterization of packings of non-overlapping cubic particles. Such packings arise in a variety of problems, ranging from biological materials, to colloids and fabrication of porous scaffolds using salt powders. In addition, packing of cubic salt crystals arise in various problems involving preservation of pavements, paintings, and historical monuments, mineral-fluid interactions,... 

We consider the O (n) theory in the n → 0 limit. We show that the theory is described by logarithmic conformal field theory, and that the correlation functions have logarithmic singularities. The explicit forms of the two-, three- and four-point correlation functions of the scaling fields and the corresponding logarithmic partners are derived. © 2004 Elsevier B.V. All rights reserved  

The formation of atomic nanoclusters on suspended graphene sheets has been investigated by employing a molecular dynamics simulation at finite temperature. Our systematic study is based on temperature-dependent molecular dynamics simulations of some transition and alkali atoms on suspended graphene sheets. We find that the transition atoms aggregate and make various size nanoclusters distributed randomly on graphene surfaces. We also report that most alkali atoms make one atomic layer on graphene sheets. Interestingly, the potassium atoms almost deposit regularly on the surface at low temperature. We expect from this behavior that the electrical conductivity of a suspended graphene doped by... 

It is well known that the ballistic deposition and the restricted solid-on-solid models belong to the same universality class, having the same roughness and growth exponents. In this paper, we determine some new statistical properties of the two models, such as the Kramers-Moyal coefficients and the Markov length scale, and show them to be distinct for the two models. © 2008 IOP Publishing Ltd  

Article Outline: Glossary Definition of the Subject Introduction Stochastic Processes Stochastic Time Series Analysis Applications: Processes in Time Applications: Processes in Scale Future Directions Further Reading Acknowledgment Bibliography  

The single and multiple scattering regimes of electromagnetic waves in a disordered system with fluctuating permittivity are studied by numerical simulations of Maxwell's equations. For an array of emitters and receivers in front of a medium with randomly varying dielectric constant, we calculate the backscattering matrix from the signal responses at all receiver points j to electromagnetic pulses generated at each emitter point i. We show that the statistical properties of the backscattering matrix are in agreement with the recent experimental results for ultrasonic waves (Aubry A. and Derode A., Phys. Rev. Lett., 102 (2009) 084301) and light (Popoff S. M. et al., Phys. Rev. Lett., 104... 

Stochastic differential equations (SDE) play an important role in a range of application areas, including biology, physics, chemistry, epidemiology, mechanics, microelectronics, economics, and finance [1]. However, most SDEs, especially nonlinear SDEs, do not have analytical solutions, so that one must resort to numerical approximation schemes in order to simulate trajectories of the solutions to the given equation. The simplest effective computational method for approximation of ordinary differential equations is the Euler’s method. The Euler–Maruyama method is the analogue of the Euler’s method for ordinary differential equations for numerical simulation of the SDEs [2]. Another numerical... 

In this chapter we study stochastic processes in the presence of jump discontinuity, and discuss the meaning of non-vanishing higher-order Kramers–Moyal coefficients. We describe in details the stochastic properties of Poisson jump processes. We derive the statistical moments of the Poisson process and the Kramers–Moyal coefficients for pure Poisson jump events. Growing evidence shows that continuous stochastic modeling (white noise-driven Langevin equation) of time series of complex systems should account for the presence of discontinuous jump components [1–6]. Such time series have some distinct important characteristics, such as heavy tails and occasionally sudden large jumps.... 

In this chapter we reconstruct stochastic dynamical equations with known drift and diffusion coefficients, as well as known properties of jumps, jump amplitude and jump rate from synthetic time series, sampled with time interval τ. The examples have Langevin (white noise- and Lévy noise-driven) and jump-diffusion dynamical equations. We also study the estimation of the Kramers–Moyal coefficients for “phase” dynamics that enable us to investigate the phenomenon of synchronisation in systems with interaction. © 2019, Springer Nature Switzerland AG  

Data sampled at discrete times appear as successions of discontinuous jump events, even if the underlying trajectory is continuous. In this chapter we study finite sampling τ expansion of the Kramers-Moyal conditional moments for the Langevin and jump-diffusion dynamics. Using the expansion for the Langevin dynamics, we introduce a criterion to validate the method numerically, namely, the Pawula theorem, to judge whether the fourth-order KM moment tends to zero. The criterion is a relation between the fourth- and second-order KM conditional moments for small time lag τ [1]. © 2019, Springer Nature Switzerland AG  

The method outlined in the Chaps. 15 – 21 has been used for revealing nonlinear deterministic and stochastic behaviors in a variety of problems, ranging from physics, to neuroscience, biology and medicine. In most cases, alternative procedures with strong emphasis on deterministic features have been only partly successful, due to their inappropriate treatment of the dynamical fluctuations [1]. In this chapter, we provide a list of the investigated phenomena using the introduced reconstruction method. In the “outlook” possible research directions for future are discussed. © 2019, Springer Nature Switzerland AG  

In Chaps. 16 – 21 we address a central question in the field of complex systems: Given a fluctuating (in time or space), sequentially uni- or multi-variant measured set of experimental data (even noisy data), how should one analyse the data non-parametrically, assess their underlying trends, discover the characteristics of the fluctuations, including diffusion and jump parts, and construct stochastic evolution equation for the data? © 2019, Springer Nature Switzerland AG  

In this chapter we show the equivalence between the Langevin approach and the Fokker–Planck equation, and derive the equation for the statistical moments of the process whose dynamics is described by the Langevin equation. © 2019, Springer Nature Switzerland AG  

In this chapter we introduce the Langevin equation and Wiener process. All the statistical properties of Wiener process will be presented and discussed. © 2019, Springer Nature Switzerland AG  

Complex systems are composed of a large number of subsystems that may interact with each other. The typically nonlinear and multiscale interactions often lead to large-scale behaviors, which are not easily predicted from the knowledge of only the behavior of individual subsystems. © 2019, Springer Nature Switzerland AG  

As an example of the analysis and reconstruction of nonlinear stochastic time series, we consider an important medical problem, namely, epileptic brain dynamics. © 2019, Springer Nature Switzerland AG  

Most real world time series have transient behaviours and are non-stationary. They exhibit different type of non-stationarities, such as trends, cycles, random-walking, and generally exhibit strong intermittency. Therefore local stochastic characteristics of time series, such as the drift and diffusion coefficients, as well as the jump rate and jump amplitude, will provide very important information for understanding and quantifying “real time” variability of time series. For diffusive processes the systems have a longer memory and a higher correlation time scale and, therefore, one expects the stochastic features of dynamics to change slowly. In contrast, a rapid change of dynamics with...