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This volume presents a pedagogical review of the functional
distribution of anomalous and nonergodic diffusion and its
numerical simulations, starting from the studied stochastic
processes to the deterministic partial differential equations
governing the probability density function of the functionals.
Since the remarkable theory of Brownian motion was proposed by
Einstein in 1905, it had a sustained and broad impact on diverse
fields, such as physics, chemistry, biology, economics, and
mathematics. The functionals of Brownian motion are later widely
attractive for their extensive applications. It was Kac, who
firstly realized the statistical properties of these functionals
can be studied by using Feynman's path integrals.In recent decades,
anomalous and nonergodic diffusions which are non-Brownian become
topical issues, such as fractional Brownian motion, Levy process,
Levy walk, among others. This volume examines the statistical
properties of the non-Brownian functionals, derives the governing
equations of their distributions, and shows some algorithms for
solving these equations numerically.
This book investigates statistical observables for anomalous and
nonergodic dynamics, focusing on the dynamical behaviors of
particles modelled by non-Brownian stochastic processes in the
complex real-world environment. Statistical observables are widely
used for anomalous and nonergodic stochastic systems, thus serving
as a key to uncover their dynamics. This study explores the cutting
edge of anomalous and nonergodic diffusion from the perspectives of
mathematics, computer science, statistical and biological physics,
and chemistry. With this interdisciplinary approach, multiple
physical applications and mathematical issues are discussed,
including stochastic and deterministic modelling, analyses of
(stochastic) partial differential equations (PDEs), scientific
computations and stochastic analyses, etc. Through regularity
analysis, numerical scheme design and numerical experiments, the
book also derives the governing equations for the probability
density function of statistical observables, linking stochastic
processes with PDEs. The book will appeal to both researchers of
electrical engineering expert in the niche area of statistical
observables and stochastic systems and scientists in a broad range
of fields interested in anomalous diffusion, especially applied
mathematicians and statistical physicists.
The aim of this book is to extend the application field of
'anomalous diffusion', and describe the newly built models and the
simulation techniques to the models.The book first introduces
'anomalous diffusion' from the statistical physics point of view,
then discusses the models characterizing anomalous diffusion and
its applications, including the Fokker-Planck equation, the
Feymann-Kac equations describing the functional distribution of the
anomalous trajectories of the particles, and also the microscopic
model - Langevin type equation. The second main part focuses on
providing the high accuracy schemes for these kinds of models, and
the corresponding convergence and stability analysis.
This book focuses on modeling the anomalous diffusion phenomena,
being ubiquitous in the natural world. Both the microscopic models
(stochastic processes) and macroscopic models (partial differential
equations) have been built up. The relationships between the two
kinds of models are clarified, and based on these models, some
statistical observables are analyzed. From statistics to
mathematics, the built models show their power with their
associated applications.This book is important for students to
develop basic skills to be able to succeed in their future
research. In addition to introducing the related models or methods,
it also provides the corresponding applications and simulation
results, which will attract more readers ranging from
mathematicians to physicists or chemists, to name a few.
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