Functional Distribution Of Anomalous And Nonergodic Diffusion: From Stochastic Processes To Pdes

This volume discusses the functional distribution of anomalous and nonergodic diffusion and its numerical simulations, covering topics such as Brownian motion, Feynman's path integrals, and non-Brownian functionals. It highlights the applications of these functionals in various fields and explores their statistical properties and numerical solutions.

Format: Hardback
Length: 260 pages
Publication date: 21 July 2022
Publisher: World Scientific Publishing Co Pte Ltd

Since Einstein's groundbreaking theory of Brownian motion was introduced in 1905, it has had a profound and far-reaching impact on a wide range of disciplines, including physics, chemistry, biology, economics, and mathematics. The functionals of Brownian motion have subsequently gained widespread appeal due to their extensive applications. It was Kac who first recognized the statistical properties of these functionals and proposed using Feynman's path integrals to study them. In recent decades, anomalous and nonergodic diffusions, which are non-Brownian, have become increasingly relevant topics, including fractional Brownian motion, Lévy processes, Lévy walks, and others.

This volume explores the statistical properties of non-Brownian functionals, derives the governing equations for their distributions, and presents algorithms for solving these equations numerically. It provides a comprehensive pedagogical review of the functional distribution of anomalous and nonergodic diffusion and its numerical simulations, covering the studied stochastic processes and the deterministic partial differential equations governing the probability density function of the functionals.

The study of Brownian motion has played a crucial role in developing our understanding of many physical phenomena, including heat transfer, chemical reactions, and particle motion. By analyzing the motion of particles in a liquid or gas, scientists can study the effects of friction, gravity, and other forces on their trajectory. Brownian motion has also been used to model financial market behavior, where it is used to describe the random movement of stock prices.

In addition to its theoretical significance, Brownian motion has had numerous practical applications. For example, it is used in drug discovery to study the behavior of molecules in the bloodstream and to identify potential new drugs. It is also used in imaging techniques, such as MRI and CT scans, to create detailed images of the body's internal structures.

Despite its many successes, Brownian motion is not without its challenges. One of the main challenges is the fact that it is a stochastic process, which means that its behavior is unpredictable and difficult to model accurately. This has led to the development of new mathematical techniques, such as stochastic differential equations, to better understand and predict the behavior of Brownian motion.

Another challenge is the fact that Brownian motion is often studied in high-dimensional spaces, which can make it difficult to analyze and interpret. This has led to the development of new computational techniques, such as Monte Carlo methods, to simulate the behavior of Brownian motion in complex systems.

Despite these challenges, Brownian motion remains a fundamental concept in physics and has had a profound impact on our understanding of the world around us. Its applications in fields such as drug discovery, finance, and imaging have made it a valuable tool for scientists and researchers alike. As our understanding of Brownian motion continues to evolve, it is likely to remain an important area of research and development for years to come.

In conclusion, 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 has 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, Lévy process, Lévy 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.

ISBN-13: 9789811250491