Focusing on the intersection of probability theory and applied mathematics, this book demonstrates how to express various problems through differential, difference, integral, and partial differential equations. It provides a comprehensive guide to solving these equations using approximation methods, making complex concepts accessible to readers interested in mathematical applications in probability.
and their Applications in Statistical Physics, Computational Neuroscience, and Biophysics
This is a monograph on the emerging branch of mathematical biophysics combining asymptotic analysis with numerical and stochastic methods to analyze partial differential equations arising in biological and physical sciences. In more detail, the book presents the analytic methods and tools for approximating solutions of mixed boundary value problems, with particular emphasis on the narrow escape problem. Informed throughout by real-world applications, the book includes topics such as the Fokker-Planck equation, boundary layer analysis, WKB approximation, applications of spectral theory, as well as recent results in narrow escape theory. Numerical and stochastic aspects, including mean first passage time and extreme statistics, are discussed in detail and relevant applications are presented in parallel with the theory. Including background on the classical asymptotic theory of differential equations, this book is written for scientists of various backgrounds interested inderiving solutions to real-world problems from first principles.
In Physics, Chemistry, and Biology
The book explores the significance of Brownian dynamics as mathematical models for the diffusive motion of microscopic particles across various environments. It emphasizes their importance in molecular and cellular biophysics, particularly in the diffusion processes that sustain life. The text delves into Brownian dynamics simulations, which apply stochastic differential equations to model biological micro devices like protein channels and neuronal synapses. Additionally, it highlights the broader applications of these models in fields such as physics, chemistry, and finance, addressing complex mathematical challenges beyond traditional theories.
Analysis and Applications
Focusing on the mathematical narrow escape problem, this book explores recent advancements in non-standard asymptotics within stochastic theory, highlighting its relevance to cell biology. The first section delves into advanced asymptotic methods in partial equations, catering to applied mathematicians and theoretical physicists. The second section provides computational biologists and other scientists with output formulas from the mathematical discussions, emphasizing practical applications in modeling various theoretical molecular and cellular biology challenges.
Focusing on stochastic processes, this book provides an analytical framework that bridges probability theory and function space, catering specifically to scientists accustomed to traditional applied mathematics. It aims to enhance accessibility to complex concepts, making it a valuable resource for those in the physical and life sciences seeking to deepen their understanding of probability.
"Brownian dynamics serve as mathematical models for the diffusive motion of microscopic particles of various shapes in gaseous, liquid, or solid environments. The renewed interest in Brownian dynamics is due primarily to their key role in molecular and cellular biophysics: diffusion of ions and molecules is the driver of all life. Brownian dynamics simulations are the numerical realizations of stochastic differential equations that model the functions of biological micro devices such as protein ionic channels of biological membranes, cardiac myocytes, neuronal synapses, and many more. Stochastic differential equations are ubiquitous models in computational physics, chemistry, biophysics, computer science, communications theory, mathematical finance theory, and many other disciplines."--Cover
This book focuses on the modeling and mathematical analysis of stochastic dynamical systems along with their simulations. The collected chapters will review fundamental and current topics and approaches to dynamical systems in cellular biology. This text aims to develop improved mathematical and computational methods with which to study biological processes. At the scale of a single cell, stochasticity becomes important due to low copy numbers of biological molecules, such as mRNA and proteins that take part in biochemical reactions driving cellular processes. When trying to describe such biological processes, the traditional deterministic models are often inadequate, precisely because of these low copy numbers. This book presents stochastic models, which are necessary to account for small particle numbers and extrinsic noise sources. The complexity of these models depend upon whether the biochemical reactions are diffusion-limited or reaction-limited. In the former case, one needs to adopt the framework of stochastic reaction-diffusion models, while in the latter, one can describe the processes by adopting the framework of Markov jump processes and stochastic differential equations. Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology will appeal to graduate students and researchers in the fields of applied mathematics, biophysics, and cellular biology.