Brownian dynamics at boundaries and interfaces : in physics, chemistry, and biology
 Responsibility
 Zeev Schuss.
 Language
 English.
 Imprint
 [New York] : Springer, [2013]
 Physical description
 xx, 322 pages : illustrations (some color) ; 24 cm.
 Series
 Applied mathematical sciences (SpringerVerlag New York Inc.) ; v. 186.
Access
Creators/Contributors
 Author/Creator
 Schuss, Zeev, 1937 author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 285306) and index.
 Contents

 Mathematical Brownian motion
 Definition of Mathematical Brownian motion
 Mathematical Brownian motion in Rd
 Construction of Mathematical Brownian motions
 Analytical and statistical properties of Brownian paths
 Integration with respect to MBM : the Itô integral
 Stochastic differentials
 The chain rule and Itô's formula
 Stochastic differential equations
 The Langevin equation
 Itô stochastic differential equations
 SDEs of Itô type
 Diffusion processes
 SDEs and PDEs
 The Kolmogorov representation
 The FeynmanKac representation and terminating Trajectories
 The PontryaginAndronovVitt equation for the MFPT
 The exit distribution
 The PDF of the FPT
 The FokkerPlanck equation
 The backward Kolmogorov equation
 The survival probability and the PDF of the FPT
 Euler's scheme and Wiener's measure
 Euler's scheme for Itô SDEs and its convergence
 The pdf of Euler's scheme in R and the FPE
 Euler's scheme in Rd
 The convergence of the pdf in Euler's scheme in Rd
 Unidirectional and net probability Flux
 Brownian dynamics at Boundaries
 Absorbing boundaries
 Unidirectional Flux and the survival probability
 Reflecting and partially reflecting boundaries
 Reflection and partial reflection in one dimension
 Partially reflected diffusion in Rd
 Partial reflection in a halfspace : constant diffusion matrix
 Statedependent diffusion and partial oblique reflection
 Curved boundary
 Boundary conditions for the backward equation
 Discussion and annotations
 Brownian simulation of Langevin's
 Diffusion limit of physical Brownian motion
 The Overdamped Langevin equation
 Diffusion approximation to the FokkerPlanck equation
 The Unidirectional current in the Smoluchowski equation
 Trajectories between fixed concentrations
 Trajectories, Fluxes, and boundary concentrations
 Connecting a simulation to the continuum
 The interface between simulation and the continuum
 Brownian dynamics simulations
 Application to channel simulation
 Annotation
 The first passage time to a boundary
 The FPT and escape from a domain
 The PDF of the FPT and the density of the mean time spent at a point
 The exit density and probability Flux density
 Conditioning
 Conditioning on Trajectories that reach A before B
 Application of the FPT to diffusion theory
 Stationary absorption flux in one dimension
 The probability law of the first arrival time
 The first arrival time for steadystate diffusion in R³
 The next arrival times
 The exponential decay of G(r, t)
 Brownian models of Chemical reactions in microdomains
 A stochastic model of a nonarrhenius reaction
 Calcium dynamics in Dendritic spines
 Dendritic spines and their function
 Modeling dendritic spine dynamics
 Biological simplifications of the model
 A simplified physical model of the spine
 A schematic model of spine twitching
 Final model simplifications
 The mathematical model
 Mathematical simplifications
 The Langevin equations
 Reactiondiffusion model of binding and unbinding
 Specification of the Hydrodynamic flow
 Chemical Kinetics of binding and unbinding Reactions
 Simulation of calcium kinetics in dendritic spines
 A Langevin (Brownian) dynamics simulation
 An estimate of a decay rate
 Summary and discussion
 Annotations
 Interfacing at the Stochastic separatrix
 Transition state theory of thermal activation
 The diffusion model of activation
 The FPE and TST
 Reaction rate and the principal eigenvalue
 MFPT
 The rate k abs (D), MFPT (...(D)), an eigenvalue ... (D)
 MFPT for domains of types I and II in Rd
 Recrossing, Stochastic Separatrix, Eigenfunctions
 The Eigenvalue problem
 Can recrossings be neglected?
 Accounting for recrossings and the MFPT
 The transmission coefficient Ktr
 Summary and discussion
 Annotations
 Narrow escape in R²
 Introduction
 The NET problem in Neuroscience
 NET, Eigenvalues, and timescale separation
 A neumannDirichlet boundary value problem
 The Neumann function and an integral equation
 The NET problem in two dimensions
 Brownian motion in Dire straits
 The MFPT to a bottleneck
 Exit from several bottlenecks
 Diffusion and NET on a surface of revolution
 A composite domain with a bottleneck
 The NET from domains with bottlenecks in R² and R³
 The principal Eigenvalue and bottlenecks
 Connecting head and neck
 The principal Eigenvalue in DumbbellShaped domains
 A Brownian needle in Dire straits
 The diffusion law of a Brownian needle in a planar strip
 The turnaround time ...LR
 Applications of the NET
 Annotations
 Annotation to the NET problem
 Narrow escape in R³
 The Neumann function in Regulär domains in R³
 Elliptic absorbing window
 Secondorder asymptotics for a circular window
 Leakage in a conductor of Brownian particles
 Activation through a narrow opening
 The Neumann function
 Narrow escape
 Deep well : a Markov chain model
 The NET in a solid funnelshaped domain
 Selected applications in molecular biophysics
 Leakage from a cylinder
 Applications of the NET
 Annotations
 Bibliography
 Index.
 Publisher's Summary
 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. Brownian dynamics simulations of the random motion of particles, be it molecules or stock prices, give rise to mathematical problems that neither the kinetic theory of Maxwell and Boltzmann, nor Einstein's and Langevin's theories of Brownian motion could predict. This book takes the readers on a journey that starts with the rigorous definition of mathematical Brownian motion, and ends with the explicit solution of a series of complex problems that have immediate applications. It is aimed at applied mathematicians, physicists, theoretical chemists, and physiologists who are interested in modeling, analysis, and simulation of micro devices of microbiology. The book contains exercises and worked out examples throughout.
(source: Nielsen Book Data)
Subjects
 Subject
 Brownian motion processes.
Bibliographic information
 Publication date
 2013
 Series
 Applied mathematical sciences, 00665452 ; Volume 186
 Note
 Also issued online.
 Available in another form
 9781461476870 (online)
 (GyWOH)har130418244
 ISBN
 1461476860
 9781461476863
 9781461476870 (online)
 1461476879 (eBook)