Brownian dynamics at boundaries and interfaces : in physics, chemistry, and biology / Zeev Schuss.

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Bibliographic Details
Uniform Title:Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 186.
Main Author: Schuss, Zeev, 1937- (Author)
Language:English
Published: New York : Springer, [2013]
Series:Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 186.
Subjects:
Brownian motion processes > Computer simulation.
Boundary value problems > Computer simulation.
Stochastic differential equations > Computer simulation.
Physical Description:xx, 322 pages : illustrations (some color) ; 24 cm.
Also issued online.
Format: Book
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 Feynman-Kac representation and terminating Trajectories
  • The Pontryagin-Andronov-Vitt equation for the MFPT
  • The exit distribution
  • The PDF of the FPT
  • The Fokker-Planck 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 half-space : constant diffusion matrix
  • State-dependent 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 Fokker-Planck 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 steady-state 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 non-arrhenius 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
  • Reaction-diffusion 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 time-scale separation
  • A neumann-Dirichlet 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 Dumbbell-Shaped domains
  • A Brownian needle in Dire straits
  • The diffusion law of a Brownian needle in a planar strip
  • The turnaround time ...L-R
  • 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
  • Second-order 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 funnel-shaped domain
  • Selected applications in molecular biophysics
  • Leakage from a cylinder
  • Applications of the NET
  • Annotations
  • Bibliography
  • Index.