Holcman D., Schuss Z. Asymptotics of Elliptic and Parabolic PDEs: and their applications in statistical physics, computational neuroscience, and biophysics

Springer International Publishing AG, 2018. — xxiii+444 p. — (Applied Mathematical Sciences, 199). — ISBN: 978-3-319-76895-3.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 in deriving solutions to real-world problems from first principles.
(перевод)
Это монография о формирующейся отрасли математической биофизики, сочетающей асимптотический анализ с численными и стохастическими методами анализа уравнений в частных производных, возникающих в биологических и физических науках.
Более подробно в книге представлены аналитические методы и инструменты аппроксимации решений смешанных краевых задач с особым акцентом на задаче окна побега. На основе реальных приложений книга включает в себя такие темы, как уравнение Фоккера-Планка, анализ пограничного слоя, ВКБ-приближения, приложения спектральной теории, а также последние результаты в теории окна побега. Подробно обсуждаются численные и стохастические аспекты, включая среднее время первого прохождения и экстремальную статистику и, параллельно с теорией, представлены соответствующие приложения.
Основываясь на классической теории асимптотических дифференциальных уравнений, эта книга адресована ученым различных областей, интересующихся в получении решений реальных проблем из базовых принципов.Singular Perturbations of Elliptic Boundary ProblemsSecond-Order Elliptic Boundary Value Problems with a Small Leading PartApplication to Stochastic Differential Equations
The Survival Probability and the Eigenvalue Problem
DiscussionA Primer of Asymptotics for ODEs
The Laplace Expansion of Integrals
The Asymptotics of a First-Order Initial Value Problem
Matched Asymptotic Expansions
An Application to Stochastic Differential Equations: The Exit Problem in R¹
Asymptotics of a Second-Order Boundary Value Problem
Asymptotics of a Homogeneous Second-Order Boundary Value Problem
Asymptotics of the Inhomogeneous Boundary Value Problem
Examples and Applications to Stochastic Equations
Small Diffusion with the Flow: The Homogeneous Boundary Value Problem
Small Diffusion Against the Flow
Small Diffusion Against the Flow: The Inhomogeneous Boundary Value Problem
The Boundary Value Problem with a Sharp Potential Barrier
The Problem for a Smooth Potential Barrier at the Boundary
The Second Eigenvalue of the Fokker–Planck Operator
A Diffusion Model of Random Signals
Loss of Lock in a First-Order Phase-Locked Loop in Phase-Modulated Radio Signals
AnnotationsSingular Perturbations in Higher DimensionsThe WKB Method
The Eikonal Equation
The Transport Equation
The Characteristic Equations
Boundary Layers at Non-characteristic Boundaries
Boundary Layers at Characteristic Boundaries in the Plane
The Boundary Value Problem With Non-characteristic Boundaries
The Boundary Value Problem in Planar Domains With Characteristic Boundaries
Loss of Lock in a Second-Order Phase-Locked Loop
The Phase Plane of the Reduced Problem
The Mean Time to Lose Lock
The Boundary Layer Structure of u_ε(x)
Asymptotic Solution of the Stationary Fokker – Planck Equation
The Eikonal Equation for (3.136)
The Eikonal on the Separatrix
The Transport Equation
Derivation of (3.114)
Green’s Function for the Boundary Value Problem is the Exit Density
Annotations
An Attractor Inside an Unstable Limit Cycle
The Reduced Equation: an Underdamped Forced Pendulum
Asymptotics of the Fokker – Planck Equation Near the Limit Cycle
The Boundary Value Problem for the Fokker – Planck Equation in Ω_S
AnnotationsEigenvalues of a Non-self-adjoint Elliptic OperatorEigenvalues and the Survival Probability
The Principal Eigenvalue and the Structure of the Field a(x)
The Precise WKB Structure of the Principal Eigenfunction
The Eikonal Equation
The Transport Equation
The Boundary Layer Equation
The First Eigenfunction of the Adjoint Problem
Higher-Order Eigenvalues
Oscillatory Escape Time
Spontaneous Activity in the Cerebral Cortex
Numerical Study of Oscillatory Decay
A Model of Up-State Dynamics in a Neuronal Network
The Phase Space of the Model
Brownian Simulations of Oscillation Phenomena in (4.80)
The Exit Density from a Focus Near a Limit Cycle
The Mean First Passage Time τ_ε(x)
Numerical Study of the Eikonal Equation
Normal Flux on ∂Ω: the Exit Time Density
Computation of the Second Eigenvalue
Brownian Dynamics Simulations
A Two-Term Approximation of the Exit-Time Density
Exit Time Densities in Three Ranges of Noise Amplitude
Appendices
The Density of Exit Points
Expansion of the Field Near the Boundary and No Cycling
The Jacobian of b(ζ) at ζ₀
The Real Part ω₁
AnnotationsShort-Time Asymptotics of the Heat Kernel
The One-Dimensional Case
The Ray Method for Short Time Asymptotics of Green’s Function
The Trace
Simply Connected Domains
Multiply Connected Domains
Recovering δ₁ from P(t)
Discussion
Construction of the Short-Time Asymptotic of the Fokker – Planck Equation with a Periodic Potential
AnnotationsMixed Boundary Conditions for Elliptic and Parabolic EquationsThe Mixed Boundary Value ProblemFormulation of the Mixed Boundary Value Problem
The Narrow Escape Time Problem
A Pathological Example
The Matched Asymptotics Approach
Higher-Order Asymptotics in the Unit Ball
The Narrow Escape Time Through Multiple Absorbing Windows
AnnotationsThe Mixed Boundary Value Problem in R²
A Neumann – Dirichlet Boundary Value Problem
The Neumann Function
The Mixed Boundary Value Problem on a Riemannian Manifold in R²
Exit though Several Windows
The Helmholtz Equation for Two Windows
Asymptotic Solution of the Helmholtz Equation
The Mixed Boundary Value Problem for Poisson’s Equation in Dire Straits
The Case of a Bottleneck
The Case of Several Bottlenecks
The Mixed Boundary Value Problem on a Surface of Revolution
A Composite Domain with a Bottleneck
The Narrow Escape Time from Domains in R² and R³ with Bottlenecks
The Principal Eigenvalue and Bottlenecks
Connecting Head and Neck
The Principal Eigenvalue in Dumbbell-Shaped Domains
Diffusion of a Needle in Dire Straits
The Diffusion Law of a Needle in a Planar Strip
The Turnaround Time τ_L→R
Applications of the Narrow Escape Time
Annotation to the Narrow Escape Time ProblemNarrow Escape in R³
The Neumann Function in Regular Domains in R³
Elliptic Absorbing Window
Second-Order Asymptotics for a Circular Window
The First Eigenvalue for Two Small Dirichlet Windows
Multiple Absorbing Windows
Higher-Order Expansion of the NET Through Many Small Windows on a Sphere
Application to Leakage in a Conductor of Brownian Particles
Activation Through a Narrow Opening
The Neumann Function
Solution of the Mixed Boundary Value Problem
Deep Well – A Markov Chain Model
The Mixed Boundary Value Problem in a Solid Funnel-Shaped Domain
The Mixed Boundary Value Problem with a Dirichlet Ribbon
Selected Applications in Molecular Biophysics
Leakage from a Cylinder
Applications of the Mixed Boundary Value Problem
AnnotationsShort-Time Asymptotics of the Heat Kernel and Extreme Statistics of the NETThe PDF of the First Escape Time
The PDF of the First Arrival Time in an Interval
Asymptotics of the Expected Shortest Time τ¹
Escape from a Ray
Escape from an Interval [0, a]
The FAT in a Bounded Domain in R^2,3
Asymptotics in R³
Asymptotics in R²
Statistics of the Arrival Time of the Second Particle
Poissonian-Like Approximation
Pr{τ⁽²⁾} of N Brownian i.i.d. Trajectories in a Segment
Applications of the FAT in Biophysics
AnnotationsThe Poisson – Nernst – Planck Equations in a BallSynopsis of Results
Poisson – Nernst – Planck Equations in a Ball
The Steady-State Solution
Existence of Solutions
The Distribution of Voltage and Charge in a Dielectric Ball
Scaling Laws for the Maximal Number of Charges
Ionic Flux in a Small Window at High Charge
Flow Through a Narrow Window at High Charge
Current in a Voltage-Clamped Dendritic Spine
Appendix 1: Reverse Liouville – Gelfand – Bratú Equation
Small λ Expansion of u_λ(x)
Numerical Scheme for the Solution of (10.13)
Steady Solution in a Ball with a Cusp-Shaped Funnel
Reduced Equations in an Uncharged Cusp-Shaped Funnel
Asymptotics of Voltage Between Funnel and Center
Poisson – Nernst – Planck Solutions in a 3D Cusp-Shaped Funnel
Asymptotic Analysis of the PNP Equations in a Cusp-Shaped Funnel
The Potential Drop in Ω_w
AnnotationsReconstruction of Surface Diffusion from Projected Data
Projection of Diffusion from a Curve to a Line
Driftless Diffusion on a Curve
The Case of Diffusion with Drift
Reconstruction of a Parabola from Projected Diffusion Data
Appendix 2
Reconstruction of Projected Stochastic Dynamics
Reconstruction of a Surface from Planar Projections of Diffusion Trajectories
The Drift Field
The Reconstruction Procedure
11.3 AnnotationsAsymptotic Formulas in Molecular and Cellular BiologyFrom Molecular to Cellular Description
Flux Through Narrow Passages Identifies Cellular Compartments
Examples of Asymptotic Formulas: Fluxes into Small Targets
Formulas in Two Dimensions
Narrow Escape Formulas in Three-Dimensions
Cusp-Shaped Funnel: Hidden Targets Control Rates in R³
DNA Repair in a Confined Chromatin Structure in R²
Asymmetric Dumbbell-Shaped Cell Division
AnnotationsBibliography
Index