Diffusion Under Confinement - A Journey Through Counterintuition

Preface
Contents
1 History of Brownian Motion in a Nutshell
	Further Reading and References
Part I Brownian Motion and the Random Elevator Game
	2 The Random Elevator Game
		2.1 Introduction
		2.2 Mathematical Model of the Game: Discrete Version
		2.3 Continuous Version of the Game
			2.3.1 Solution to the Continuous Model
		2.4 Duration of the Game
			2.4.1 Survival Probability and First-Passage Time
		2.5 Moments of the Mean First-Passage Time
		2.6 The Backward Equation
		2.7 Going to the Observatory or the Lobby
			2.7.1 Probability Flux
			2.7.2 Probability Flux at the Observatory and the Lobby
		2.8 Moments of Mean First-Passage Time: Revisited
		2.9 Splitting Probability
		2.10 Concluding Remarks
		Further Reading and References
Part II Diffusion: Free Particle
	3 Solution of the Diffusion Equation in Free Space
		3.1 Fourier Transform
		3.2 Laplace Transform
		3.3 Mean and Standard Deviation of Gaussian Distribution
		3.4 The Central Limit Theorem and Moments of Displacement
		3.5 Green\'s Function Method
			3.5.1 An Application of Green\'s Function: The Forced Undamped Harmonic Oscillator
			3.5.2 The Inhomogeneous Diffusion Equation and Green\'s Function
		3.6 Free Diffusion on d-Dimensional Space
		3.7 Concluding Remarks
		3.A Galton Board Simulation
		Further Reading and References
Part III One-Dimensional Diffusion and Boundary Conditions
	4 One-Dimensional Semi-infinite Systems Solutions
		4.1 Boundary Conditions
		4.2 Derivation of Boundary Conditions
			4.2.1 Useful Formulas for Trapping Rate Coefficients and Rate Constants
			4.2.2 Perfectly Absorbing Sphere
			4.2.3 Partially Absorbing Sphere
			4.2.4 Perfectly Absorbing Circular Disk
			4.2.5 Boundary Homogenization
				4.2.5.1 Patchy Surface
				4.2.5.2 Cylindrical Capillary: One-Dimensional Reduction
		4.3 Semi-infinite: Perfectly Absorbing Endpoint
			4.3.1 The Fourier Transform Solution
			4.3.2 The Laplace Transform Solution
			4.3.3 From the Propagator of Two Absorbing Endpoints
			4.3.4 Method of Images
			4.3.5 Survival Probability and First-Passage Time
			4.3.6 Survival Probability: Revisited
		4.4 Perfectly Reflecting Endpoint
			4.4.1 The Fourier Transform Solution
			4.4.2 The Laplace Transform Solution
			4.4.3 Method of Images
			4.4.4 Survival Probability and First-Passage Time
		4.5 Partially Absorbing Endpoint
			4.5.1 Survival Probability and First-Passage Time
		4.6 Concluding Remarks
		4.A Mathematical Computations
			4.A.1 Derivation of Eq. (4.88)
			4.A.2 Derivation of Eq. (4.98)
			4.A.3 Derivation of Eq. (4.104)
			4.A.4 Derivation of Eq. (4.107)
		Further Reading and References
	5 Diffusion Between Two Targets
		5.1 Separation of Variables
		5.2 Reflecting-Reflecting
			5.2.1 The Separation of Variables Method
			5.2.2 The Laplace Transformation Solution
		5.3 Final Value Theorem
		5.4 Absorbing-Absorbing: Revisited
			5.4.1 The Separation of Variables Method
			5.4.2 The Laplace Transform Solution
			5.4.3 Survival Probability and Moments of MFPT: Revisited
			5.4.4 Splitting Probability: Revisited
		5.5 Absorbing-Absorbing: Uniformly Distributed Initial Position
			5.5.1 Survival Probability and Mean First-Passage Time
			5.5.2 Moments of MFPT and Splitting Probability
		5.6 Absorbing-Reflecting
			5.6.1 The Separation of Variables Method
			5.6.2 Survival Probability and First-Passage Time
			5.6.3 Moments of MFPT
			5.6.4 The Laplace Transform Solution
			5.6.5 Survival Probability and Moments of MFPT: Revisited
			5.6.6 First-Passage Time and Splitting Probability: Revisited
		5.7 Partially Absorbing-Reflecting
			5.7.1 The Separation of Variables Method
			5.7.2 Survival Probability and First-Passage Time
			5.7.3 Moments of MFPT and Splitting Probability
			5.7.4 The Laplace Transform Solution
			5.7.5 Survival Probability and Moments of MFPT
			5.7.6 Density of Mean First-Passage Time and Splitting Probability
		5.8 Absorbing-Partially Absorbing
			5.8.1 The Laplace Transform Solution
			5.8.2 Survival Probability and Moments of First-Passage Time
			5.8.3 Probability Density and Splitting Probability
		5.9 Partially Absorbing-Partially Absorbing
			5.9.1 The Laplace Transform Solution
			5.9.2 Survival Probability and Moments of First-Passage Time
			5.9.3 Probability Density of MFPT and SplittingProbability
		5.10 Concluding Remarks
		5.A Numerical Laplace Inversion: Gaver-Stehfest Method
		Further Reading and References
	6 Diffusion in the Presence of a Force Field
		6.1 The Smoluchowski Equation
		6.2 The Backward Smoluchowski Equation
		6.3 Survival Probability and the Moments of MFPT in the Presence of a Force Field
		6.4 Fluctuation-Dissipation Theorem
		6.5 Diffusion in a Linear Potential: Constant Drift
			6.5.1 Diffusion with Constant Drift Revisited: Integral Transforms
				6.5.1.1 Fourier Transform
				6.5.1.2 Laplace Transform
		6.6 Diffusion in a Harmonic Potential
		6.7 Ionic Diffusion Through Membrane: The Nernst Potential
		6.8 Concluding Remarks
		6.A The Adjoint Operator of the Smoluchowski Operator
		Further Reading and References
	7 Trapping Particles Influenced by External Forces
		7.1 Semi-infinite: Perfectly Absorbing Endpoint, U(x)=-Fx
			7.1.1 Survival Probability and First-Passage Time
		7.2 Drift and Diffusion into Partially Absorbing and Absorbing Endpoints, U(x)=-Fx
			7.2.1 Flux and Splitting Probability
			7.2.2 Reflecting-Absorbing and Absorbing-Absorbing
		7.3 Drift and Diffusion into Two Partially Absorbing Points, U(x)=-Fx
		7.4 Perfectly Absorbent Target: Harmonic Potential
			7.4.1 Survival Probability and First-Passage Time
		7.5 Drift and Diffusion for the Periodic PotentialU(x)=V(x)-Fx
		7.6 Concluding Remarks
		Further Reading and References
	8 Splitting and Breaking Brownian Pathways: Conditional Processes
		8.1 Conditional Propagators: A First Glance
		8.2 Conditional Probability Fluxes and Densities of the First-Passage Time
		8.3 Conditional Mean First-Passage Time
			8.3.1 Direct-Transit Time and Looping Time
		8.4 Conditional Survival Probabilities
			8.4.1 Direct-Transit Time and Looping Time in the Presence of a Constant External Force
		8.5 Concluding Remarks
		Further Reading and References
	9 Diffusion with Stochastic Resetting
		9.1 Introduction
		9.2 Diffusion Equation with Stochastic Resetting
		9.3 Steady-State Solution
		9.4 Mean First-Passage Time Under Resetting: Semi-Infinite Line
		9.5 Renewal Equation Approach for Poissonian Resetting
		9.6 Mean First-Passage Time Under Resetting: Absorbing-Absorbing
		9.7 Optimal Restart Rate
		9.8 Semi-Infinite Revisited: x_0 = x_r U x_0 != x_r
		9.9 Concluding Remarks
		Further Reading and References
	10 Langevin Equation and Brownian Dynamics Simulations
		10.1 Discrete Equations of Brownian Dynamics
			10.1.1 Equipartition Theorem
			10.1.2 Langevin Equation
				10.1.2.1 Analysis of Velocity
				10.1.2.2 Correlation of Velocities and the Diffusion Coefficient
				10.1.2.3 Long Time Limit
				10.1.2.4 Analysis of Position
				10.1.2.5 Displacement Variance
				10.1.2.6 Overdamped Langevin Equation
			10.1.3 Brownian Dynamics Simulations
		10.2 Random and Pseudorandom Numbers
			10.2.1 Middle Square Method
			10.2.2 Linear Congruential Generator
				10.2.2.1 An Implementation of an LCG
			10.2.3 Inverse Transform Sampling
			10.2.4 Box-Müller Method
		10.3 Simulation Helpers and Programs
		10.4 Computational Experiments
			10.4.1 Absorbing-Absorbing
			10.4.2 Absorbing-Absorbing: Initial Position Uniformly Distributed
		10.5 Concluding Remarks
		10.A helpers.f90 Companion File
		Further Reading and References
	11 Numerical Solutions of the Diffusion Equation
		11.1 Differences Construction
			11.1.1 Discretization and Mesh
		11.2 Forward Time-Centered Space Method
			11.2.1 Numerical Implementation
		11.3 Backward Time-Centered Space Method
			11.3.1 Numerical Implementation
		11.4 Stability Analysis
			11.4.1 Stability of the FTCS FDM
			11.4.2 Stability of the BTCS FDM
		11.5 Concluding Remarks
		Further Reading and References
Part IV Two-Dimensional Diffusion and Reaction-Diffusion Equations
	12 Two-Dimensional Systems
		12.1 Partially Absorbent Disk: Internal Problem
		12.2 Perfectly Absorbent Disk: External Problem
		12.3 Partially Absorbent Disk
		12.4 Concentric Disks
			12.4.1 Steady State: Effect of Dimensionality
				12.4.1.1 Constant Concentration
				12.4.1.2 Constant Concentration (Partially Absorbing)
			12.4.2 Mean First-Passage Times in 2D: Perfectly Reflecting (Partially Absorbing)
		12.5 Concluding Remarks
		Further Reading and References
	13 Reaction-Diffusion Equations
		13.1 Turing-Like Reaction-Diffusion Equations
			13.1.1 Turing Mechanism
		13.2 Turing Conditions
		13.3 Gierer-Meinhardt Model
			13.3.1 Turing Domain
		13.4 Pattern Formation: One-Dimensional Model
			13.4.1 Pattern Formation: Two-Dimensional Model
		13.5 Concluding Remarks
		13.A Stability Matrix and Principles of Linearization
		13.B Linearization
		13.C Numerical Solution of Reaction-Diffusion Equations
			13.C.1 One-Dimensional Gierer-Meinhardt Model
			13.C.2 Two-Dimensional Gierer-Meinhardt Model
		Further Reading and References
Part V Three-Dimensional Diffusion
	14 Three-Dimensional Systems
		14.1 Perfectly Absorbent Sphere
			14.1.1 Perfectly Absorbing Sphere: Internal Problem
			14.1.2 Perfectly Absorbing Sphere: External Problem
		14.2 Concentric Spheres
			14.2.1 Absorbing-Absorbing
		14.3 Concentric Spheres Propagator Revisited: The Effect of Dimensionality
			14.3.1 Perfectly Absorbing-Perfectly Absorbing
			14.3.2 Perfectly Absorbing-Perfectly Reflecting
			14.3.3 Splitting Probability Absorbing-Absorbing: The Effect of Dimensionality
			14.3.4 Mean First-Passage Time: The Effect of Dimensionality
			14.3.5 Partially Absorbing and Reflecting: MFPT
				14.3.5.1 Circular Disk on a Reflecting Flat Surface: Absorbing Hemisphere Approximation
		14.4 Diffusion to an Absorbent Circular Disk: Weber\'s Disk
		14.5 Absorbing Patches of Arbitrary Shape
		14.6 Hyperboloidal Cone
		14.7 Single Exponential Decay
		14.8 Computational Experiments: Perfectly Absorbent Sphere from the Inside
		14.9 Computational Experiments: Perfectly Absorbent Sphere from the Inside with Uniformly Distributed Particles
		14.10 Concluding Remarks
		Further Reading and References
Part VI Trapping Rate Coefficient and Boundary Homogenization
	15 Trapping Rate Coefficient
		15.1 The Rate Coefficient
			15.1.1 Smoluchowski Formula: Perfectly Absorbing Sphere
			15.1.2 Collins-Kimball Formula: Partially Absorbing Sphere
		15.2 Berg-Purcell Formula: The Patchy Sphere
		15.3 Zwanzig-Szabo Formula: The Partially Absorbing Circular Disk and Diffusing Interference Between Binding Sites
		15.4 Chemoreceptors over a Spherical Cell
			15.4.1 A Counterintuitive Experiment: Circular and Elliptical Absorbing Patches
		15.5 Reaction Between Charged Particles: Debye-Smoluchowski Formula
			15.5.1 Debye-Smoluchowski Equation
			15.5.2 Steady-State Rate Constant
		15.6 Concluding Remarks
		Further Reading and References
	16 Boundary Homogenization
		16.1 Sphere with an Absorbing Cap
		16.2 Absorbing Circular Spot at a Cylinder End Wall
		16.3 Cylinder with Absorbing Stripes
			16.3.1 Stripes Perpendicular to the Tube Axis
			16.3.2 Stripes Parallel to the Tube Axis
		16.4 Trapping of Particles Diffusing in a Two-Dimensional Rectangular Chamber by an Absorbing Strip
		16.5 Binding Site Hidden in a Tunnel
		16.6 Table of Useful Trapping Rates
		16.7 Concluding Remarks
		Further Reading and References
Part VII Quasi-one-dimensional Diffusion: Channel/Tube
	17 Fick-Jacobs 1D Reduction
		17.1 Introduction
		17.2 The Fick-Jacobs Equation
		17.3 Fick\'s Funnel
		17.4 Concluding Remarks
		Further Reading and References
	18 Zwanzig 1D Reduction
		18.1 Zwanzig\'s Derivation of the FJ Equation in 2D
		18.2 Effective Diffusion Coefficient
			18.2.1 Harmonic Potential
			18.2.2 Box-Like Potential
		18.3 Zwanzig\'s Derivation of the FJ Equation in 3D
		18.4 3D Hyperboloidal Cone
		18.5 The Effective Diffusion Coefficient
			18.5.1 Exact Formula for the Hyperboloidal Cone
			18.5.2 Mean Square and Transient Behavior
		18.6 Concluding Remarks
		18.A Mathematical Computations
			18.A.1 Derivation of Eq.(18.29)
			18.A.2 Derivation of Eq.(18.39)
		Further Reading and References
	19 Reguera and Rubi Kinetic Equation
		19.1 Introduction
		19.2 Entropy Production
			19.2.1 Continuity Equations
			19.2.2 Kinetic Coefficients
			19.2.3 Kinetic Equation
		19.3 Reduction of the Kinetic Equation
			19.3.1 Reduced Equation for a Gravitational-Like Field
				19.3.1.1 Equilibrium Solution
			19.3.2 Diffusion Coefficient
		19.4 Concluding Remarks
		Further Reading and References
	20 Kalinay and Percus Projection Method
		20.1 2D Asymmetric Channel: Projection Method
			20.1.1 The Projection Method
			20.1.2 Recurrence Formula for the Operators j (x,y,∂x)
			20.1.3 First- and Second-Order Corrections
			20.1.4 The Position-Dependent Effective Diffusion Coefficient
		20.2 Trapezoidal 2D Channel
		20.3 First-Passage Time in Conical Channels
		20.4 3D Tube: Projection Method
			20.4.1 The Projection Method
		20.5 Recurrence Formula for Operators j (x,r,∂x)
			20.5.1 First- and Second-Order Corrections
			20.5.2 The Position-Dependent Effective Diffusion Coefficient
		20.6 First Passage in Conical Tubes
		20.7 Position-Dependent Diffusion Coefficient Formulas
		20.8 Concluding Remarks
		Further Reading and References
	21 External Transverse Field: 2D Narrow Channel
		21.1 Projection of the Smoluchowski Equation
			21.1.1 The Projection Method
				21.1.1.1 Equilibrium Solution
				21.1.1.2 General Solution of c(x,y,t)
			21.1.2 Recurrence Formula for the Operators k (x,y,∂x)
				21.1.2.1 Boundary Conditions
			21.1.3 First-Order Correction
				21.1.3.1 Limiting Case
			21.1.4 The Position-Dependent Effective Diffusion Coefficient
		21.2 2D Asymmetric Channel: Projection Method
			21.2.1 Recurrence Formula for Operators k (x,y,∂x)
				21.2.1.1 General Solution
			21.2.2 First-Order Correction
			21.2.3 The Position-Dependent Effective Diffusion Coefficient
		21.3 Interpolation Formula
		21.4 Limiting Cases
			21.4.1 Symmetric Channel with a Transverse Force
			21.4.2 Dominant g
			21.4.3 Small Values of g
			21.4.4 Asymmetric Channel Without an External Field
		21.5 First-Passage Times in Conical Channels
			21.5.1 Asymmetrical Cone Under the Influence of Gravity
				21.5.1.1 Wide to Narrow
		21.6 Concluding Remarks
		21.A Mathematical Computations
			21.A.1 Derivation of Eq.(21.61)
			21.A.2 Derivation of Eq.(21.64)
			21.A.3 Derivation of Eq.(21.84)
			21.A.4 Derivation of Eq.(21.91)
		Further Reading and References
	22 Periodical Systems
		22.1 Lifson-Jackson Formula
		22.2 Diffusion into a Periodic Tube Formed by Contacting Spheres
		22.3 Diffusion in a Periodic Channel with Corrugated Walls
		22.4 Diffusion in the Presence of Cylindrical Obstacles
		22.5 Concluding Remarks
		Further Reading and References
	23 Active Brownian Particles
		23.1 Low Reynolds Number
		23.2 Rotational Diffusion: Debye\'s Problem
		23.3 Fick-Jacobs-Zwanzig Equation for Active Brownian Particles
		23.4 Effective Diffusivity for Active Brownian Particles
		23.5 Corrugated Periodical Channel
		23.6 Concluding Remarks
		23.A Translational and Rotational Friction Coefficients
		Further Reading and References
	24 Diffusion in Narrow Channels Embedded on Curved Manifolds
		24.1 Transformation of Differential Operators
		24.2 Covariant Form of the Diffusion Equation
		24.3 2D Asymmetric Channel in Curved Surfaces: Projection Method
			24.3.1 Fick-Jacobs Equation on Curved Surfaces
			24.3.2 Recurrence Formula for Operators j (ξ,η,∂ξ)
			24.3.3 The Position-Dependent Effective Diffusion Coefficient
		24.4 Narrow Channels on Curved Manifolds
			24.4.1 Cylindrical Surface
			24.4.2 Spherical Surface
			24.4.3 Torus Surface
		24.5 Mean First-Passage Time
			24.5.1 Mean First-Passage Time on a Cylinder
		24.6 Concluding Remarks
		Further Reading and References
	25 Representation of a Channel as a Tubular Manifold: Frenet-Serret Moving Frame
		25.1 Fick\'s Laws in General Coordinates
		25.2 Representation of a Channel as a Tubular Manifold
		25.3 Generalized Fick-Jacobs-Like Equation: 3D Frenet-Serret Moving Frame
		25.4 Straight Tube with Circular-Shaped Cross-Section
		25.5 Tilted Straight Tube with Circular-Shaped Cross-Section
		25.6 Generalized Fick-Jacobs-Like Equation: 2D Frenet-Serret Moving Frame
		25.7 Diffusivity Coefficient for an Asymmetric and Curved Midline Channel
		25.8 Concluding Remarks
		Further Reading and References
A Mathematical Requirements
	A.1 Useful Trigonometric Identities
	A.2 Hyperbolic Function Relations
	A.3 Leibniz Rule for Integrals
	A.4 Table of Integrals
	A.5 Gaussian Integral and the Feynman Rule
	A.6 Series and Products
		A.6.1 Taylor Series
			A.6.1.1 Series of Hyperbolic Functions
		A.6.2 Euler Formula to Fourier Series Coefficients
		A.6.3 Table of Series
		A.6.4 Cauchy Product for Power Series
		A.6.5 Generalized Binomial Theorem
	A.7 Fourier Transform
		A.7.1 Table of Transforms
	A.8 Laplace Transform
		A.8.1 Change of Laplace Variable
		A.8.2 Table of Transforms
	A.9 Convolution of Functions
		A.9.1 Convolution Theorem
	A.10 Special Functions
		A.10.1 Gamma Function
		A.10.2 Error Functions
		A.10.3 Dirac Delta Function (Distribution)
		A.10.4 Heaviside Function
		A.10.5 Riemann Zeta Function
		A.10.6 The Sign Function
	A.11 Bessel Differential Equation
		A.11.1 Recurrence Formulas: Derivatives
	A.12 Solution of Differential Equations by Quadratures
B Vector Analysis of Differential Operators
	B.1 Rectangular (Cartesian) Coordinates
	B.2 Circular Cylindrical Coordinates
	B.3 Spherical Coordinates
C Differential Geometry in a Nutshell
Index