Stochastic Processes in Polymeric Fluids

1 Stochastic Processes, Polymer Dynamics, and Fluid Mechanics.- 1.1 Approach to Kinetic Theory Models.- 1.2 Flow Calculation and Material Functions.- 1.2.1 Shear Flows.- 1.2.2 General Extensional Flows.- 1.2.3 The CONNFFESSIT Idea.- References.- I Stochastic Processes.- 2 Basic Concepts from Stochastics.- 2.1 Events and Probabilities.- 2.1.1 Events and ?-Algebras.- 2.1.2 Probability Axioms.- 2.1.3 Gaussian Probability Measures.- 2.2 Random Variables.- 2.2.1 Definitions and Examples.- 2.2.2 Expectations and Moments.- 2.2.3 Joint Distributions and Independence.- 2.2.4 Conditional Expectations and Probabilities.- 2.2.5 Gaussian Random Variables.- 2.2.6 Convergence of Random Variables.- 2.3 Basic Theory of Stochastic Processes.- 2.3.1 Definitions and Distributions.- 2.3.2 Gaussian Processes.- 2.3.3 Markov Processes.- 2.3.4 Martingales.- References.- 3 Stochastic Calculus.- 3.1 Motivation.- 3.1.1 Naive Approach to Stochastic Differential Equations.- 3.1.2 Criticism of the Naive Approach.- 3.2 Stochastic Integration.- 3.2.1 Definition of the Itô Integral.- 3.2.2 Properties of the Itô Integral.- 3.2.3 Itô's Formula.- 3.3 Stochastic Differential Equations.- 3.3.1 Definitions and Basic Theorems.- 3.3.2 Linear Stochastic Differential Equations.- 3.3.3 Fokker-Planck Equations.- 3.3.4 Mean Field Interactions.- 3.3.5 Boundary Conditions.- 3.3.6 Stratonovich's Stochastic Calculus.- 3.4 Numerical Integration Schemes.- 3.4.1 Euler's Method.- 3.4.2 Mil'shtein's Method.- 3.4.3 Weak Approximation Schemes.- 3.4.4 More Sophisticated Methods.- References.- II Polymer Dynamics.- 4 Bead-Spring Models for Dilute Solutions.- 4.1 Rouse Model.- 4.1.1 Analytical Solution for the Equations of Motion.- 4.1.2 Stress Tensor.- 4.1.3 Material Functions in Shear and Extensional Flows.- 4.1.4 A Primer in Brownian Dynamics Simulations.- 4.1.5 Variance Reduced Simulations.- 4.2 Hydrodynamic Interaction.- 4.2.1 Description of Hydrodynamic Interaction.- 4.2.2 Zimm Model.- 4.2.3 Long Chain Limit and